diff --git a/books/bookvol10.2.pamphlet b/books/bookvol10.2.pamphlet index 3e48804..7727b56 100644 --- a/books/bookvol10.2.pamphlet +++ b/books/bookvol10.2.pamphlet @@ -350,6 +350,8 @@ digraph pic { ArcHyperbolicFunctionCategory examples ==================================================================== +This is the Category for the inverse hyperbolic trigonometric functions + See Also: o )show ArcHyperbolicFunctionCategory @@ -383,9 +385,6 @@ These are directly exported but not implemented: \begin{chunk}{category AHYP ArcHyperbolicFunctionCategory} )abbrev category AHYP ArcHyperbolicFunctionCategory -++ Category for the inverse hyperbolic trigonometric functions -++ Author: ??? -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the inverse hyperbolic trigonometric functions; @@ -461,6 +460,8 @@ intermediate test to check that the argument has a reciprocal values. ArcTrigonometricFunctionCategory examples ==================================================================== +This is the Category for the inverse trigonometric functions + See Also: o )show ArcTrigonometricFunctionCategory @@ -497,9 +498,6 @@ These are implemented by this category: \begin{chunk}{category ATRIG ArcTrigonometricFunctionCategory} )abbrev category ATRIG ArcTrigonometricFunctionCategory -++ Category for the inverse trigonometric functions -++ Author: ??? -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the inverse trigonometric functions; @@ -549,6 +547,17 @@ digraph pic { %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagehead{AttributeRegistry}{ATTREG} \pagepic{ps/v102attributeregistry.ps}{ATTREG}{1.00} +\begin{chunk}{AttributeRegistry.help} +==================================================================== +AttributeRegistry examples +==================================================================== + +This category exports the attributes in the AXIOM Library. + +See Also: +o )show BasicType + +\end{chunk} {\bf See:} @@ -748,6 +757,9 @@ digraph pic { BasicType examples ==================================================================== +BasicType is the basic category for describing a collection +of elements with = (equality). + See Also: o )show BasicType @@ -778,18 +790,9 @@ These are implemented by this category: \begin{chunk}{category BASTYPE BasicType} )abbrev category BASTYPE BasicType --% BasicType -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: -++ \spadtype{BasicType} is the basic category for describing a collection -++ of elements with \spadop{=} (equality). +++ BasicType is the basic category for describing a collection +++ of elements with = (equality). BasicType(): Category == with "=": (%,%) -> Boolean ++ x=y tests if x and y are equal. @@ -856,6 +859,9 @@ digraph pic { CoercibleTo examples ==================================================================== +A is coercible to B means any element of A can automatically be +converted into an element of B by the interpreter. + See Also: o )show CoercibleTo @@ -882,9 +888,7 @@ This is directly exported but not implemented: \begin{chunk}{category KOERCE CoercibleTo} )abbrev category KOERCE CoercibleTo -++ Category for coerce ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ A is coercible to B means any element of A can automatically be @@ -963,6 +967,8 @@ digraph pic { CombinatorialFunctionCategory examples ==================================================================== +This is the Category for the usual combinatorial functions + See Also: o )show CombinatorialFunctionCategory @@ -990,9 +996,7 @@ These are directly exported but not implemented: \begin{chunk}{category CFCAT CombinatorialFunctionCategory} )abbrev category CFCAT CombinatorialFunctionCategory -++ Category for the usual combinatorial functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the usual combinatorial functions; @@ -1072,6 +1076,9 @@ digraph pic { ConvertibleTo examples ==================================================================== +A is convertible to B means any element of A can be converted into +an element of B, but not automatically by the interpreter. + See Also: o )show ConvertibleTo @@ -1098,9 +1105,7 @@ This is directly exported but not implemented: \begin{chunk}{category KONVERT ConvertibleTo} )abbrev category KONVERT ConvertibleTo -++ Category for convert ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ A is convertible to B means any element of A @@ -1231,6 +1236,8 @@ digraph pic { ElementaryFunctionCategory examples ==================================================================== +This is the Category for the elementary functions. + See Also: o )show ElementaryFunctionCategory @@ -1263,7 +1270,6 @@ These are implemented by this category: )abbrev category ELEMFUN ElementaryFunctionCategory ++ Category for the elementary functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the elementary functions; @@ -1334,6 +1340,11 @@ digraph pic { Eltable examples ==================================================================== +An eltable over domains D and I is a structure which can be viewed +as a function from D to I. Examples of eltable structures range from +data structures, e.g. those of type List, to algebraic structures like +Polynomial. + See Also: o )show Eltable @@ -1361,18 +1372,11 @@ This is directly exported but not implemented: ++ Author: Michael Monagan; revised by Manuel Bronstein and Manuel Bronstein ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An eltable over domains D and I is a structure which can be viewed ++ as a function from D to I. ++ Examples of eltable structures range from data structures, e.g. those -++ of type \spadtype{List}, to algebraic structures like -++ \spadtype{Polynomial}. +++ of type List, to algebraic structures like Polynomial. Eltable(S:SetCategory, Index:Type): Category == with elt : (%, S) -> Index @@ -1456,6 +1460,8 @@ intermediate test to check that the argument has a reciprocal values. HyperbolicFunctionCategory examples ==================================================================== +This is the Category for the hyperbolic trigonometric functions. + See Also: o )show HyperbolicFunctionCategory @@ -1488,11 +1494,7 @@ These are implemented by this category: \begin{chunk}{category HYPCAT HyperbolicFunctionCategory} )abbrev category HYPCAT HyperbolicFunctionCategory -++ Category for the hyperbolic trigonometric functions -++ Author: ??? -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 -++ Description: ++ Category for the hyperbolic trigonometric functions; HyperbolicFunctionCategory(): Category == with @@ -1579,6 +1581,12 @@ digraph pic { InnerEvalable examples ==================================================================== +This category provides eval operations. A domain may belong to this +category if it is possible to make "evaluation" substitutions. The +difference between this and Evalable is that the operations in this +category specify the substitution as a pair of arguments rather than +as an equation. + See Also: o )show InnerEvalable @@ -1611,16 +1619,7 @@ These are implemented by this category: \begin{chunk}{category IEVALAB InnerEvalable} )abbrev category IEVALAB InnerEvalable -- FOR THE BENEFIT OF LIBAX0 GENERATION -++ Author: -++ Date Created: ++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: Evalable -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This category provides \spadfun{eval} operations. ++ A domain may belong to this category if it is possible to make @@ -1870,6 +1869,8 @@ digraph pic { OpenMath examples ==================================================================== +OpenMath provides operations for exporting an object in OpenMath format. + See Also: o )show OpenMath @@ -1897,11 +1898,6 @@ These are directly exported but not implemented: )abbrev category OM OpenMath ++ Author: Mike Dewar & Vilya Harvey ++ Basic Functions: OMwrite -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{OpenMath} provides operations for exporting an object ++ in OpenMath format. @@ -1997,6 +1993,9 @@ digraph pic { PartialTranscendentalFunctions examples ==================================================================== +This is the description of any package which provides partial +functions on a domain belonging to TranscendentalFunctionCategory. + See Also: o )show PartialTranscendentalFunctions @@ -2070,15 +2069,12 @@ These are directly exported but not implemented: \begin{chunk}{category PTRANFN PartialTranscendentalFunctions} )abbrev category PTRANFN PartialTranscendentalFunctions -++ Description of a package which provides partial transcendental -++ functions, i.e. functions which return an answer or "failed" ++ Author: Clifton J. Williamson ++ Date Created: 12 February 1990 ++ Date Last Updated: 14 February 1990 -++ Keywords: -++ Examples: -++ References: ++ Description: +++ A package which provides partial transcendental +++ functions, i.e. functions which return an answer or "failed" ++ This is the description of any package which provides partial ++ functions on a domain belonging to TranscendentalFunctionCategory. @@ -2222,6 +2218,10 @@ digraph pic { Patternable examples ==================================================================== +Category of sets that can be converted to useful patterns. An object +S is Patternable over an object R if S can lift the conversions from +R into Pattern(Integer) and Pattern(Float) to itself. + See Also: o )show Patternable @@ -2252,12 +2252,11 @@ These exports come from \refto{ConvertibleTo}(Pattern(Float)): \begin{chunk}{category PATAB Patternable} )abbrev category PATAB Patternable -++ Category of sets that can be converted to useful patterns ++ Author: Manuel Bronstein ++ Date Created: 29 Nov 1989 ++ Date Last Updated: 29 Nov 1989 -++ Keywords: pattern, matching. ++ Description: +++ Category of sets that can be converted to useful patterns ++ An object S is Patternable over an object R if S can ++ lift the conversions from R into \spadtype{Pattern(Integer)} and ++ \spadtype{Pattern(Float)} to itself; @@ -2340,6 +2339,8 @@ digraph pic { PrimitiveFunctionCategory examples ==================================================================== +This is the Category for the functions defined by integrals. + See Also: o )show PrimitiveFunctionCategory @@ -2363,9 +2364,7 @@ These are directly exported but not implemented: \begin{chunk}{category PRIMCAT PrimitiveFunctionCategory} )abbrev category PRIMCAT PrimitiveFunctionCategory -++ Category for the integral functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the functions defined by integrals; @@ -2437,6 +2436,8 @@ digraph pic { RadicalCategory examples ==================================================================== +The RadicalCategory is a model for the rational numbers. + See Also: o )show RadicalCategory @@ -2473,14 +2474,7 @@ These are implemented by this category: \begin{chunk}{category RADCAT RadicalCategory} )abbrev category RADCAT RadicalCategory -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: nthRoot, sqrt, ** -++ Related Constructors: -++ Keywords: rational numbers ++ Description: -++ The \spad{RadicalCategory} is a model for the rational numbers. RadicalCategory(): Category == with sqrt : % -> % @@ -2553,6 +2547,10 @@ digraph pic { RetractableTo examples ==================================================================== +A is retractable to B means that some elementsif A can be converted +into elements of B and any element of B can be converted into an +element of A. + See Also: o )show RetractableTo @@ -2601,12 +2599,9 @@ These are implemented by this category: \begin{chunk}{category RETRACT RetractableTo} )abbrev category RETRACT RetractableTo -++ Category for retract -++ Author: ??? -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: -++ A is retractable to B means that some elementsif A can be converted +++ A is retractable to B means that some elements if A can be converted ++ into elements of B and any element of B can be converted into an ++ element of A. @@ -2740,6 +2735,8 @@ digraph pic { SpecialFunctionCategory examples ==================================================================== +This is the Category for the other special functions. + See Also: o )show SpecialFunctionCategory @@ -2782,9 +2779,7 @@ These are directly exported but not implemented: \begin{chunk}{category SPFCAT SpecialFunctionCategory} )abbrev category SPFCAT SpecialFunctionCategory -++ Category for the other special functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 11 May 1993 ++ Description: ++ Category for the other special functions; @@ -2880,6 +2875,8 @@ intermediate test to check that the argument has a reciprocal values. TrigonometricFunctionCategory examples ==================================================================== +This is the Category for the trigonometric functions. + See Also: o )show TrigonometricFunctionCategory @@ -2916,9 +2913,6 @@ These are implemented by this category: \begin{chunk}{category TRIGCAT TrigonometricFunctionCategory} )abbrev category TRIGCAT TrigonometricFunctionCategory -++ Category for the trigonometric functions -++ Author: ??? -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the trigonometric functions; @@ -2971,6 +2965,17 @@ digraph pic { \pagehead{Type}{TYPE} \pagepic{ps/v102type.ps}{TYPE}{1.00} +\begin{chunk}{Type.help} +==================================================================== +Type examples +==================================================================== + +The fundamental Type. + +See Also: +o )show Type + +\end{chunk} {\bf See:} \pageto{Aggregate}{AGG} @@ -2986,7 +2991,6 @@ digraph pic { \begin{chunk}{category TYPE Type} )abbrev category TYPE Type -++ The new fundamental Type (keeping Object for 1.5 as well) ++ Author: Richard Jenks ++ Date Created: 14 May 1992 ++ Date Last Updated: 14 May 1992 @@ -3058,6 +3062,16 @@ digraph pic { Aggregate examples ==================================================================== +The notion of aggregate serves to model any data structure aggregate, +designating any collection of objects, with heterogenous or homogeneous +members, with a finite or infinite number of members, explicitly or +implicitly represented. An aggregate can in principle represent +everything from a string of characters to abstract sets such +as "the set of x satisfying relation r(x)" + +An attribute "finiteAggregate" is used to assert that a domain +has a finite number of elements. + See Also: o )show Aggregate @@ -3104,12 +3118,6 @@ These are implemented by this category: ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The notion of aggregate serves to model any data structure aggregate, ++ designating any collection of objects, with heterogenous or homogeneous @@ -3117,7 +3125,7 @@ These are implemented by this category: ++ implicitly represented. An aggregate can in principle represent ++ everything from a string of characters to abstract sets such ++ as "the set of x satisfying relation r(x)" -++ An attribute \spadatt{finiteAggregate} is used to assert that a domain +++ An attribute "finiteAggregate" is used to assert that a domain ++ has a finite number of elements. Aggregate: Category == Type with @@ -3218,6 +3226,9 @@ digraph pic { CombinatorialOpsCategory examples ==================================================================== +CombinatorialOpsCategory is the category obtaining by adjoining +summations and products to the usual combinatorial operations; + See Also: o )show CombinatorialOpsCategory @@ -3256,9 +3267,7 @@ These exports come from \refto{CombinatorialFunctionCategory}(): \begin{chunk}{category COMBOPC CombinatorialOpsCategory} )abbrev category COMBOPC CombinatorialOpsCategory -++ Category for summations and products ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 22 February 1993 (JHD/BMT) ++ Description: ++ CombinatorialOpsCategory is the category obtaining by adjoining @@ -3349,6 +3358,12 @@ digraph pic { EltableAggregate examples ==================================================================== +An eltable aggregate is one which can be viewed as a function. +For example, the list [1,7,4] can applied to 0,1, and 2 respectively +will return the integers 1, 7, and 4; thus this list may be viewed +as mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate +can map members of a domain Dom to an image domain Im. + See Also: o )show EltableAggregate @@ -3398,17 +3413,11 @@ These exports come from \refto{Eltable}(): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An eltable aggregate is one which can be viewed as a function. -++ For example, the list \axiom{[1,7,4]} can applied to 0,1, and 2 -++ respectively will return the integers 1,7, and 4; thus this list may -++ be viewed as mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate +++ For example, the list [1,7,4] can applied to 0,1, and 2 respectively +++ will return the integers 1, 7, and 4; thus this list may be viewed as +++ mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate ++ can map members of a domain Dom to an image domain Im. EltableAggregate(Dom:SetCategory, Im:Type): Category == @@ -3507,6 +3516,9 @@ digraph pic { Evalable examples ==================================================================== +This category provides eval operations. A domain may belong to this +category if it is possible to make "evaluation" substitutions. + See Also: o )show Evalable @@ -3542,16 +3554,7 @@ These exports come from \refto{InnerEvalable}(R:SetCategory,R:SetCategory): \begin{chunk}{category EVALAB Evalable} )abbrev category EVALAB Evalable -++ Author: -++ Date Created: ++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: FullyEvalable -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This category provides \spadfun{eval} operations. ++ A domain may belong to this category if it is possible to make @@ -3645,6 +3648,9 @@ digraph pic { FortranProgramCategory examples ==================================================================== +FortranProgramCategory provides various models of FORTRAN subprograms. +These can be transformed into actual FORTRAN code. + See Also: o )show FortranProgramCategory @@ -3685,16 +3691,9 @@ These exports come from \refto{CoercibleTo}(OutputForm): )abbrev category FORTCAT FortranProgramCategory ++ Author: Mike Dewar ++ Date Created: November 1992 -++ Date Last Updated: -++ Basic Operations: -++ Related Constructors: FortranType, FortranCode, Switch -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: -++ \axiomType{FortranProgramCategory} provides various models of -++ FORTRAN subprograms. These can be transformed into actual FORTRAN code. +++ FortranProgramCategory provides various models of FORTRAN subprograms. +++ These can be transformed into actual FORTRAN code. FortranProgramCategory():Category == Join(Type,CoercibleTo OutputForm) with outputAsFortran : $ -> Void @@ -3779,6 +3778,11 @@ digraph pic { FullyRetractableTo examples ==================================================================== +A is fully retractable to B means that A is retractable to B and +if B is retractable to the integers or rational numbers then so is A. +In particular, what we are asserting is that there are no integers +(rationals) in A which don't retract into B. + See Also: o )show FullyRetractableTo @@ -3946,6 +3950,10 @@ digraph pic { FullyPatternMatchable examples ==================================================================== +A set S is PatternMatchable over R if S can lift the pattern-matching +functions of S over the integers and float to itself (necessary for +matching in towers). + See Also: o )show FullyPatternMatchable @@ -4008,11 +4016,9 @@ These exports come from \refto{Type}(): \begin{chunk}{category FPATMAB FullyPatternMatchable} )abbrev category FPATMAB FullyPatternMatchable -++ Category of sets that can be pattern-matched on ++ Author: Manuel Bronstein ++ Date Created: 28 Nov 1989 ++ Date Last Updated: 29 Nov 1989 -++ Keywords: pattern, matching. ++ Description: ++ A set S is PatternMatchable over R if S can lift the ++ pattern-matching functions of S over the integers and float @@ -4100,6 +4106,8 @@ digraph pic { Logic examples ==================================================================== +Logic provides the basic operations for lattices, e.g., boolean algebra. + See Also: o )show Logic @@ -4138,12 +4146,6 @@ These exports come from \refto{BasicType}(): \begin{chunk}{category LOGIC Logic} )abbrev category LOGIC Logic -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: ~, /\, \/ -++ Related Constructors: -++ Keywords: boolean ++ Description: ++ `Logic' provides the basic operations for lattices, e.g., boolean algebra. @@ -4222,6 +4224,12 @@ digraph pic { PlottablePlaneCurveCategory examples ==================================================================== +PlottablePlaneCurveCategory is the category of curves in the plane +which may be plotted via the graphics facilities. Functions are +provided for obtaining lists of lists of points, representing the +branches of the curve, and for determining the ranges of the +x-coordinates and y-coordinates of the points on the curve. + See Also: o )show PlottablePlaneCurveCategory @@ -4256,18 +4264,7 @@ These exports come from \refto{CoercibleTo}(OutputForm): ++ Author: Clifton J. Williamson ++ Date Created: 11 January 1990 ++ Date Last Updated: 15 June 1990 -++ Basic Operations: listBranches, xRange, yRange -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: plot, graphics -++ References: ++ Description: -++ PlottablePlaneCurveCategory is the category of curves in the -++ plane which may be plotted via the graphics facilities. Functions are -++ provided for obtaining lists of lists of points, representing the -++ branches of the curve, and for determining the ranges of the -++ x-coordinates and y-coordinates of the points on the curve. PlottablePlaneCurveCategory(): Category == Definition where L ==> List @@ -4356,6 +4353,12 @@ digraph pic { PlottableSpaceCurveCategory examples ==================================================================== +PlottableSpaceCurveCategory is the category of curves in 3-space which +may be plotted via the graphics facilities. Functions are provided for +obtaining lists of lists of points, representing the branches of the +curve, and for determining the ranges of the x-, y-, and z-coordinates +of the points on the curve. + See Also: o )show PlottableSpaceCurveCategory @@ -4392,12 +4395,6 @@ These exports come from \refto{CoercibleTo}(OutputForm): ++ Author: Clifton J. Williamson ++ Date Created: 11 January 1990 ++ Date Last Updated: 15 June 1990 -++ Basic Operations: listBranches, xRange, yRange, zRange -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: plot, graphics -++ References: ++ Description: ++ PlottableSpaceCurveCategory is the category of curves in ++ 3-space which may be plotted via the graphics facilities. Functions are @@ -4490,6 +4487,8 @@ digraph pic { RealConstant examples ==================================================================== +The category of real numeric domains, i.e. convertible to floats. + See Also: o )show RealConstant @@ -4519,15 +4518,6 @@ These exports come from \refto{ConvertibleTo}(Float): \begin{chunk}{category REAL RealConstant} )abbrev category REAL RealConstant -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of real numeric domains, i.e. convertible to floats. @@ -4608,6 +4598,8 @@ digraph pic { SegmentCategory examples ==================================================================== +This category provides operations on ranges, or segments as they are called. + See Also: o )show SegmentCategory @@ -4654,13 +4646,6 @@ These are directly exported but not implemented: ++ Author: Stephen M. Watt ++ Date Created: December 1986 ++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: range, segment -++ Examples: -++ References: ++ Description: ++ This category provides operations on ranges, or segments ++ as they are called. @@ -4760,6 +4745,11 @@ digraph pic { SetCategory examples ==================================================================== +SetCategory is the basic category for describing a collection +of elements with = (equality) and coerce to output form. + +Conditional Attributes canonical data structure equality is the same as = + See Also: o )show SetCategory @@ -4820,15 +4810,7 @@ These exports come from \refto{CoercibleTo}(OutputForm): \begin{chunk}{category SETCAT SetCategory} )abbrev category SETCAT SetCategory -++ Author: -++ Date Created: ++ Date Last Updated: November 10, 2009 tpd happy birthday -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{SetCategory} is the basic category for describing a collection ++ of elements with \spadop{=} (equality) and \spadfun{coerce} to @@ -4931,6 +4913,8 @@ digraph pic { TranscendentalFunctionCategory examples ==================================================================== +This is the Category for the transcendental elementary functions. + See Also: o )show TranscendentalFunctionCategory @@ -5039,9 +5023,7 @@ These exports come from \refto{ElementaryFunctionCategory}(): \begin{chunk}{category TRANFUN TranscendentalFunctionCategory} )abbrev category TRANFUN TranscendentalFunctionCategory -++ Category for the transcendental elementary functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the transcendental elementary functions; @@ -5170,6 +5152,13 @@ digraph pic { AbelianSemiGroup examples ==================================================================== +This is the class of all additive (commutative) semigroups, i.e. +a set with a commutative and associative operation +. + +Axioms: + associative("+":(%,%)->%) (x+y)+z = x+(y+z) + commutative("+":(%,%)->%) x+y = y+x + See Also: o )show AbelianSemiGroup @@ -5213,15 +5202,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category ABELSG AbelianSemiGroup} )abbrev category ABELSG AbelianSemiGroup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The class of all additive (commutative) semigroups, i.e. ++ a set with a commutative and associative operation \spadop{+}. @@ -5510,6 +5490,8 @@ digraph pic { DesingTreeCategory examples ==================================================================== +This category is part of the PAFF package. + See Also: o )show DesingTreeCategory @@ -5728,6 +5710,9 @@ digraph pic { FortranFunctionCategory examples ==================================================================== +FortranFunctionCategory is the category of arguments to NAG Library +routines which return (sets of) function values. + See Also: o )show FortranFunctionCategory @@ -5780,7 +5765,6 @@ These exports come from \refto{FortranProgramCategory}(): ++ Author: Mike Dewar ++ Date Created: 13 January 1994 ++ Date Last Updated: 18 March 1994 -++ Related Constructors: FortranProgramCategory. ++ Description: ++ \axiomType{FortranFunctionCategory} is the category of arguments to ++ NAG Library routines which return (sets of) function values. @@ -5911,6 +5895,10 @@ digraph pic { FortranMatrixCategory examples ==================================================================== +FortranMatrixCategory provides support for producing Functions and +Subroutines when the input to these is an AXIOM object of type Matrix +or in domains involving FortranCode. + See Also: o )show FortranMatrixCategory @@ -5949,8 +5937,6 @@ These exports come from \refto{FortranProgramCategory}(): )abbrev category FMC FortranMatrixCategory ++ Author: Mike Dewar ++ Date Created: 21 March 1994 -++ Date Last Updated: -++ Related Constructors: FortranProgramCategory. ++ Description: ++ \axiomType{FortranMatrixCategory} provides support for ++ producing Functions and Subroutines when the input to these @@ -6057,6 +6043,9 @@ digraph pic { FortranMatrixFunctionCategory examples ==================================================================== +FortranMatrixFunctionCategory provides support for producing Functions +and Subroutines representing matrices of expressions. + See Also: o )show FortranMatrixFunctionCategory @@ -6108,8 +6097,6 @@ These exports come from \refto{FortranProgramCategory}(): )abbrev category FMFUN FortranMatrixFunctionCategory ++ Author: Mike Dewar ++ Date Created: March 18 1994 -++ Date Last Updated: -++ Related Constructors: FortranProgramCategory. ++ Description: ++ \axiomType{FortranMatrixFunctionCategory} provides support for ++ producing Functions and Subroutines representing matrices of @@ -6241,6 +6228,10 @@ digraph pic { FortranVectorCategory examples ==================================================================== +FortranVectorCategory provides support for producing Functions and +Subroutines when the input to these is an AXIOM object of type +Vector or in domains involving FortranCode. + See Also: o )show FortranVectorCategory @@ -6278,7 +6269,6 @@ These exports come from \refto{FortranProgramCategory}(): ++ Author: Mike Dewar ++ Date Created: October 1993 ++ Date Last Updated: 18 March 1994 -++ Related Constructors: FortranProgramCategory. ++ Description: ++ \axiomType{FortranVectorCategory} provides support for ++ producing Functions and Subroutines when the input to these @@ -6385,6 +6375,9 @@ digraph pic { FortranVectorFunctionCategory examples ==================================================================== +FortranVectorFunctionCategory is the catagory of arguments +to NAG Library routines which return the values of vectors of functions. + See Also: o )show FortranVectorFunctionCategory @@ -6437,7 +6430,6 @@ These exports come from \refto{FortranProgramCategory}(): ++ Author: Mike Dewar ++ Date Created: 11 March 1994 ++ Date Last Updated: 18 March 1994 -++ Related Constructors: FortranProgramCategory. ++ Description: ++ \axiomType{FortranVectorFunctionCategory} is the catagory of arguments ++ to NAG Library routines which return the values of vectors of functions. @@ -6572,6 +6564,9 @@ digraph pic { FullyEvalableOver examples ==================================================================== +This category provides a selection of evaluation operations depending +on what the argument type R provides. + See Also: o )show FullyEvalableOver @@ -6618,16 +6613,7 @@ These exports come from \refto{InnerEvalable}(a:Symbol,b:SetCategory): \begin{chunk}{category FEVALAB FullyEvalableOver} )abbrev category FEVALAB FullyEvalableOver -++ Author: -++ Date Created: ++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: Equation -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This category provides a selection of evaluation operations ++ depending on what the argument type R provides. @@ -6748,6 +6734,11 @@ digraph pic { FileCategory examples ==================================================================== +This category provides an interface to operate on files in the +computer's file system. The precise method of naming files +is determined by the Name parameter. The type of the contents +of the file is determined by S. + See Also: o )show FileCategory @@ -6797,15 +6788,7 @@ These exports come from SetCategory(): \begin{chunk}{category FILECAT FileCategory} )abbrev category FILECAT FileCategory ++ Author: Stephen M. Watt, Victor Miller -++ Date Created: ++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: File -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This category provides an interface to operate on files in the ++ computer's file system. The precise method of naming files @@ -6929,6 +6912,14 @@ digraph pic { Finite examples ==================================================================== +The category of domains composed of a finite set of elements. We include +the functions lookup and index to give a bijection between the finite set +and an initial segment of positive integers. + +Axioms: + lookup(index(n)) = n + index(lookup(s)) = s + See Also: o )show Finite @@ -6973,15 +6964,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category FINITE Finite} )abbrev category FINITE Finite -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of domains composed of a finite set of elements. ++ We include the functions \spadfun{lookup} and \spadfun{index} @@ -7086,6 +7068,8 @@ digraph pic { FileNameCategory examples ==================================================================== +This category provides an interface to names in the file system. + See Also: o )show FileNameCategory @@ -7140,13 +7124,6 @@ These exports come from \refto{SetCategory}(): ++ Author: Stephen M. Watt ++ Date Created: 1985 ++ Date Last Updated: June 20, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This category provides an interface to names in the file system. @@ -7262,6 +7239,15 @@ digraph pic { GradedModule examples ==================================================================== +GradedModule(R,E) denotes "E-graded R-module", i.e. collection of +R-modules indexed by an abelian monoid E. An element g of G[s] for +some specific s in E is said to be an element of G with degree s. +Sums are defined in each module G[s] so two elements of G have a +sum if they have the same degree. + +Morphisms can be defined and composed by degree to give the mathematical +category of graded modules. + See Also: o )show GradedModule @@ -7316,12 +7302,6 @@ These exports come from \refto{SetCategory}(): ++ Author: Stephen M. Watt ++ Date Created: May 20, 1991 ++ Date Last Updated: May 20, 1991 -++ Basic Operations: +, *, degree -++ Related Domains: CartesianTensor(n,dim,R) -++ Also See: -++ AMS Classifications: -++ Keywords: graded module, tensor, multi-linear algebra -++ Examples: ++ References: Algebra 2d Edition, MacLane and Birkhoff, MacMillan 1979 ++ Description: ++ GradedModule(R,E) denotes ``E-graded R-module'', i.e. collection of @@ -7456,6 +7436,14 @@ digraph pic { HomogeneousAggregate examples ==================================================================== +A homogeneous aggregate is an aggregate of elements all of the same type. + +In the current system, all aggregates are homogeneous. Two attributes +characterize classes of aggregates. Aggregates from domains with +attribute "finiteAggregate" have a finite number of members. Those +with attribute "shallowlyMutable" allow an element to be modified +or updated without changing its overall value. + See Also: o )show HomogeneousAggregate @@ -7575,12 +7563,6 @@ These exports come from \refto{SetCategory}(): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991, May 1995 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A homogeneous aggregate is an aggregate of elements all of the ++ same type. @@ -7737,6 +7719,9 @@ digraph pic { IndexedDirectProductCategory examples ==================================================================== +This category represents the direct product of some set with respect +to an ordered indexing set. + See Also: o )show IndexedDirectProductCategory @@ -7781,14 +7766,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category IDPC IndexedDirectProductCategory} )abbrev category IDPC IndexedDirectProductCategory ++ Author: James Davenport -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category represents the direct product of some set with ++ respect to an ordered indexing set. @@ -7909,6 +7886,8 @@ digraph pic { LiouvillianFunctionCategory examples ==================================================================== +This is the Category for the transcendental Liouvillian functions. + See Also: o )show LiouvillianFunctionCategory @@ -8014,7 +7993,6 @@ These exports come from \refto{TranscendentalFunctionCategory}(): )abbrev category LFCAT LiouvillianFunctionCategory ++ Category for the transcendental Liouvillian functions ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Category for the transcendental Liouvillian functions; @@ -8142,6 +8120,9 @@ digraph pic { Monad examples ==================================================================== +Monad is the class of all multiplicative monads, i.e. sets +with a binary operation. + See Also: o )show Monad @@ -8191,11 +8172,6 @@ These exports come from \refto{SetCategory}(): ++ Authors: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 11 June 1991 -++ Basic Operations: *, ** -++ Related Constructors: SemiGroup, Monoid, MonadWithUnit -++ Also See: -++ AMS Classifications: -++ Keywords: Monad, binary operation ++ Reference: ++ N. Jacobson: Structure and Representations of Jordan Algebras ++ AMS, Providence, 1968 @@ -8323,6 +8299,9 @@ digraph pic { NumericalIntegrationCategory examples ==================================================================== +NumericalIntegrationCategory is the category for describing the set of +Numerical Integration domains with measure and numericalIntegration. + See Also: o )show NumericalIntegrationCategory @@ -8520,6 +8499,9 @@ digraph pic { NumericalOptimizationCategory examples ==================================================================== +NumericalOptimizationCategory is the category for describing the set of +Numerical Optimization domains with measure and optimize. + See Also: o )show NumericalOptimizationCategory @@ -8709,6 +8691,9 @@ digraph pic { OrdinaryDifferentialEquationsSolverCategory examples ==================================================================== +OrdinaryDifferentialEquationsSolverCategory is the category for describing +the set of ODE solver domains with measure and ODEsolve. + See Also: o )show OrdinaryDifferentialEquationsSolverCategory @@ -8768,7 +8753,6 @@ These exports come from \refto{SetCategory}(): ++ Author: Brian Dupee ++ Date Created: February 1995 ++ Date Last Updated: June 1995 -++ Basic Operations: ++ Description: ++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} is the ++ \axiom{category} for describing the set of ODE solver \axiom{domains} @@ -8879,6 +8863,10 @@ digraph pic { OrderedSet examples ==================================================================== +The class of totally ordered sets, that is, sets such that for each +pair of elements (a,b) exactly one of the following relations holds +a a%) (x*y)*z = x*(y*z) + +Conditional attributes: + commutative("*":(%,%)->%) x*y = y*x + See Also: o )show SemiGroup @@ -9836,15 +9819,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category SGROUP SemiGroup} )abbrev category SGROUP SemiGroup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ the class of all multiplicative semigroups, i.e. a set ++ with an associative operation \spadop{*}. @@ -9950,6 +9924,8 @@ digraph pic { SetCategoryWithDegree examples ==================================================================== +This is part of the PAFF package, related to projective space. + See Also: o )show SetCategoryWithDegree @@ -9992,7 +9968,6 @@ These exports come from \refto{SetCategory}(): ++ Author: Gaetan Hache ++ Date Created: 17 nov 1992 ++ Date Last Updated: May 2010 by Tim Daly -++ Keywords: ++ Description: ++ This is part of the PAFF package, related to projective space. SetCategoryWithDegree:Category == SetCategory with @@ -10088,6 +10063,9 @@ digraph pic { SExpressionCategory examples ==================================================================== +This category allows the manipulation of Lisp values while keeping +the grunge fairly localized. + See Also: o )show SExpressionCategory @@ -10167,7 +10145,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category SEXCAT SExpressionCategory} )abbrev category SEXCAT SExpressionCategory -++ Category for Lisp values ++ Author: S.M.Watt ++ Date Created: July 1987 ++ Date Last Modified: 23 May 1991 @@ -10320,6 +10297,18 @@ digraph pic { StepThrough examples ==================================================================== +A class of objects which can be 'stepped through'. + +Repeated applications of nextItem is guaranteed never to return +duplicate items and only return "failed" after exhausting all +elements of the domain. This assumes that the sequence starts +with init(). For infinite domains, repeated application of nextItem +is not required to reach all possible domain elements starting from +any initial element. + +Conditional attributes: + infinite -- repeated nextItem's are never "failed". + See Also: o )show StepThrough @@ -10361,15 +10350,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category STEP StepThrough} )abbrev category STEP StepThrough -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A class of objects which can be 'stepped through'. ++ Repeated applications of \spadfun{nextItem} is guaranteed never to @@ -10497,6 +10477,10 @@ digraph pic { ThreeSpaceCategory examples ==================================================================== +The category ThreeSpaceCategory is used for creating three dimensional +objects using functions for defining points, curves, polygons, +constructs and the subspaces containing them. + See Also: o )show ThreeSpaceCategory @@ -10615,19 +10599,6 @@ These exports come from \refto{SetCategory}(): \begin{chunk}{category SPACEC ThreeSpaceCategory} )abbrev category SPACEC ThreeSpaceCategory -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Operations: create3Space, numberOfComponents, numberOfComposites, -++ merge, composite, components, copy, enterPointData, modifyPointData, -++ point, point?, curve, curve?, closedCurve, closedCurve?, polygon, -++ polygon? mesh, mesh?, lp, lllip, lllp, llprop, lprop, objects, -++ check, subspace, coerce -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category ThreeSpaceCategory is used for creating ++ three dimensional objects using functions for defining points, curves, @@ -10986,6 +10957,13 @@ digraph pic { AbelianMonoid examples ==================================================================== +The class of multiplicative monoids, i.e. semigroups with an +additive identity element. + +Axioms: + leftIdentity("+":(%,%)->%,0) 0+x=x + rightIdentity("+":(%,%)->%,0) x+0=x + See Also: o )show AbelianMonoid @@ -11038,15 +11016,6 @@ These exports come from \refto{AbelianSemiGroup}(): \begin{chunk}{category ABELMON AbelianMonoid} )abbrev category ABELMON AbelianMonoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The class of multiplicative monoids, i.e. semigroups with an ++ additive identity element. @@ -11174,6 +11143,9 @@ digraph pic { AffineSpaceCategory examples ==================================================================== +The following is all the categories and domains related to projective +space and part of the PAFF package + See Also: o )show AffineSpaceCategory @@ -11415,6 +11387,10 @@ digraph pic { BagAggregate examples ==================================================================== +A bag aggregate is an aggregate for which one can insert and extract +objects, and where the order in which objects are inserted determines the +order of extraction. Examples of bags are stacks, queues, and dequeues. + See Also: o )show BagAggregate @@ -11524,12 +11500,6 @@ These exports come from \refto{HomogeneousAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A bag aggregate is an aggregate for which one can insert and extract ++ objects, and where the order in which objects are inserted determines @@ -11631,6 +11601,9 @@ digraph pic { CachableSet examples ==================================================================== +A cachable set is a set whose elements keep an integer as part +of their structure. + See Also: o )show CachableSet @@ -11680,7 +11653,6 @@ These exports come from \refto{OrderedSet}(): \begin{chunk}{category CACHSET CachableSet} )abbrev category CACHSET CachableSet -++ Sets whose elements can cache an integer ++ Author: Manuel Bronstein ++ Date Created: 31 Oct 1988 ++ Date Last Updated: 14 May 1991 @@ -11802,6 +11774,13 @@ digraph pic { Collection examples ==================================================================== +A collection is a homogeneous aggregate which can built from +list of members. The operation used to build the aggregate is +generically named construct. However, each collection provides +its own special function with the same name as the data type, +except with an initial lower case letter, e.g. +list for List, flexibleArray for FlexibleArray, and so on. + See Also: o )show Collection @@ -11930,12 +11909,6 @@ These exports come from \refto{ConvertibleTo}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A collection is a homogeneous aggregate which can built from ++ list of members. The operation used to build the aggregate is @@ -12109,6 +12082,42 @@ digraph pic { DifferentialVariableCategory examples ==================================================================== +DifferentialVariableCategory constructs the set of derivatives of a +given set of (ordinary) differential indeterminates. If x,...,y is +an ordered set of differential indeterminates, and the prime notation +is used for differentiation, then the set of derivatives (including +zero-th order) of the differential indeterminates is + x, x', x'',..., y, y', y'',... +(Note that in the interpreter, the n-th derivative of y is displayed as +y with a subscript n.) This set is viewed as a set of algebraic +indeterminates, totally ordered in a way compatible with differentiation +and the given order on the differential indeterminates. Such a total +order is called a ranking of the differential indeterminates. + +A domain in this category is needed to construct a differential +polynomial domain. Differential polynomials are ordered by a ranking +on the derivatives, and by an order (extending the ranking) on the set +of differential monomials. One may thus associate a domain in this +category with a ranking of the differential indeterminates, just as +one associates a domain in the category OrderedAbelianMonoidSup with +an ordering of the set of monomials in a set of algebraic indeterminates. +The ranking is specified through the binary relation <. For example, one +may define one derivative to be less than another by lexicographically +comparing first the order, then the given order of the differential +indeterminates appearing in the derivatives. This is the default +implementation. + +The notion of weight generalizes that of degree. A polynomial domain +may be made into a graded ring if a weight function is given on the set +of indeterminates. Very often, a grading is the first step in ordering +the set of monomials. For differential polynomial domains, this +constructor provides a function \spadfun{weight}, which allows the +assignment of a non-negative number to each derivative of a differential +indeterminate. For example, one may define the weight of a derivative +to be simply its order (this is the default assignment). This weight +function can then be extended to the set of all differential polynomials, +providing a graded ring structure. + See Also: o )show DifferentialVariableCategory @@ -12182,13 +12191,6 @@ These exports come from \refto{RetractableTo}(S:OrderedSet): ++ Author: William Sit ++ Date Created: 19 July 1990 ++ Date Last Updated: 13 September 1991 -++ Basic Operations: -++ Related Constructors:DifferentialPolynomialCategory -++ See Also:OrderedDifferentialVariable, -++ SequentialDifferentialVariable, -++ DifferentialSparseMultivariatePolynomial. -++ AMS Classifications:12H05 -++ Keywords: differential indeterminates, ranking, order, weight ++ References:Ritt, J.F. "Differential Algebra" (Dover, 1950). ++ Description: ++ \spadtype{DifferentialVariableCategory} constructs the @@ -12422,6 +12424,8 @@ digraph pic { ExpressionSpace examples ==================================================================== +An expression space is a set which is closed under certain operators. + See Also: o )show ExpressionSpace @@ -12563,7 +12567,6 @@ These exports come from \refto{Evalable}(a:SetCategory): ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 27 May 1994 -++ Keywords: operator, kernel, expression, space. ++ Description: ++ An expression space is a set which is closed under certain operators; @@ -12991,6 +12994,13 @@ digraph pic { GradedAlgebra examples ==================================================================== +GradedAlgebra(R,E) denotes "E-graded R-algebra". A graded algebra is a +graded module together with a degree preserving R-linear map, called +the product. + +The name "product" is written out in full so inner and outer products +with the same mapping type can be distinguished by name. + See Also: o )show GradedAlgebra @@ -13059,12 +13069,6 @@ These exports come from \refto{RetractableTo}(R:CommutativeRing): ++ Author: Stephen M. Watt ++ Date Created: May 20, 1991 ++ Date Last Updated: May 20, 1991 -++ Basic Operations: +, *, degree -++ Related Domains: CartesianTensor(n,dim,R) -++ Also See: -++ AMS Classifications: -++ Keywords: graded module, tensor, multi-linear algebra -++ Examples: ++ References: Encyclopedic Dictionary of Mathematics, MIT Press, 1977 ++ Description: ++ GradedAlgebra(R,E) denotes ``E-graded R-algebra''. @@ -13215,6 +13219,11 @@ digraph pic { IndexedAggregate examples ==================================================================== +An indexed aggregate is a many-to-one mapping of indices to entries. +For example, a one-dimensional-array is an indexed aggregate where +the index is an integer. Also, a table is an indexed aggregate +where the indices and entries may have any type. + See Also: o )show IndexedAggregate @@ -13356,12 +13365,6 @@ These exports come from \refto{EltableAggregate}(Index:SetCategory,Entry:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An indexed aggregate is a many-to-one mapping of indices to entries. ++ For example, a one-dimensional-array is an indexed aggregate where @@ -13540,6 +13543,16 @@ digraph pic { MonadWithUnit examples ==================================================================== +MonadWithUnit is the class of multiplicative monads with unit, +i.e. sets with a binary operation and a unit element. + +Axioms: + leftIdentity("*":(%,%)->%,1) 1*x=x + rightIdentity("*":(%,%)->%,1) x*1=x + +Common Additional Axioms: + unitsKnown - if "recip" says "failed", it PROVES input wasn't a unit + See Also: o )show MonadWithUnit @@ -13602,12 +13615,6 @@ These exports come from \refto{Monad}(): ++ Authors: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 11 June 1991 -++ Basic Operations: *, **, 1 -++ Related Constructors: SemiGroup, Monoid, Monad -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Keywords: Monad with unit, binary operation ++ Reference: ++ N. Jacobson: Structure and Representations of Jordan Algebras ++ AMS, Providence, 1968 @@ -13759,6 +13766,16 @@ digraph pic { Monoid examples ==================================================================== +The class of multiplicative monoids, i.e. semigroups with a +multiplicative identity element. + +Axioms: + leftIdentity("*":(%,%)->%,1) 1*x=x + rightIdentity("*":(%,%)->%,1) x*1=x + +Conditional attributes: + unitsKnown - \spadfun{recip} only returns "failed" on non-units + See Also: o )show Monoid @@ -13817,15 +13834,6 @@ These exports come from \refto{SemiGroup}(): \begin{chunk}{category MONOID Monoid} )abbrev category MONOID Monoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The class of multiplicative monoids, i.e. semigroups with a ++ multiplicative identity element. @@ -13956,6 +13964,8 @@ digraph pic { OrderedFinite examples ==================================================================== +This is the category of Ordered finite sets. + See Also: o )show OrderedFinite @@ -14010,15 +14020,6 @@ These exports come from \refto{Finite}(): \begin{chunk}{category ORDFIN OrderedFinite} )abbrev category ORDFIN OrderedFinite -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered finite sets. @@ -14120,6 +14121,8 @@ digraph pic { PlacesCategory examples ==================================================================== +This is part of the PAFF package, related to projective space. + See Also: o )show PlacesCategory @@ -14340,6 +14343,8 @@ digraph pic { ProjectiveSpaceCategory examples ==================================================================== +This is part of the PAFF package, related to projective space. + See Also: o )show ProjectiveSpaceCategory @@ -14606,6 +14611,13 @@ digraph pic { RecursiveAggregate examples ==================================================================== +A recursive aggregate over a type S is a model for a a directed graph +containing values of type S. Recursively, a recursive aggregate is a node +consisting of a value from S and 0 or more children which are recursive +aggregates. A node with no children is called a leaf node. A recursive +aggregate may be cyclic for which some operations as noted may go into +an infinite loop. + See Also: o )show RecursiveAggregate @@ -14736,12 +14748,6 @@ These exports come from \refto{HomogeneousAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A recursive aggregate over a type S is a model for a ++ a directed graph containing values of type S. @@ -14906,6 +14912,8 @@ first column in an array and vice versa. TwoDimensionalArrayCategory examples ==================================================================== +This is the category of two dimensional array categories and domains. + See Also: o )show TwoDimensionalArrayCategory @@ -15039,12 +15047,9 @@ These exports come from \refto{HomogeneousAggregate}(R:Type) \begin{chunk}{category ARR2CAT TwoDimensionalArrayCategory} )abbrev category ARR2CAT TwoDimensionalArrayCategory -++ Author: ++ Date Created: 27 October 1989 ++ Date Last Updated: 27 June 1990 ++ Keywords: array, data structure -++ Examples: -++ References: ++ Description: ++ Two dimensional array categories and domains @@ -15521,6 +15526,9 @@ digraph pic { BinaryRecursiveAggregate examples ==================================================================== +A binary-recursive aggregate has 0, 1 or 2 children and serves +as a model for a binary tree or a doubly-linked aggregate structure + See Also: o )show BinaryRecursiveAggregate @@ -15665,12 +15673,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type) ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A binary-recursive aggregate has 0, 1 or 2 children and serves ++ as a model for a binary tree or a doubly-linked aggregate structure @@ -15854,6 +15856,12 @@ digraph pic { CancellationAbelianMonoid examples ==================================================================== +This is an AbelianMonoid with the cancellation property, i.e. + a+b = a+c => b=c +This is formalised by the partial subtraction operator, which satisfies +the Axiom + c = a+b <=> c-b = a + See Also: o )show CancellationAbelianMonoid @@ -15903,15 +15911,6 @@ These exports come from \refto{AbelianMonoid}(): \begin{chunk}{category CABMON CancellationAbelianMonoid} )abbrev category CABMON CancellationAbelianMonoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: Davenport & Trager I ++ Description: ++ This is an \spadtype{AbelianMonoid} with the cancellation property, i.e.\br ++ \tab{5}\spad{ a+b = a+c => b=c }.\br @@ -16051,6 +16050,9 @@ digraph pic { DictionaryOperations examples ==================================================================== +This category is a collection of operations common to both +categories Dictionary and MultiDictionary. + See Also: o )show DictionaryOperations @@ -16196,12 +16198,6 @@ These exports come from \refto{Collection}(S:SetCategory) ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category is a collection of operations common to both ++ categories \spadtype{Dictionary} and \spadtype{MultiDictionary} @@ -16360,6 +16356,10 @@ digraph pic { DoublyLinkedAggregate examples ==================================================================== +A doubly-linked aggregate serves as a model for a doubly-linked +list, that is, a list which can has links to both next and previous +nodes and thus can be efficiently traversed in both directions. + See Also: o )show DoublyLinkedAggregate @@ -16499,12 +16499,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A doubly-linked aggregate serves as a model for a doubly-linked ++ list, that is, a list which can has links to both next and previous @@ -16621,6 +16615,12 @@ digraph pic { Group examples ==================================================================== +The class of multiplicative groups, i.e. monoids with multiplicative inverses. + +Axioms: + leftInverse("*":(%,%)->%,inv) inv(x)*x = 1 + rightInverse("*":(%,%)->%,inv) x*inv(x) = 1 + See Also: o )show Group @@ -16694,15 +16694,6 @@ These exports come from \refto{Monoid}(): \begin{chunk}{category GROUP Group} )abbrev category GROUP Group -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The class of multiplicative groups, i.e. monoids with ++ multiplicative inverses. @@ -16883,6 +16874,17 @@ digraph pic { LinearAggregate examples ==================================================================== +A linear aggregate is an aggregate whose elements are indexed by integers. +Examples of linear aggregates are strings, lists, and arrays. + +Most of the exported operations for linear aggregates are non-destructive +but are not always efficient for a particular aggregate. + +For example, concat of two lists needs only to copy its first argument, +whereas concat of two arrays needs to copy both arguments. Most of the +operations exported here apply to infinite objects (e.g. streams) as well +to finite ones. For finite linear aggregates, see FiniteLinearAggregate. + See Also: o )show LinearAggregate @@ -17058,12 +17060,6 @@ These exports come from \refto{Collection}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A linear aggregate is an aggregate whose elements are indexed by integers. ++ Examples of linear aggregates are strings, lists, and @@ -18093,6 +18089,16 @@ inverse matrix [[j**i for i in 0..4] for j in 1..5] MatrixCategory examples ==================================================================== +MatrixCategory is a general matrix category which allows different +representations and indexing schemes. Rows and columns may be +extracted with rows returned as objects of type Row and colums +returned as objects of type Col. A domain belonging to this category +will be shallowly mutable. The index of the 'first' row may be +obtained by calling the function minRowIndex. The index of the +'first' column may be obtained by calling the function minColIndex. +The index of the first element of a Row is the same as the index of the +first column in a matrix and vice versa. + Predicates: square?(m) returns true if m is a square matrix @@ -18595,13 +18601,6 @@ Col:FiniteLinearAggregate(R): ++ Authors: Grabmeier, Gschnitzer, Williamson, Gabriel Dos Reis ++ Date Created: 1987 ++ Date Last Updated: July 1990 -++ Basic Operations: -++ Related Domains: Matrix(R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, linear algebra -++ Examples: -++ References: ++ Description: ++ \spadtype{MatrixCategory} is a general matrix category which allows ++ different representations and indexing schemes. Rows and @@ -19456,6 +19455,12 @@ digraph pic { OrderedAbelianSemiGroup examples ==================================================================== +Ordered sets which are also abelian semigroups, such that the addition +preserves the ordering. + +Axiom: + x < y => x+z < y+z + See Also: o )show OrderedAbelianSemiGroup @@ -19514,15 +19519,6 @@ These exports come from \refto{AbelianMonoid}(): \begin{chunk}{category OASGP OrderedAbelianSemiGroup} )abbrev category OASGP OrderedAbelianSemiGroup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also abelian semigroups, such that the addition ++ preserves the ordering.\br @@ -19635,6 +19631,13 @@ digraph pic { OrderedMonoid examples ==================================================================== +Ordered sets which are also monoids, such that multiplication +preserves the ordering. + +Axioms: + x < y => x*z < y*z + x < y => z*x < z*y + See Also: o )show OrderedMonoid @@ -19697,15 +19700,6 @@ These exports come from \refto{OrderedSet}(): \begin{chunk}{category ORDMON OrderedMonoid} )abbrev category ORDMON OrderedMonoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also monoids, such that multiplication ++ preserves the ordering. @@ -19859,6 +19853,16 @@ digraph pic { PolynomialSetCategory examples ==================================================================== +A category for finite subsets of a polynomial ring. Such a set is +only regarded as a set of polynomials and not identified to the ideal +it generates. So two distinct sets may generate the same the ideal. +Furthermore, for R being an integral domain, a set of polynomials may +be viewed as a representation of the ideal it generates in the polynomial +ring (R)^(-1) P, or the set of its zeros (described for instance by the +radical of the previous ideal, or a split of the associated affine +variety) and so on. So this category provides operations about +those different notions. + See Also: o )show PolynomialSetCategory @@ -20023,12 +20027,6 @@ These exports come from \refto{IntegralDomain}(): ++ Author: Marc Moreno Maza ++ Date Created: 04/26/1994 ++ Date Last Updated: 12/15/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set -++ References: ++ Description: ++ A category for finite subsets of a polynomial ring. ++ Such a set is only regarded as a set of polynomials and not @@ -20523,6 +20521,9 @@ digraph pic { PriorityQueueAggregate examples ==================================================================== +A priority queue is a bag of items from an ordered set where the item +extracted is always the maximum element. + See Also: o )show PriorityQueueAggregate @@ -20632,12 +20633,6 @@ These exports come from \refto{BagAggregate}(S:OrderedSet): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A priority queue is a bag of items from an ordered set where the item ++ extracted is always the maximum element. @@ -20761,6 +20756,8 @@ digraph pic { QueueAggregate examples ==================================================================== +A queue is a bag where the first item inserted is the first item extracted. + See Also: o )show QueueAggregate @@ -20876,12 +20873,6 @@ These exports come from \refto{BagAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A queue is a bag where the first item inserted is the first ++ item extracted. @@ -21021,6 +21012,12 @@ digraph pic { SetAggregate examples ==================================================================== +A set category lists a collection of set-theoretic operations useful +for both finite sets and multisets. Note however that finite sets are +distinct from multisets. Although the operations defined for set +categories are common to both, the relationship between the two cannot +be described by inclusion or inheritance. + See Also: o )show SetAggregate @@ -21179,12 +21176,6 @@ These exports come from \refto{Collection}(S:SetCategory): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: 14 Oct, 1993 by RSS -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A set category lists a collection of set-theoretic operations ++ useful for both finite sets and multisets. @@ -21371,6 +21362,8 @@ digraph pic { StackAggregate examples ==================================================================== +A stack is a bag where the last item inserted is the first item extracted. + See Also: o )show StackAggregate @@ -21482,12 +21475,6 @@ These exports come from \refto{BagAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A stack is a bag where the last item inserted is the first item extracted. @@ -21650,6 +21637,16 @@ digraph pic { UnaryRecursiveAggregate examples ==================================================================== +A unary-recursive aggregate is a one where nodes may have either +0 or 1 children. This aggregate models, though not precisely, a linked +list possibly with a single cycle. + +A node with one children models a non-empty list, with the value of the +list designating the head, or first, of the list, and the child +designating the tail, or rest, of the list. A node with no child then +designates the empty list. Since these aggregates are recursive aggregates, +they may be cyclic. + See Also: o )show UnaryRecursiveAggregate @@ -21826,12 +21823,6 @@ These exports come from \refto{RecursiveAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A unary-recursive aggregate is a one where nodes may have either ++ 0 or 1 children. @@ -22163,6 +22154,13 @@ digraph pic { AbelianGroup examples ==================================================================== +The class of abelian groups, i.e. additive monoids where each element +has an additive inverse. + +Axioms: + -(-x) = x + x+(-x) = 0 + See Also: o )show AbelianGroup @@ -22225,15 +22223,6 @@ These exports come from \refto{CancellationAbelianMonoid}(): \begin{chunk}{category ABELGRP AbelianGroup} )abbrev category ABELGRP AbelianGroup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The class of abelian groups, i.e. additive monoids where ++ each element has an additive inverse. @@ -22384,6 +22373,10 @@ digraph pic { BinaryTreeCategory examples ==================================================================== +BinaryTreeCategory(S) is the category of binary trees: a tree which +is either empty or else is a node consisting of a value and a left and +right, both binary trees. + See Also: o )show BinaryTreeCategory @@ -22514,14 +22507,6 @@ These exports come from \refto{BinaryRecursiveAggregate}(S:SetCategory): )abbrev category BTCAT BinaryTreeCategory ++ Author:W. H. Burge ++ Date Created:17 Feb 1992 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ \spadtype{BinaryTreeCategory(S)} is the category of ++ binary trees: a tree which is either empty or else is a @@ -22685,6 +22670,13 @@ digraph pic { Dictionary examples ==================================================================== +A dictionary is an aggregate in which entries can be inserted, +searched for and removed. Duplicates are thrown away on insertion. +This category models the usual notion of dictionary which involves +large amounts of data where copying is impractical. + +Principal operations are thus destructive (non-copying) ones. + See Also: o )show Dictionary @@ -22821,12 +22813,6 @@ These exports come from \refto{DictionaryOperations}(S:SetCategory): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A dictionary is an aggregate in which entries can be inserted, ++ searched for and removed. Duplicates are thrown away on insertion. @@ -22983,6 +22969,10 @@ digraph pic { DequeueAggregate examples ==================================================================== +A dequeue is a doubly ended stack, that is, a bag where first items +inserted are the first items extracted, at either the front or +the back end of the data structure. + See Also: o )show DequeueAggregate @@ -23130,12 +23120,6 @@ These exports come from \refto{QueueAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A dequeue is a doubly ended stack, that is, a bag where first items ++ inserted are the first items extracted, at either the front or @@ -23313,8 +23297,16 @@ digraph pic { ExtensibleLinearAggregate examples ==================================================================== +An extensible aggregate is one which allows insertion and deletion of +entries. These aggregates are models of lists and streams which are +represented by linked structures so as to make insertion, deletion, and +concatenation efficient. However, access to elements of these +extensible aggregates is generally slow since access is made from the end. +See FlexibleArray for an exception. + See Also: o )show ExtensibleLinearAggregate +o )show FlexibleArray \end{chunk} {\bf See:} @@ -23500,12 +23492,6 @@ These exports come from \refto{LinearAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An extensible aggregate is one which allows insertion and deletion of ++ entries. These aggregates are models of lists and streams which are @@ -23715,6 +23701,10 @@ digraph pic { FiniteLinearAggregate examples ==================================================================== +A finite linear aggregate is a linear aggregate of finite length. +The finite property of the aggregate adds several exports to the +list of exports from LinearAggregate such as reverse, sort, and so on. + See Also: o )show FiniteLinearAggregate @@ -23937,12 +23927,6 @@ These exports come from \refto{OrderedSet}: ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A finite linear aggregate is a linear aggregate of finite length. ++ The finite property of the aggregate adds several exports to the @@ -24111,6 +24095,10 @@ digraph pic { FreeAbelianMonoidCategory examples ==================================================================== +A free abelian monoid on a set S is the monoid of finite sums of +the form reduce(+,[ni * si]) where the si's are in S, and the ni's +are in a given abelian monoid. The operation is commutative. + See Also: o )show FreeAbelianMonoidCategory @@ -24372,6 +24360,10 @@ digraph pic { MultiDictionary examples ==================================================================== +A multi-dictionary is a dictionary which may contain duplicates. +As for any dictionary, its size is assumed large so that +copying (non-destructive) operations are generally to be avoided. + See Also: o )show MultiDictionary @@ -24509,12 +24501,6 @@ These exports come from \refto{DictionaryOperations}(S:SetCategory): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A multi-dictionary is a dictionary which may contain duplicates. ++ As for any dictionary, its size is assumed large so that @@ -24619,6 +24605,9 @@ digraph pic { OrderedAbelianMonoid examples ==================================================================== +Ordered sets which are also abelian monoids, such that the addition +preserves the ordering. + See Also: o )show OrderedAbelianMonoid @@ -24677,15 +24666,6 @@ These exports come from \refto{AbelianMonoid}(): \begin{chunk}{category OAMON OrderedAbelianMonoid} )abbrev category OAMON OrderedAbelianMonoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also abelian monoids, such that the addition ++ preserves the ordering. @@ -24783,6 +24763,9 @@ digraph pic { PermutationCategory examples ==================================================================== +PermutationCategory provides a categorial environment for subgroups +of bijections of a set (i.e. permutations) + See Also: o )show PermutationCategory @@ -24879,12 +24862,6 @@ These exports come from \refto{OrderedSet}(): ++ Authors: Holger Gollan, Johannes Grabmeier, Gerhard Schneider ++ Date Created: 27 July 1989 ++ Date Last Updated: 29 March 1990 -++ Basic Operations: cycle, cycles, eval, orbit -++ Related Constructors: PermutationGroup, PermutationGroupExamples -++ Also See: RepresentationTheoryPackage1 -++ AMS Classifications: -++ Keywords: permutation, symmetric group -++ References: ++ Description: ++ PermutationCategory provides a categorial environment ++ for subgroups of bijections of a set (i.e. permutations) @@ -25094,8 +25071,15 @@ digraph pic { StreamAggregate examples ==================================================================== +A stream aggregate is a linear aggregate which possibly has an infinite +number of elements. A basic domain constructor which builds stream +aggregates is Stream. From streams, a number of infinite structures +such power series can be built. A stream aggregate may also be infinite +since it may be cyclic. For example, see DecimalExpansion. + See Also: o )show StreamAggregate +o )show DecimalExpansion \end{chunk} {\bf See:} @@ -25350,12 +25334,6 @@ These exports come from \refto{LinearAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A stream aggregate is a linear aggregate which possibly has an infinite ++ number of elements. A basic domain constructor which builds stream @@ -25595,6 +25573,28 @@ digraph pic { TriangularSetCategory examples ==================================================================== +The category of triangular sets of multivariate polynomials with +coefficients in an integral domain. + +Let R be an integral domain and V a finite ordered set of variables, + X1 < X2 < ... < Xn + +A set S of polynomials in R[X1,X2,...,Xn] is triangular if no elements +of S lies in R, and if two distinct elements of S have distinct main +variables. + +Note that the empty set is a triangular set. A triangular set is not +necessarily a (lexicographical) Groebner basis and the notion of +reduction related to triangular sets is based on the recursive view +of polynomials. We recall this notion here. For details see + P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories + of Triangular Sets" Journal of Symbol. Comp. + +A polynomial P is reduced with respect to a non-constant polynomial Q +if the degree of P in the main variable of Q is less than the main +degree of Q. A polynomial P is reduced with respect to a triangular +set T if it is reduced with respect to every polynomial of T. + See Also: o )show TriangularSetCategory @@ -25825,11 +25825,6 @@ V:OrderedSet, P:RecursivePolynomialCategory(R,E,V)): ++ Author: Marc Moreno Maza (marc@nag.co.uk) ++ Date Created: 04/26/1994 ++ Date Last Updated: 12/15/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set ++ References : ++ [1] P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories ++ of Triangular Sets" Journal of Symbol. Comp. (to appear) @@ -26457,6 +26452,10 @@ digraph pic { FiniteDivisorCategory examples ==================================================================== +This category describes finite rational divisors on a curve, that +is finite formal sums SUM(n * P) where the n's are integers and the +P's are finite rational points on the curve. + See Also: o )show FiniteDivisorCategory @@ -26533,7 +26532,6 @@ These exports come from \refto{AbelianGroup}(): ++ Author: Manuel Bronstein ++ Date Created: 19 May 1993 ++ Date Last Updated: 19 May 1993 -++ Keywords: divisor, algebraic, curve. ++ Description: ++ This category describes finite rational divisors on a curve, that ++ is finite formal sums SUM(n * P) where the n's are integers and the @@ -26716,8 +26714,13 @@ digraph pic { FiniteSetAggregate examples ==================================================================== +A finite-set aggregate models the notion of a finite set, that is, +a collection of elements characterized by membership, but not +by order or multiplicity. See Set for an example. + See Also: o )show FiniteSetAggregate +o )show Set \end{chunk} {\bf See:} @@ -26910,12 +26913,6 @@ These exports come from \refto{SetAggregate}(S:SetCategory): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: 14 Oct, 1993 by RSS -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A finite-set aggregate models the notion of a finite set, that is, ++ a collection of elements characterized by membership, but not @@ -27123,6 +27120,9 @@ digraph pic { KeyedDictionary examples ==================================================================== +A keyed dictionary is a dictionary of key-entry pairs for which there is +a unique entry for each key. + See Also: o )show KeyedDictionary @@ -27282,12 +27282,6 @@ and S=Record(key: Key,entry: Entry) ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A keyed dictionary is a dictionary of key-entry pairs for which there is ++ a unique entry for each key. @@ -27484,6 +27478,12 @@ digraph pic { LazyStreamAggregate examples ==================================================================== +LazyStreamAggregate is the category of streams with lazy +evaluation. It is understood that the function 'empty?' will +cause lazy evaluation if necessary to determine if there are +entries. Functions which call 'empty?', e.g. 'first' and 'rest', +will also cause lazy evaluation if necessary. + See Also: o )show LazyStreamAggregate @@ -27737,11 +27737,9 @@ These exports come from \refto{StreamAggregate}(S:Type): \begin{chunk}{category LZSTAGG LazyStreamAggregate} )abbrev category LZSTAGG LazyStreamAggregate -++ Category of streams with lazy evaluation ++ Author: Clifton J. Williamson ++ Date Created: 22 November 1989 ++ Date Last Updated: 20 July 1990 -++ Keywords: stream, infinite list, infinite sequence ++ Description: ++ LazyStreamAggregate is the category of streams with lazy ++ evaluation. It is understood that the function 'empty?' will @@ -28347,6 +28345,10 @@ digraph pic { LeftModule examples ==================================================================== +The category of left modules over an rng (ring not necessarily with unit). +This is an abelian group which supports left multiplation by elements of +the rng. + See Also: o )show LeftModule @@ -28402,15 +28404,6 @@ These exports come from \refto{AbelianGroup}(): \begin{chunk}{category LMODULE LeftModule} )abbrev category LMODULE LeftModule -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of left modules over an rng (ring not necessarily with unit). ++ This is an abelian group which supports left multiplation by elements of @@ -28611,6 +28604,10 @@ digraph pic { ListAggregate examples ==================================================================== +A list aggregate is a model for a linked list data structure. A linked +list is a versatile data structure. Insertion and deletion are efficient +and searching is a linear operation. + See Also: o )show ListAggregate @@ -28919,12 +28916,6 @@ These exports come from \refto{ExtensibleLinearAggregate}(S:Type): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A list aggregate is a model for a linked list data structure. ++ A linked list is a versatile @@ -29267,6 +29258,9 @@ digraph pic { MultisetAggregate examples ==================================================================== +A multi-set aggregate is a set which keeps track of the multiplicity +of its elements. + See Also: o )show MultisetAggregate @@ -29428,12 +29422,6 @@ These exports come from \refto{SetAggregate}(S:SetCategory): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A multi-set aggregate is a set which keeps track of the multiplicity ++ of its elements. @@ -29527,6 +29515,16 @@ digraph pic { NonAssociativeRng examples ==================================================================== +NonAssociativeRng is a basic ring-type structure, not necessarily +commutative or associative, and not necessarily with unit. + +Axioms: + x*(y+z) = x*y + x*z + (x+y)*z = x*z + y*z + +Common Additional Axioms + noZeroDivisors ab = 0 => a=0 or b=0 + See Also: o )show NonAssociativeRng @@ -29601,11 +29599,6 @@ These exports come from \refto{Monad}(): ++ Author: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 03 July 1991 -++ Basic Operations: +, *, -, ** -++ Related Constructors: Rng, Ring, NonAssociativeRing -++ Also See: -++ AMS Classifications: -++ Keywords: not associative ring ++ Reference: ++ R.D. Schafer: An Introduction to Nonassociative Algebras ++ Academic Press, New York, 1966 @@ -29784,8 +29777,24 @@ digraph pic { OneDimensionalArrayAggregate examples ==================================================================== +One-dimensional-array aggregates serves as models for one-dimensional +arrays. Categorically, these aggregates are finite linear aggregates +with the shallowlyMutable property, that is, any component of the array +may be changed without affecting the identity of the overall array. +Array data structures are typically represented by a fixed area in +storage and cannot efficiently grow or shrink on demand as can list +structures (see however FlexibleArray for a data structure +which is a cross between a list and an array). + +Iteration over, and access to, elements of arrays is extremely fast +(and often can be optimized to open-code). + +Insertion and deletion however is generally slow since an entirely new +data structure must be created for the result. + See Also: o )show OneDimensionalArrayAggregate +o )show FlexibleArray \end{chunk} {\bf See:} @@ -30323,6 +30332,9 @@ digraph pic { OrderedCancellationAbelianMonoid examples ==================================================================== +Ordered sets which are also abelian cancellation monoids, such that +the addition preserves the ordering. + See Also: o )show OrderedCancellationAbelianMonoid @@ -30384,15 +30396,6 @@ These exports come from \refto{CancellationAbelianMonoid}(): \begin{chunk}{category OCAMON OrderedCancellationAbelianMonoid} )abbrev category OCAMON OrderedCancellationAbelianMonoid -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also abelian cancellation monoids, ++ such that the addition preserves the ordering. @@ -30572,6 +30575,47 @@ digraph pic { RegularTriangularSetCategory examples ==================================================================== +The category of regular triangular sets was introduced under the name +regular chains in M. KALKBRENER "Three contributions to elimination theory". + +In P. AUBRY, D. LAZARD and M. MORENO MAZA "On the Theories of Triangular Sets" it is proved that regular triangular sets and towers of simple +extensions of a field are equivalent notions. + +In the following definitions, all polynomials and ideals are taken from +the polynomial ring k[x1,...,xn] where k is the fraction field of R. + +The triangular set [t1,...,tm] is regular iff for every i the initial +of ti+1 is invertible in the tower of simple extensions associated +with [t1,...,ti]. + +A family [T1,...,Ts] of regular triangular sets is a split of +Kalkbrener of a given ideal I iff the radical of I is equal to the +intersection of the radical ideals generated by the saturated ideals +of the [T1,...,Ti]. + +A family [T1,...,Ts] of regular triangular sets is a split of Kalkbrener +of a given triangular set T iff it is a split of Kalkbrener of the +saturated ideal of T. Let K be an algebraic closure of k. + +Assume that V is finite with cardinality n and let A be the affine +space K^n. + +For a regular triangular set T let denote by W(T) the set of regular +zeros of T. A family [T1,...,Ts] of regular triangular sets is a split +of Lazard of a given subset S of A iff the union of the W(Ti) contains +S and is contained in the closure of S (w.r.t. Zariski topology). + +A family [T1,...,Ts] of regular triangular sets is a split of Lazard +of a given triangular set T if it is a split of Lazard of W(T). +Note that if [T1,...,Ts] is a split of Lazard of T then it is also a +split of Kalkbrener of T. The converse is false. + +This category provides operations related to both kinds of splits, the +former being related to ideals decomposition whereas the latter deals +with varieties decomposition. See the example illustrating the +RegularTriangularSet constructor for more explanations about +decompositions by means of regular triangular sets. + See Also: o )show RegularTriangularSetCategory @@ -30838,11 +30882,6 @@ P:RecursivePolynomialCategory(R,E,V)): ++ Author: Marc Moreno Maza ++ Date Created: 09/03/1998 ++ Date Last Updated: 12/15/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: essai Graphisme -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set ++ References : ++ [1] M. KALKBRENER "Three contributions to elimination theory" ++ Phd Thesis, University of Linz, Austria, 1991. @@ -31273,6 +31312,15 @@ digraph pic { RightModule examples ==================================================================== +The category of right modules over an rng (ring not necessarily with unit). +This is an abelian group which supports right multiplication by elements +of the rng. + +Axioms: + x*(a*b) = (x*a)*b + x*(a+b) = (x*a)+(x*b) + (x+y)*x = (x*a)+(y*a) + See Also: o )show RightModule @@ -31326,15 +31374,6 @@ These exports come from \refto{AbelianGroup}(): \begin{chunk}{category RMODULE RightModule} )abbrev category RMODULE RightModule -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of right modules over an rng (ring not necessarily ++ with unit). This is an abelian group which supports right @@ -31431,6 +31470,17 @@ Rng is a Ring that does not necessarily have a unit. Rng examples ==================================================================== +The category of associative rings, not necessarily commutative, and not +necessarily with a 1. This is a combination of an abelian group +and a semigroup, with multiplication distributing over addition. + +Axioms: + x*(y+z) = x*y + x*z + (x+y)*z = x*z + y*z + +Conditional attributes: + noZeroDivisors ab = 0 => a=0 or b=0 + See Also: o )show Rng @@ -31490,15 +31540,6 @@ These exports come from \refto{SemiGroup}(): \begin{chunk}{category RNG Rng} )abbrev category RNG Rng -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of associative rings, not necessarily commutative, and not ++ necessarily with a 1. This is a combination of an abelian group @@ -31602,6 +31643,12 @@ digraph pic { BiModule examples ==================================================================== +A BiModule is both a left and right module with respect to potentially +different rings. + +Axiom: + r*(x*s) = (r*x)*s + See Also: o )show BiModule @@ -31675,15 +31722,6 @@ These exports come from \refto{RightModule}(S:Ring): \begin{chunk}{category BMODULE BiModule} )abbrev category BMODULE BiModule -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A \spadtype{BiModule} is both a left and right module with respect ++ to potentially different rings. @@ -31861,6 +31899,9 @@ digraph pic { BitAggregate examples ==================================================================== +The bit aggregate category models aggregates representing large +quantities of Boolean data. + See Also: o )show BitAggregate @@ -32105,12 +32146,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(Boolean): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The bit aggregate category models aggregates representing large ++ quantities of Boolean data. @@ -32269,6 +32304,9 @@ digraph pic { NonAssociativeRing examples ==================================================================== +A NonAssociativeRing is a non associative rng which has a unit, +the multiplication is not necessarily commutative or associative. + See Also: o )show NonAssociativeRing @@ -32361,11 +32399,6 @@ These exports come from \refto{MonadWithUnit}(): ++ Author: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 11 June 1991 -++ Basic Operations: +, *, -, ** -++ Related Constructors: NonAssociativeRng, Rng, Ring -++ Also See: -++ AMS Classifications: -++ Keywords: non-associative ring with unit ++ Reference: ++ R.D. Schafer: An Introduction to Nonassociative Algebras ++ Academic Press, New York, 1966 @@ -32565,6 +32598,15 @@ digraph pic { NormalizedTriangularSetCategory examples ==================================================================== +The category of normalized triangular sets. A triangular set ts is said +normalized if for every algebraic variable v of ts the polynomial +select(ts,v) is normalized w.r.t. every polynomial in collectUnder(ts,v). + +A polynomial p is said normalized w.r.t. a non-constant polynomial q +if p is constant or degree(p,mdeg(q)) = 0 and init(p) is normalized +w.r.t. q. One of the important features of normalized triangular sets +is that they are regular sets. + See Also: o )show NormalizedTriangularSetCategory @@ -32824,11 +32866,6 @@ P:RecursivePolynomialCategory(R,E,V)): ++ Author: Marc Moreno Maza ++ Date Created: 10/07/1998 ++ Date Last Updated: 12/12/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: essai Graphisme -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -32960,6 +32997,9 @@ digraph pic { OrderedAbelianGroup examples ==================================================================== +Ordered sets which are also abelian groups, such that the +addition preserves the ordering. + See Also: o )show OrderedAbelianGroup @@ -33025,15 +33065,6 @@ These exports come from \refto{AbelianGroup}(): \begin{chunk}{category OAGROUP OrderedAbelianGroup} )abbrev category OAGROUP OrderedAbelianGroup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also abelian groups, such that the ++ addition preserves the ordering. @@ -33122,6 +33153,16 @@ digraph pic { OrderedAbelianMonoidSup examples ==================================================================== +This domain is an OrderedAbelianMonoid with a sup operation added. +The purpose of the sup operator in this domain is to act as a +supremum with respect to the partial order imposed by `-`, rather +than with respect to the total > order (since that is "max"). + +Axioms: + sup(a,b)-a ~= "failed" + sup(a,b)-b ~= "failed" + x-a ~= "failed" and x-b ~= "failed" => x >= sup(a,b) + See Also: o )show OrderedAbelianMonoidSup @@ -33184,15 +33225,6 @@ These exports come from \refto{OrderedCancellationAbelianMonoid}(): \begin{chunk}{category OAMONS OrderedAbelianMonoidSup} )abbrev category OAMONS OrderedAbelianMonoidSup -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This domain is an OrderedAbelianMonoid with a sup ++ operation added. The purpose of the sup operator @@ -33339,6 +33371,10 @@ digraph pic { OrderedMultisetAggregate examples ==================================================================== +An ordered-multiset aggregate is a multiset built over an ordered set S +so that the relative sizes of its entries can be assessed. +These aggregates serve as models for priority queues. + See Also: o )show OrderedMultisetAggregate @@ -33518,12 +33554,6 @@ These exports come from \refto{PriorityQueueAggregate}(S:OrderedSet): ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An ordered-multiset aggregate is a multiset built over an ordered set S ++ so that the relative sizes of its entries can be assessed. @@ -33640,6 +33670,9 @@ digraph pic { Ring examples ==================================================================== +The category of rings with unity, always associative, but not +necessarily commutative. + See Also: o )show Ring @@ -33742,15 +33775,6 @@ These exports come from \refto{Monoid}(): \begin{chunk}{category RING Ring} )abbrev category RING Ring -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of rings with unity, always associative, but ++ not necessarily commutative. @@ -33954,6 +33978,12 @@ digraph pic { SquareFreeRegularTriangularSetCategory examples ==================================================================== +The category of square-free regular triangular sets. A regular +triangular set ts is square-free if the gcd of any polynomial p in ts +and differentiate(p,mvar(p)) w.r.t. collectUnder(ts,mvar(p)) +has degree zero w.r.t. mvar(p). Thus any square-free regular +set defines a tower of square-free simple extensions. + See Also: o )show SquareFreeRegularTriangularSetCategory @@ -34213,11 +34243,6 @@ P:RecursivePolynomialCategory(R,E,V)): ++ Author: Marc Moreno Maza ++ Date Created: 09/03/1996 ++ Date Last Updated: 09/10/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: essai Graphisme -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -34423,6 +34448,9 @@ digraph pic { StringAggregate examples ==================================================================== +A string aggregate is a category for strings, that is, one dimensional +arrays of characters. + See Also: o )show StringAggregate @@ -34691,12 +34719,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(Character): ++ revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A string aggregate is a category for strings, that is, ++ one dimensional arrays of characters. @@ -34941,6 +34963,9 @@ digraph pic { TableAggregate examples ==================================================================== +A table aggregate is a model of a table, i.e. a discrete many-to-one +mapping from keys to entries. + See Also: o )show TableAggregate @@ -35197,12 +35222,6 @@ and RecKE=Record(key: Key,entry: Entry): ++ revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A table aggregate is a model of a table, i.e. a discrete many-to-one ++ mapping from keys to entries. @@ -35505,6 +35524,13 @@ digraph pic { VectorCategory examples ==================================================================== +VectorCategory represents the type of vector like objects, +i.e. finite sequences indexed by some finite segment of the +integers. The operations available on vectors depend on the structure +of the underlying components. Many operations from the component domain +are defined for vectors componentwise. It can by assumed that extraction or +updating components can be done in constant time. + See Also: o )show VectorCategory @@ -35706,15 +35732,6 @@ These exports come from \refto{OneDimensionalArrayAggregate}(R:Type): \begin{chunk}{category VECTCAT VectorCategory} )abbrev category VECTCAT VectorCategory -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: DirectProductCategory, Vector, IndexedVector -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{VectorCategory} represents the type of vector like objects, ++ i.e. finite sequences indexed by some finite segment of the @@ -36059,6 +36076,10 @@ digraph pic { AssociationListAggregate examples ==================================================================== +An association list is a list of key entry pairs which may be viewed +as a table. It is a poor mans version of a table; searching for a key +is a linear operation. + See Also: o )show AssociationListAggregate @@ -36501,12 +36522,6 @@ and RecKE=Record(key: Key,entry: Entry) ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An association list is a list of key entry pairs which may be viewed ++ as a table. It is a poor mans version of a table: @@ -36622,6 +36637,8 @@ digraph pic { CharacteristicNonZero examples ==================================================================== +The category of Rings of Characteristic Non Zero + See Also: o )show CharacteristicNonZero @@ -36703,15 +36720,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category CHARNZ CharacteristicNonZero} )abbrev category CHARNZ CharacteristicNonZero -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Rings of Characteristic Non Zero @@ -36821,6 +36829,8 @@ digraph pic { CharacteristicZero examples ==================================================================== +The category of Rings of Characteristic Zero. + See Also: o )show CharacteristicZero @@ -36899,15 +36909,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category CHARZ CharacteristicZero} )abbrev category CHARZ CharacteristicZero -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Rings of Characteristic Zero. @@ -37014,6 +37015,9 @@ digraph pic { CommutativeRing examples ==================================================================== +The category of commutative rings with unity, i.e. rings where * is +commutative, and which have a multiplicative identity element. + See Also: o )show CommutativeRing @@ -37102,15 +37106,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category COMRING CommutativeRing} )abbrev category COMRING CommutativeRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of commutative rings with unity, i.e. rings where ++ \spadop{*} is commutative, and which have a multiplicative identity @@ -37237,6 +37232,13 @@ digraph pic { DifferentialRing examples ==================================================================== +An ordinary differential ring, that is, a ring with an operation +differentiate. + +Axioms: + differentiate(x+y) = differentiate(x)+differentiate(y) + differentiate(x*y) = x*differentiate(y) + differentiate(x)*y + See Also: o )show DifferentialRing @@ -37325,15 +37327,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category DIFRING DifferentialRing} )abbrev category DIFRING DifferentialRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An ordinary differential ring, that is, a ring with an operation ++ \spadfun{differentiate}. @@ -37463,6 +37456,13 @@ digraph pic { EntireRing examples ==================================================================== +Entire Rings (non-commutative Integral Domains), i.e. a ring +not necessarily commutative which has no zero divisors. + +Axioms: + ab=0 => a=0 or b=0 -- known as noZeroDivisors + not(1=0) + See Also: o )show EntireRing @@ -37543,15 +37543,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category ENTIRER EntireRing} )abbrev category ENTIRER EntireRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Entire Rings (non-commutative Integral Domains), i.e. a ring ++ not necessarily commutative which has no zero divisors. @@ -37680,8 +37671,16 @@ digraph pic { FreeModuleCat examples ==================================================================== +A domain of this category implements formal linear combinations +of elements from a domain Basis with coefficients in a domain R. +The domain Basis needs only to belong to the category SetCategory +and R to the category Ring. Thus the coefficient ring may be +non-commutative. See the XDistributedPolynomial constructor for +examples of domains built with the FreeModuleCat category constructor. + See Also: o )show FreeModuleCat +o )show XDistributedPolynomial \end{chunk} {\bf See:} @@ -37780,12 +37779,6 @@ These exports come from \refto{RetractableTo}(Basis:SetCategory): ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 ++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A domain of this category ++ implements formal linear combinations @@ -37946,6 +37939,8 @@ digraph pic { LeftAlgebra examples ==================================================================== +The category of all left algebras over an arbitrary ring. + See Also: o )show LeftAlgebra @@ -38137,6 +38132,8 @@ digraph pic { LinearlyExplicitRingOver examples ==================================================================== +An extension ring with an explicit linear dependence test. + See Also: o )show LinearlyExplicitRingOver @@ -38218,15 +38215,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category LINEXP LinearlyExplicitRingOver} )abbrev category LINEXP LinearlyExplicitRingOver -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ An extension ring with an explicit linear dependence test. @@ -38344,6 +38332,14 @@ digraph pic { Module examples ==================================================================== +The category of modules over a commutative ring. + +Axioms: + 1*x = x + (a*b)*x = a*(b*x) + (a+b)*x = (a*x)+(b*x) + a*(x+y) = (a*x)+(a*y) + See Also: o )show Module @@ -38410,15 +38406,6 @@ These exports come from \refto{BiModule}(a:Ring,b:Ring): \begin{chunk}{category MODULE Module} )abbrev category MODULE Module -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of modules over a commutative ring. ++ @@ -38534,6 +38521,12 @@ digraph pic { OrderedRing examples ==================================================================== +Ordered sets which are also rings, that is, domains where the ring +operations are compatible with the ordering. + +Axiom: + 0 ab < ac + See Also: o )show OrderedRing @@ -38639,15 +38632,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category ORDRING OrderedRing} )abbrev category ORDRING OrderedRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Ordered sets which are also rings, that is, domains where the ring ++ operations are compatible with the ordering. @@ -38793,6 +38777,13 @@ digraph pic { PartialDifferentialRing examples ==================================================================== +A partial differential ring with differentiations indexed by a +parameter type S. + +Axioms: + differentiate(x+y,e)=differentiate(x,e)+differentiate(y,e) + differentiate(x*y,e)=x*differentiate(y,e)+differentiate(x,e)*y + See Also: o )show PartialDifferentialRing @@ -38886,15 +38877,6 @@ These exports come from \refto{Ring}(): \begin{chunk}{category PDRING PartialDifferentialRing} )abbrev category PDRING PartialDifferentialRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A partial differential ring with differentiations indexed by a ++ parameter type S. @@ -39130,6 +39112,10 @@ digraph pic { PointCategory examples ==================================================================== +PointCategory is the category of points in space which may be plotted +via the graphics facilities. Functions are provided for defining +points and handling elements of points. + See Also: o )show PointCategory @@ -39338,16 +39324,6 @@ These exports come from \refto{VectorCategory}(R:Ring): \begin{chunk}{category PTCAT PointCategory} )abbrev category PTCAT PointCategory -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Operations: point, elt, setelt, copy, dimension, minIndex, maxIndex, -++ convert -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ PointCategory is the category of points in space which ++ may be plotted via the graphics facilities. Functions are provided for @@ -39505,6 +39481,10 @@ The RectangularMatrix domain is matrices of fixed dimension. RectangularMatrixCategory examples ==================================================================== +RectangularMatrixCategory is a category of matrices of fixed dimensions. +The dimensions of the matrix will be parameters of the domain. +Domains in this category will be R-modules and will be non-mutable. + See Also: o )show RectangularMatrixCategory @@ -39668,13 +39648,6 @@ These exports come from \refto{HomogeneousAggregate}(Ring)" ++ Authors: Grabmeier, Gschnitzer, Williamson ++ Date Created: 1987 ++ Date Last Updated: July 1990 -++ Basic Operations: -++ Related Domains: RectangularMatrix(m,n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ \spadtype{RectangularMatrixCategory} is a category of matrices of fixed ++ dimensions. The dimensions of the matrix will be parameters of the @@ -40003,6 +39976,11 @@ digraph pic { SquareFreeNormalizedTriangularSetCategory examples ==================================================================== +The category of square-free and normalized triangular sets. +Thus, up to the primitivity axiom of D. LAZARD +"A new method for solving algebraic systems of positive dimension", +these sets are Lazard triangular sets. + See Also: o )show SquareFreeNormalizedTriangularSetCategory @@ -40262,11 +40240,6 @@ P:RecursivePolynomialCategory(R,E,V)): ++ Author: Marc Moreno Maza ++ Date Created: 10/07/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: essai Graphisme -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -40446,6 +40419,8 @@ digraph pic { StringCategory examples ==================================================================== +A category for string-like objects + See Also: o )show StringCategory @@ -40725,8 +40700,6 @@ These exports come from \refto{OpenMath}(): \begin{chunk}{category STRICAT StringCategory} )abbrev category STRICAT StringCategory --- Note that StringCategory is built into the old compiler --- redundant SetCategory added to help A# compiler ++ Description: ++ A category for string-like objects @@ -40869,6 +40842,12 @@ digraph pic { UnivariateSkewPolynomialCategory examples ==================================================================== +This is the category of univariate skew polynomials over an Ore +coefficient ring. The multiplication is given by + x a = \sigma(a) x + \delta a +This category is an evolution of the types MonogenicLinearOperator, +OppositeMonogenicLinearOperator, and NonCommutativeOperatorDivision + See Also: o )show UnivariateSkewPolynomialCategory @@ -41414,6 +41393,10 @@ digraph pic { XAlgebra examples ==================================================================== +This is the category of algebras over non-commutative rings. +It is used by constructors of non-commutative algebras such as +XPolynomialRing and XFreeAlgebra + See Also: o )show XAlgebra @@ -41508,12 +41491,6 @@ These exports come from \refto{BiModule}(R:Ring,R:Ring): ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 ++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is the category of algebras over non-commutative rings. ++ It is used by constructors of non-commutative algebras such as @@ -41639,6 +41616,15 @@ digraph pic { Algebra examples ==================================================================== +The category of associative algebras (modules which are themselves rings). + +Axioms: + (b+c)::% = (b::%) + (c::%) + (b*c)::% = (b::%) * (c::%) + (1::R)::% = 1::% + b*x = (b::%)*x + r*(a*b) = (r*a)*b = a*(r*b) + See Also: o )show Algebra @@ -41735,15 +41721,6 @@ These exports come from \refto{Module}(R:CommutativeRing): \begin{chunk}{category ALGEBRA Algebra} )abbrev category ALGEBRA Algebra -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of associative algebras (modules which are themselves rings). ++ @@ -41913,6 +41890,9 @@ digraph pic { DifferentialExtension examples ==================================================================== +Differential extensions of a ring R. Given a differentiation on R, +extend it to a differentiation on %. + See Also: o )show DifferentialExtension @@ -42032,15 +42012,6 @@ These exports come from \refto{PartialDifferentialRing}(Symbol): \begin{chunk}{category DIFEXT DifferentialExtension} )abbrev category DIFEXT DifferentialExtension -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Differential extensions of a ring R. ++ Given a differentiation on R, extend it to a differentiation on %. @@ -42197,6 +42168,10 @@ digraph pic { FullyLinearlyExplicitRingOver examples ==================================================================== +S is FullyLinearlyExplicitRingOver R means that S is a +LinearlyExplicitRingOver R and, in addition, if R is a +LinearlyExplicitRingOver Integer, then so is S + See Also: o )show FullyLinearlyExplicitRingOver @@ -42289,15 +42264,6 @@ These exports come from \refto{LinearlyExplicitRingOver}(a:Ring): \begin{chunk}{category FLINEXP FullyLinearlyExplicitRingOver} )abbrev category FLINEXP FullyLinearlyExplicitRingOver -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ S is \spadtype{FullyLinearlyExplicitRingOver R} means that S is a ++ \spadtype{LinearlyExplicitRingOver R} and, in addition, if R is a @@ -42431,6 +42397,9 @@ digraph pic { LieAlgebra examples ==================================================================== +The category of Lie Algebras. It is used by the domains of non-commutative +algebra, LiePolynomial and XPBWPolynomial. + See Also: o )show LieAlgebra @@ -42509,8 +42478,6 @@ These exports come from \refto{Module}(R:Ring): ++ Author: Michel Petitot (petitot@lifl.fr). ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 -++ Keywords: -++ References: ++ Description: ++ The category of Lie Algebras. ++ It is used by the domains of non-commutative algebra, @@ -42665,6 +42632,13 @@ digraph pic { LinearOrdinaryDifferentialOperatorCategory examples ==================================================================== +LinearOrdinaryDifferentialOperatorCategory is the category +of differential operators with coefficients in a ring A with a given +derivation. + +Multiplication of operators corresponds to functional composition: + (L1 * L2).(f) = L1 L2 f + See Also: o )show LinearOrdinaryDifferentialOperatorCategory @@ -42844,7 +42818,6 @@ These exports come from \refto{Eltable}(A:Ring,A:Ring): ++ Author: Manuel Bronstein ++ Date Created: 9 December 1993 ++ Date Last Updated: 15 April 1994 -++ Keywords: differential operator ++ Description: ++ LinearOrdinaryDifferentialOperatorCategory is the category ++ of differential operators with coefficients in a ring A with a given @@ -43043,6 +43016,12 @@ digraph pic { NonAssociativeAlgebra examples ==================================================================== +NonAssociativeAlgebra is the category of non associative algebras +(modules which are themselves non associative rngs).\br + +Axiom: + r*(a*b) = (r*a)*b = a*(r*b) + See Also: o )show NonAssociativeAlgebra @@ -43128,11 +43107,6 @@ These exports come from \refto{Module}(R:CommutativeRing): ++ Author: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 11 June 1991 -++ Basic Operations: +, -, *, ** -++ Related Constructors: Algebra -++ Also See: -++ AMS Classifications: -++ Keywords: nonassociative algebra ++ Reference: ++ R.D. Schafer: An Introduction to Nonassociative Algebras ++ Academic Press, New York, 1966 @@ -43261,6 +43235,8 @@ digraph pic { VectorSpace examples ==================================================================== +Vector Spaces (not necessarily finite dimensional) over a field. + See Also: o )show VectorSpace @@ -43331,15 +43307,6 @@ These exports come from \refto{Module}(): \begin{chunk}{category VSPACE VectorSpace} )abbrev category VSPACE VectorSpace -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Vector Spaces (not necessarily finite dimensional) over a field. @@ -43470,6 +43437,9 @@ digraph pic { XFreeAlgebra examples ==================================================================== +This category specifies opeations for polynomials and formal series +with non-commutative variables. + See Also: o )show XFreeAlgebra @@ -43617,12 +43587,6 @@ where WORD:OrderedFreeMonoid(OrderedSet)) ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 ++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category specifies opeations for polynomials ++ and formal series with non-commutative variables. @@ -43926,6 +43890,9 @@ digraph pic { DirectProductCategory examples ==================================================================== +This category represents a finite cartesian product of a given type. +Many categorical properties are preserved under this construction. + See Also: o )show DirectProductCategory @@ -44252,18 +44219,6 @@ These exports come from \refto{OrderedAbelianMonoidSup}(): -- all direct product category domains must be compiled -- without subsumption, set SourceLevelSubset to EQUAL --)bo $noSubsumption := true - ---% DirectProductCategory - -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: DirectProduct -++ Also See: VectorCategory -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category represents a finite cartesian product of a given type. ++ Many categorical properties are preserved under this construction. @@ -44458,6 +44413,9 @@ digraph pic { DivisionRing examples ==================================================================== +A division ring (sometimes called a skew field), i.e. a not necessarily +commutative ring where all non-zero elements have multiplicative inverses. + See Also: o )show DivisionRing @@ -44557,15 +44515,6 @@ These exports come from \refto{Algebra}(Fraction(Integer)): \begin{chunk}{category DIVRING DivisionRing} )abbrev category DIVRING DivisionRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A division ring (sometimes called a skew field), ++ i.e. a not necessarily commutative ring where @@ -44739,6 +44688,9 @@ digraph pic { FiniteRankNonAssociativeAlgebra examples ==================================================================== +A FiniteRankNonAssociativeAlgebra is a non associative algebra over +a commutative ring R which is a free R-module of finite rank. + See Also: o )show FiniteRankNonAssociativeAlgebra @@ -44921,12 +44873,6 @@ These exports come from \refto{NonAssociativeAlgebra}(R:CommutativeRing): ++ Author: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 12 June 1991 -++ Basic Operations: +,-,*,**, someBasis -++ Related Constructors: FramedNonAssociativeAlgebra, FramedAlgebra, -++ FiniteRankAssociativeAlgebra -++ Also See: -++ AMS Classifications: -++ Keywords: nonassociative algebra, basis ++ References: ++ R.D. Schafer: An Introduction to Nonassociative Algebras ++ Academic Press, New York, 1966 @@ -45694,6 +45640,9 @@ digraph pic { FreeLieAlgebra examples ==================================================================== +The category of free Lie algebras. It is used by domains of +non-commutative algebra such as LiePolynomial and XPBWPolynomial. + See Also: o )show FreeLieAlgebra @@ -45791,12 +45740,6 @@ These exports come from \refto{LieAlgebra}(CommutativeRing): ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 ++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of free Lie algebras. ++ It is used by domains of non-commutative algebra: @@ -45954,6 +45897,13 @@ digraph pic { IntegralDomain examples ==================================================================== +The category of commutative integral domains, i.e. commutative +rings with no zero divisors. + +Conditional attributes: + canonicalUnitNormal - the canonical field is the same for all associates + canonicalsClosed - the product of two canonicals is itself canonical + See Also: o )show IntegralDomain @@ -46080,15 +46030,6 @@ These exports come from \refto{Algebra}(a:IntegralDomain): \begin{chunk}{category INTDOM IntegralDomain} )abbrev category INTDOM IntegralDomain -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: Davenport & Trager I ++ Description: ++ The category of commutative integral domains, i.e. commutative ++ rings with no zero divisors. @@ -46243,6 +46184,18 @@ digraph pic { MonogenicLinearOperator examples ==================================================================== +This is the category of linear operator rings with one generator. +The generator is not named by the category but can always be +constructed as monomial(1,1). + +For convenience, call the generator G. +Then each value is equal to + sum(a(i)*G**i, i = 0..n) +for some unique n and a(i) in R. + +Note that multiplication is not necessarily commutative. +In fact, if a is in R, it is quite normal to have a*G ^= G*a. + See Also: o )show MonogenicLinearOperator @@ -46350,13 +46303,6 @@ These exports come from \refto{Algebra}(R:CommutativeRing): ++ Author: Stephen M. Watt ++ Date Created: 1986 ++ Date Last Updated: May 30, 1991 -++ Basic Operations: -++ Related Domains: NonCommutativeOperatorDivision -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This is the category of linear operator rings with one generator. ++ The generator is not named by the category but can always be @@ -46562,6 +46508,10 @@ digraph pic { OctonionCategory examples ==================================================================== +OctonionCategory gives the categorial frame for the octonions, and +eight-dimensional non-associative algebra, doubling the the quaternions +in the same way as doubling the Complex numbers to get the quaternions. + See Also: o )show OctonionCategory @@ -46749,12 +46699,6 @@ These exports come from \refto{CharacteristicNonZero}(): ++ Author: R. Wisbauer, J. Grabmeier ++ Date Created: 05 September 1990 ++ Date Last Updated: 19 September 1990 -++ Basic Operations: _+, _*, octon, real, imagi, imagj, imagk, -++ imagE, imagI, imagJ, imagK -++ Related Constructors: QuaternionCategory -++ Also See: -++ AMS Classifications: -++ Keywords: octonion, non-associative algebra, Cayley-Dixon ++ References: e.g. I.L Kantor, A.S. Solodovnikov: ++ Hypercomplex Numbers, Springer Verlag Heidelberg, 1989, ++ ISBN 0-387-96980-2 @@ -47151,6 +47095,9 @@ digraph pic { QuaternionCategory examples ==================================================================== +QuaternionCategory describes the category of quaternions and implements +functions that are not representation specific. + See Also: o )show QuaternionCategory @@ -47367,15 +47314,7 @@ These exports come from \refto{CharacteristicNonZero}(): )abbrev category QUATCAT QuaternionCategory ++ Author: Robert S. Sutor ++ Date Created: 23 May 1990 -++ Change History: -++ 10 September 1990 -++ Basic Operations: (Algebra) -++ abs, conjugate, imagI, imagJ, imagK, norm, quatern, rational, -++ rational?, real -++ Related Constructors: Quaternion, QuaternionCategoryFunctions2 -++ Also See: DivisionRing -++ AMS Classifications: 11R52 -++ Keywords: quaternions, division ring, algebra +++ Change History: 10 September 1990 ++ Description: ++ \spadtype{QuaternionCategory} describes the category of quaternions ++ and implements functions that are not representation specific. @@ -47735,6 +47674,11 @@ The SquareMatrix domain is for square matrices of fixed dimension. SquareMatrixCategory examples ==================================================================== +SquareMatrixCategory is a general square matrix category which allows +different representations and indexing schemes. Rows and columns may +be extracted with rows returned as objects of type Row and colums +returned as objects of type Col. + See Also: o )show SquareMatrixCategory @@ -47991,13 +47935,6 @@ These exports come from \refto{FullyLinearlyExplicitRingOver}(R:Ring): ++ Authors: Grabmeier, Gschnitzer, Williamson ++ Date Created: 1987 ++ Date Last Updated: July 1990 -++ Basic Operations: -++ Related Domains: SquareMatrix(ndim,R) -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ \spadtype{SquareMatrixCategory} is a general square matrix category which ++ allows different representations and indexing schemes. Rows and @@ -48317,6 +48254,10 @@ digraph pic { XPolynomialsCat examples ==================================================================== +The Category of polynomial rings with non-commutative variables. +The coefficient ring may be non-commutative too. +However coefficients commute with variables. + See Also: o )show XPolynomialsCat @@ -48463,16 +48404,10 @@ These exports come from \refto{SetCategory}(): ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 ++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The Category of polynomial rings with non-commutative variables. ++ The coefficient ring may be non-commutative too. -++ However coefficients commute with vaiables. +++ However coefficients commute with variables. XPolynomialsCat(vl:OrderedSet,R:Ring):Category == Export where WORD ==> OrderedFreeMonoid(vl) @@ -48637,6 +48572,17 @@ digraph pic { AbelianMonoidRing examples ==================================================================== +Abelian monoid ring elements (not necessarily of finite support) +of this ring are of the form formal SUM (r_i * e_i) +where the r_i are coefficents and the e_i, elements of the +ordered abelian monoid, are thought of as exponents or monomials. +The monomials commute with each other, and with +the coefficients (which themselves may or may not be commutative). + +See FiniteAbelianMonoidRing for the case of finite support +a useful common model for polynomials and power series. +Conceptually at least, only the non-zero terms are ever operated on. + See Also: o )show AbelianMonoidRing @@ -48787,15 +48733,6 @@ These exports come from \refto{Algebra}(Fraction(Integer)): \begin{chunk}{category AMR AbelianMonoidRing} )abbrev category AMR AbelianMonoidRing -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Abelian monoid ring elements (not necessarily of finite support) ++ of this ring are of the form formal SUM (r_i * e_i) @@ -48986,6 +48923,9 @@ digraph pic { FortranMachineTypeCategory examples ==================================================================== +A category of domains which model machine arithmetic used by machines +in the AXIOM-NAG link. + See Also: o )show FortranMachineTypeCategory @@ -49106,14 +49046,6 @@ These exports come from \refto{RetractableTo}(Integer): )abbrev category FMTC FortranMachineTypeCategory ++ Author: Mike Dewar ++ Date Created: December 1993 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: FortranExpression, MachineInteger, MachineFloat, MachineComplex -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ A category of domains which model machine arithmetic ++ used by machines in the AXIOM-NAG link. @@ -49300,6 +49232,10 @@ digraph pic { FramedNonAssociativeAlgebra examples ==================================================================== +FramedNonAssociativeAlgebra(R) is a FiniteRankNonAssociativeAlgebra +(i.e. a non associative algebra over R which is a free R-module of +finite rank) over a commutative ring R together with a fixed R-module basis. + See Also: o )show FramedNonAssociativeAlgebra @@ -49517,12 +49453,6 @@ where R:CommutativeRing: ++ Author: J. Grabmeier, R. Wisbauer ++ Date Created: 01 March 1991 ++ Date Last Updated: 11 June 1991 -++ Basic Operations: +,-,*,**,basis -++ Related Constructors: FiniteRankNonAssociativeAlgebra, FramedAlgebra, -++ FiniteRankAssociativeAlgebra -++ Also See: -++ AMS Classifications: -++ Keywords: nonassociative algebra, basis ++ Reference: ++ R.D. Schafer: An Introduction to Nonassociative Algebras ++ Academic Press, New York, 1966 @@ -49950,6 +49880,11 @@ digraph pic { GcdDomain examples ==================================================================== +This category describes domains where gcd can be computed but where +there is no guarantee of the existence of factor operation for factorisation +into irreducibles. However, if such a factor operation exist, factorization +will be unique up to order and units. + See Also: o )show GcdDomain @@ -50064,15 +49999,6 @@ These exports come from \refto{IntegralDomain}(): \begin{chunk}{category GCDDOM GcdDomain} )abbrev category GCDDOM GcdDomain -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: Davenport & Trager 1 ++ Description: ++ This category describes domains where ++ \spadfun{gcd} can be computed but where there is no guarantee @@ -50234,6 +50160,9 @@ digraph pic { OrderedIntegralDomain examples ==================================================================== +The category of ordered commutative integral domains, where ordering +and the arithmetic operations are compatible + See Also: o )show OrderedIntegralDomain @@ -50354,12 +50283,6 @@ These exports come from \refto{OrderedRing}(): )abbrev category OINTDOM OrderedIntegralDomain ++ Author: JH Davenport (after L Gonzalez-Vega) ++ Date Created: 30.1.96 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ Description: ++ The category of ordered commutative integral domains, where ordering ++ and the arithmetic operations are compatible @@ -50507,6 +50430,10 @@ digraph pic { FiniteAbelianMonoidRing examples ==================================================================== +This category is similar to AbelianMonoidRing, except that the sum is +assumed to be finite. It is a useful model for polynomials, but is +somewhat more general. + See Also: o )show FiniteAbelianMonoidRing @@ -50678,15 +50605,7 @@ These exports come from \refto{FullyRetractableTo}(R:Ring): \begin{chunk}{category FAMR FiniteAbelianMonoidRing} )abbrev category FAMR FiniteAbelianMonoidRing -++ Author: -++ Date Created: ++ Date Last Updated: 14.08.2000 Exported pomopo! and binomThmExpt [MMM] -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category is similar to AbelianMonoidRing, except that the sum is ++ assumed to be finite. It is a useful model for polynomials, @@ -50963,6 +50882,9 @@ digraph pic { IntervalCategory examples ==================================================================== +This category implements of interval arithmetic and transcendental +functions over intervals. + See Also: o )show IntervalCategory @@ -51191,13 +51113,6 @@ These exports come from \refto{RetractableTo}(Integer): )abbrev category INTCAT IntervalCategory ++ Author: Mike Dewar ++ Date Created: November 1996 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This category implements of interval arithmetic and transcendental ++ functions over intervals. @@ -51381,6 +51296,9 @@ digraph pic { PowerSeriesCategory examples ==================================================================== +PowerSeriesCategory is the most general power series category with +exponents in an ordered abelian monoid. + See Also: o )show PowerSeriesCategory @@ -51523,13 +51441,6 @@ where Coef:Ring and Expon:OrderedAbelianMonoid: ++ Author: Clifton J. Williamson ++ Date Created: 21 December 1989 ++ Date Last Updated: 25 February 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: power series -++ Examples: -++ References: ++ Description: ++ \spadtype{PowerSeriesCategory} is the most general power series ++ category with exponents in an ordered abelian monoid. @@ -51717,6 +51628,12 @@ digraph pic { PrincipalIdealDomain examples ==================================================================== +The category of constructive principal ideal domains, i.e. where a +single generator can be constructively found for any ideal given by +a finite set of generators. Note that this constructive definition +only implies that finitely generated ideals are principal. It is not +clear what we would mean by an infinitely generated ideal. + See Also: o )show PrincipalIdealDomain @@ -51832,15 +51749,6 @@ These exports come from \refto{GcdDomain}(): \begin{chunk}{category PID PrincipalIdealDomain} )abbrev category PID PrincipalIdealDomain -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of constructive principal ideal domains, i.e. ++ where a single generator can be constructively found for @@ -51970,6 +51878,10 @@ digraph pic { UniqueFactorizationDomain examples ==================================================================== +A constructive unique factorization domain, i.e. where we can +constructively factor members into a product of a finite number +of irreducible elements. + See Also: o )show UniqueFactorizationDomain @@ -52095,15 +52007,6 @@ These exports come from \refto{GcdDomain}(): \begin{chunk}{category UFD UniqueFactorizationDomain} )abbrev category UFD UniqueFactorizationDomain -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A constructive unique factorization domain, i.e. where ++ we can constructively factor members into a product of @@ -52240,6 +52143,8 @@ digraph pic { DivisorCategory examples ==================================================================== +This category exports the function for domains. + See Also: o )show DivisorCategory @@ -52503,13 +52408,22 @@ digraph pic { EuclideanDomain examples ==================================================================== +A constructive euclidean domain, i.e. one can divide producing +a quotient and a remainder where the remainder is either zero +or is smaller (euclideanSize) than the divisor. + +Conditional attributes: + multiplicativeValuation - Size(a*b)=Size(a)*Size(b) + additiveValuation - Size(a*b)=Size(a)+Size(b) + +Principal Ideal Domains are a subset of Euclidean Domains. +Euclidean Domains are a subset of Fields. + See Also: o )show EuclideanDomain \end{chunk} -Principal Ideal Domains are a subset of Euclidean Domains. \pagefrom{PrincipalIdealDomain}{PID}. -Euclidean Domains are a subset of Fields. \pageto{Field}{FIELD} {\bf See:} @@ -52640,15 +52554,6 @@ These exports come from \refto{PrincipalIdealDomain}(): \begin{chunk}{category EUCDOM EuclideanDomain} )abbrev category EUCDOM EuclideanDomain -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A constructive euclidean domain, i.e. one can divide producing ++ a quotient and a remainder where the remainder is either zero @@ -52961,6 +52866,9 @@ digraph pic { MultivariateTaylorSeriesCategory examples ==================================================================== +MultivariateTaylorSeriesCategory is the most general multivariate +Taylor series category. + See Also: o )show MultivariateTaylorSeriesCategory @@ -53211,13 +53119,6 @@ These exports come from \refto{TranscendentalFunctionCategory}(): ++ Author: Clifton J. Williamson ++ Date Created: 6 March 1990 ++ Date Last Updated: 6 March 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: multivariate, Taylor, series -++ Examples: -++ References: ++ Description: ++ \spadtype{MultivariateTaylorSeriesCategory} is the most general ++ multivariate Taylor series category. @@ -53377,6 +53278,12 @@ digraph pic { PolynomialFactorizationExplicit examples ==================================================================== +This is the category of domains that know "enough" about themselves +in order to factor univariate polynomials over themselves. This will +be used in future releases for supporting factorization over finitely +generated coefficient fields, it is not yet available in the current +release of Axiom. + See Also: o )show PolynomialFactorizationExplicit @@ -53520,14 +53427,6 @@ These exports come from \refto{UniqueFactorizationDomain}(): \begin{chunk}{category PFECAT PolynomialFactorizationExplicit} )abbrev category PFECAT PolynomialFactorizationExplicit ++ Author: James Davenport -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is the category of domains that know "enough" about ++ themselves in order to factor univariate polynomials over themselves. @@ -53741,6 +53640,12 @@ digraph pic { UnivariatePowerSeriesCategory examples ==================================================================== +UnivariatePowerSeriesCategory is the most general univariate power +series category with exponents in an ordered abelian monoid. Note that +this category exports a substitution function if it is possible to +multiply exponents. Also note that this category exports a derivative +operation if it is possible to multiply coefficients by exponents. + See Also: o )show UnivariatePowerSeriesCategory @@ -53956,13 +53861,6 @@ These exports come from \refto{PartialDifferentialRing}(Symbol): ++ Author: Clifton J. Williamson ++ Date Created: 21 December 1989 ++ Date Last Updated: 20 September 1993 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ \spadtype{UnivariatePowerSeriesCategory} is the most general ++ univariate power series category with exponents in an ordered @@ -54218,6 +54116,14 @@ digraph pic { Field examples ==================================================================== +The category of commutative fields, i.e. commutative rings where all +non-zero elements have multiplicative inverses. The factor operation +while trivial is useful to have defined. + +Axioms: + a*(b/a) = b + inv(a) = 1/a + See Also: o )show Field @@ -54391,15 +54297,6 @@ These exports come from \refto{DivisionRing}(): \begin{chunk}{category FIELD Field} )abbrev category FIELD Field -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of commutative fields, i.e. commutative rings ++ where all non-zero elements have multiplicative inverses. @@ -54571,6 +54468,8 @@ digraph pic { IntegerNumberSystem examples ==================================================================== +An IntegerNumberSystem is a model for the integers. + See Also: o )show IntegerNumberSystem @@ -54847,7 +54746,6 @@ These exports come from \refto{LinearlyExplicitRingOver}(Integer): )abbrev category INS IntegerNumberSystem ++ Author: Stephen M. Watt ++ Date Created: January 1988 -++ Change History: ++ Description: ++ An \spad{IntegerNumberSystem} is a model for the integers. @@ -55626,6 +55524,8 @@ digraph pic { PAdicIntegerCategory examples ==================================================================== +This is the category of stream-based representations of the p-adic integers. + See Also: o )show PAdicIntegerCategory @@ -55775,13 +55675,6 @@ These exports come from \refto{EuclideanDomain}(): ++ Author: Clifton J. Williamson ++ Date Created: 15 May 1990 ++ Date Last Updated: 15 May 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: p-adic, completion -++ Examples: -++ References: ++ Description: ++ This is the category of stream-based representations of ++ the p-adic integers. @@ -56028,6 +55921,9 @@ digraph pic { PolynomialCategory examples ==================================================================== +The category for general multi-variate polynomials over a ring R, +in variables from VarSet, with exponents from the OrderedAbelianMonoidSup. + See Also: o )show PolynomialCategory @@ -56404,15 +56300,6 @@ These exports come from \refto{PolynomialFactorizationExplicit}(): \begin{chunk}{category POLYCAT PolynomialCategory} )abbrev category POLYCAT PolynomialCategory -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: Ring, monomial, coefficient, differentiate, eval -++ Related Constructors: Polynomial, DistributedMultivariatePolynomial -++ Also See: UnivariatePolynomialCategory -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category for general multi-variate polynomials over a ring ++ R, in variables from VarSet, with exponents from the @@ -57078,6 +56965,9 @@ digraph pic { UnivariateTaylorSeriesCategory examples ==================================================================== +UnivariateTaylorSeriesCategory is the category of Taylor series +in one variable. + See Also: o )show UnivariateTaylorSeriesCategory @@ -57370,13 +57260,6 @@ These exports come from \refto{RadicalCategory}(): ++ Author: Clifton J. Williamson ++ Date Created: 21 December 1989 ++ Date Last Updated: 26 May 1994 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Taylor, linebacker -++ Examples: -++ References: ++ Description: ++ \spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor ++ series in one variable. @@ -57879,6 +57762,8 @@ zerosOf(sup,x) AlgebraicallyClosedField examples ==================================================================== +This category is a model for algebraically closed fields. + Given the polynomial: pi:Polynomial(Integer):=-3*x^3+2*x+13 @@ -58139,7 +58024,6 @@ These exports come from \refto{RadicalCategory}(): ++ Author: Manuel Bronstein ++ Date Created: 22 Mar 1988 ++ Date Last Updated: 27 November 1991 -++ Keywords: algebraic, closure, field. ++ Description: ++ Model for algebraically closed fields. @@ -58545,6 +58429,28 @@ digraph pic { DifferentialPolynomialCategory examples ==================================================================== +DifferentialPolynomialCategory is a category constructor specifying +basic functions in an ordinary differential polynomial ring with a +given ordered set of differential indeterminates. In addition, it +implements defaults for the basic functions. + +The functions order and weight are extended from the set of +derivatives of differential indeterminates to the set of differential +polynomials. Other operations provided on differential polynomials are +leader, initial, separant, differentialVariables, and isobaric?. +Furthermore, if the ground ring is a differential ring, then evaluation +(substitution of differential indeterminates by elements of the ground ring +or by differential polynomials) is provided by eval. + +A convenient way of referencing derivatives is provided by the functions +makeVariable. + +To construct a domain using this constructor, one needs to provide a +ground ring R, an ordered set S of differential indeterminates, a ranking +V on the set of derivatives of the differential indeterminates, and a set +E of exponents in bijection with the set of differential monomials +in the given differential indeterminates. + See Also: o )show DifferentialPolynomialCategory @@ -58941,12 +58847,6 @@ These exports come from \refto{Evalable}(%:DPOLCAT): ++ Author: William Sit ++ Date Created: 19 July 1990 ++ Date Last Updated: 13 September 1991 -++ Basic Operations:PolynomialCategory -++ Related Constructors:DifferentialVariableCategory -++ See Also: -++ AMS Classifications:12H05 -++ Keywords: differential indeterminates, ranking, differential polynomials, -++ order, weight, leader, separant, initial, isobaric ++ References:Kolchin, E.R. "Differential Algebra and Algebraic Groups" ++ (Academic Press, 1973). ++ Description: @@ -59330,6 +59230,11 @@ digraph pic { FieldOfPrimeCharacteristic examples ==================================================================== +FieldOfPrimeCharacteristic is the category of fields of prime +characteristic, e.g. finite fields, algebraic closures of +fields of prime characteristic, transcendental extensions of +of fields of prime characteristic. + See Also: o )show FieldOfPrimeCharacteristic @@ -59500,11 +59405,6 @@ These exports come from \refto{CharacteristicNonZero}(): ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 10 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: _+, _* -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: field, finite field, prime characteristic ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. @@ -59634,6 +59534,9 @@ digraph pic { FiniteRankAlgebra examples ==================================================================== +A FiniteRankAlgebra is an algebra over a commutative ring R which +is a free R-module of finite rank. + See Also: o )show FiniteRankAlgebra @@ -59760,14 +59663,6 @@ These exports come from \refto{CharacteristicZero}(): \begin{chunk}{category FINRALG FiniteRankAlgebra} )abbrev category FINRALG FiniteRankAlgebra ++ Author: Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A FiniteRankAlgebra is an algebra over a commutative ring R which ++ is a free R-module of finite rank. @@ -60091,6 +59986,9 @@ digraph pic { FunctionSpace examples ==================================================================== +This is the category for formal functions. +A space of formal functions with arguments in an arbitrary ordered set. + See Also: o )show FunctionSpace @@ -60566,12 +60464,11 @@ These exports come from \refto{RetractableTo}(Fraction(Integer)): \begin{chunk}{category FS FunctionSpace} )abbrev category FS FunctionSpace -++ Category for formal functions ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 14 February 1994 -++ Keywords: operator, kernel, function. ++ Description: +++ Category for formal functions ++ A space of formal functions with arguments in an arbitrary ordered set. FunctionSpace(R:OrderedSet): Category == Definition where @@ -61422,6 +61319,8 @@ digraph pic { InfinitlyClosePointCategory examples ==================================================================== +This category is part of the PAFF package + See Also: o )show InfinitlyClosePointCategory @@ -61689,6 +61588,27 @@ digraph pic { PseudoAlgebraicClosureOfPerfectFieldCategory examples ==================================================================== +This category exports the function for domains which implement dynamic +extension using the simple notion of tower extensions. A tower extension +T of the ground field K is any sequence of field extensions + (T : K_0, K_1, ..., K_i...,K_n) where K_0 = K +and for + i =1,2,...,n, K_i is an extension of K_{i-1} of degree > 1 +and defined by an irreducible polynomial p(Z) in K_{i-1}. + +Two towers + (T_1: K_01, K_11,...,K_i1,...,K_n1) +and + (T_2: K_02, K_12,...,K_i2,...,K_n2) +are said to be related if + T_1 <= T_2 (or T_1 >= T_2), +that is if + K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2). + +Any algebraic operations defined for several elements are only defined +if all of the concerned elements are coming from a set of related tower +extensions. + See Also: o )show PseudoAlgebraicClosureOfPerfectFieldCategory @@ -61880,7 +61800,8 @@ These exports come from \refto{DivisionRing}(): ++ Authors: Gaetan Hache ++ Date Created: may 1997 ++ Date Last Updated: April 2010, by Tim Daly -++ Description: This category exports the function for domains +++ Description: +++ This category exports the function for domains ++ which implement dynamic extension using the simple notion of tower ++ extensions. ++ A tower extension T of the ground ++ field K is any sequence of field extension @@ -61892,8 +61813,8 @@ These exports come from \refto{DivisionRing}(): ++ are said to be related if T_1 <= T_2 (or T_1 >= T_2), ++ that is if K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2). ++ Any algebraic operations defined for several elements -++ are only defined if all of the concerned elements are comming from -++ a set of related tour extensions. +++ are only defined if all of the concerned elements are coming from +++ a set of related tower extensions. PseudoAlgebraicClosureOfPerfectFieldCategory() : Category == PUB where INT ==> Integer @@ -62099,6 +62020,8 @@ digraph pic { QuotientFieldCategory examples ==================================================================== +QuotientField(S) is the category of fractions of an Integral Domain S. + See Also: o )show QuotientFieldCategory @@ -62454,15 +62377,7 @@ These exports come from \refto{PolynomialFactorizationExplicit}(): \begin{chunk}{category QFCAT QuotientFieldCategory} )abbrev category QFCAT QuotientFieldCategory -++ Author: -++ Date Created: ++ Date Last Updated: 5th March 1996 -++ Basic Functions: + - * / numer denom -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ QuotientField(S) is the category of fractions of an Integral Domain S. @@ -62769,6 +62684,9 @@ digraph pic { RealClosedField examples ==================================================================== +RealClosedField provides common access functions for all real closed fields. +It provides computations with generic real roots of polynomials. + See Also: o )show RealClosedField @@ -63020,16 +62938,10 @@ These exports come from \refto{Algebra}(Integer): ++ Author: Renaud Rioboo ++ Date Created: may 1993 ++ Date Last Updated: January 2004 -++ Basic Functions: provides computations with generic real roots of -++ polynomials -++ Related Constructors: SimpleOrderedAlgebraicExtension, RealClosure -++ Also See: -++ AMS Classifications: -++ Keywords: Real Algebraic Numbers -++ References: ++ Description: -++ \axiomType{RealClosedField} provides common acces +++ \axiomType{RealClosedField} provides common access ++ functions for all real closed fields. +++ provides computations with generic real roots of polynomials RealClosedField : Category == PUB where @@ -63308,8 +63220,14 @@ digraph pic { RealNumberSystem examples ==================================================================== +The real number system category is intended as a model for the real +numbers. The real numbers form an ordered normed field. Note that +we have purposely not included DifferentialRing or the elementary +functions (see TranscendentalFunctionCategory) in the definition. + See Also: o )show RealNumberSystem +o )show TranscendentalFunctionCategory \end{chunk} {\bf See:} @@ -63532,11 +63450,7 @@ These exports come from \refto{CharacteristicZero}(): \begin{chunk}{category RNS RealNumberSystem} )abbrev category RNS RealNumberSystem ++ Author: Michael Monagan and Stephen M. Watt -++ Date Created: -++ January 1988 -++ Change History: -++ Related Constructors: -++ Keywords: real numbers +++ Date Created: January 1988 ++ Description: ++ The real number system category is intended as a model for the real ++ numbers. The real numbers form an ordered normed field. Note that @@ -63861,6 +63775,15 @@ digraph pic { RecursivePolynomialCategory examples ==================================================================== +A category for general multi-variate polynomials with coefficients +in a ring, variables in an ordered set, and exponents from an +ordered abelian monoid, with a sup operation. + +When not constant, such a polynomial is viewed as a univariate polynomial +in its main variable w. r. t. to the total ordering on the elements in +the ordered set, so that some operations usually defined for univariate +polynomials make sense here. + See Also: o )show RecursivePolynomialCategory @@ -64355,19 +64278,7 @@ where R:Ring, E:OrderedAbelianMonoidSup, V:OrderedSet: ++ Author: Marc Moreno Maza ++ Date Created: 04/22/1994 ++ Date Last Updated: 14/12/1998 -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: polynomial, multivariate, ordered variables set -++ References: ++ Description: -++ A category for general multi-variate polynomials with coefficients -++ in a ring, variables in an ordered set, and exponents from an -++ ordered abelian monoid, with a \axiomOp{sup} operation. -++ When not constant, such a polynomial is viewed as a univariate polynomial -++ in its main variable w. r. t. to the total ordering on the elements in -++ the ordered set, so that some operations usually defined for univariate -++ polynomials make sense here. RecursivePolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, V:OrderedSet):_ Category == @@ -65790,6 +65701,9 @@ digraph pic { UnivariateLaurentSeriesCategory examples ==================================================================== +UnivariateLaurentSeriesCategory is the category of Laurent series +in one variable. + See Also: o )show UnivariateLaurentSeriesCategory @@ -66142,13 +66056,6 @@ These exports come from \refto{Field}(): ++ Author: Clifton J. Williamson ++ Date Created: 21 December 1989 ++ Date Last Updated: 20 September 1993 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Laurent -++ Examples: -++ References: ++ Description: ++ \spadtype{UnivariateLaurentSeriesCategory} is the category of ++ Laurent series in one variable. @@ -66423,6 +66330,9 @@ digraph pic { UnivariatePuiseuxSeriesCategory examples ==================================================================== +UnivariatePuiseuxSeriesCategory is the category of Puiseux series +in one variable. + See Also: o )show UnivariatePuiseuxSeriesCategory @@ -66768,13 +66678,6 @@ These exports come from \refto{RadicalCategory}(): ++ Author: Clifton J. Williamson ++ Date Created: 21 December 1989 ++ Date Last Updated: 20 September 1993 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Puiseux -++ Examples: -++ References: ++ Description: ++ \spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux ++ series in one variable. @@ -67091,6 +66994,9 @@ digraph pic { UnivariatePolynomialCategory examples ==================================================================== +The category of univariate polynomials over a ring R. No particular +model is assumed - implementations can be either sparse or dense. + See Also: o )show UnivariatePolynomialCategory @@ -67601,16 +67507,6 @@ These exports come from \refto{PolynomialFactorizationExplicit}() \begin{chunk}{category UPOLYC UnivariatePolynomialCategory} )abbrev category UPOLYC UnivariatePolynomialCategory -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: Ring, monomial, coefficient, reductum, differentiate, -++ elt, map, resultant, discriminant -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ The category of univariate polynomials over a ring R. ++ No particular model is assumed - implementations can be either @@ -68273,6 +68169,8 @@ digraph pic { AlgebraicallyClosedFunctionSpace examples ==================================================================== +Model for algebraically closed function spaces. + See Also: o )show AlgebraicallyClosedFunctionSpace @@ -68683,7 +68581,6 @@ where R:Join(OrderedSet, IntegralDomain)): ++ Author: Manuel Bronstein ++ Date Created: 31 October 1988 ++ Date Last Updated: 7 October 1991 -++ Keywords: algebraic, closure, field. ++ Description: ++ Model for algebraically closed function spaces. @@ -68935,6 +68832,8 @@ digraph pic { ExtensionField examples ==================================================================== +ExtensionField F is the category of fields which extend the field F + See Also: o )show ExtensionField @@ -69147,11 +69046,6 @@ These exports come from \refto{FieldOfPrimeCharacteristic}(): ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 10 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: _+, _*, extensionDegree, algebraic?, transcendent? -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: field, extension field ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. @@ -69348,6 +69242,8 @@ digraph pic { FiniteFieldCategory examples ==================================================================== +FiniteFieldCategory is the category of finite fields + See Also: o )show FiniteFieldCategory @@ -69564,12 +69460,6 @@ These exports come from \refto{DifferentialRing}(): ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 11 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: _+, _*, extensionDegree, order, primitiveElement -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: field, extension field, algebraic extension, finite field -++ Galois field ++ References: ++ D.Lipson, Elements of Algebra and Algebraic Computing, The ++ Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981. @@ -69947,6 +69837,25 @@ digraph pic { FloatingPointSystem examples ==================================================================== +This category is intended as a model for floating point systems. +A floating point system is a model for the real numbers. In fact, +it is an approximation in the sense that not all real numbers are +exactly representable by floating point numbers. + +A floating point system is characterized by the following: + + 1: base of the exponent where the actual implemenations are + usually binary or decimal) + 2: precision of the mantissa (arbitrary or fixed) + 3: rounding error for operations + 4: when, and what happens if exponent overflow/underflow occurs + +Because a Float is an approximation to the real numbers, even though +it is defined to be a join of a Field and OrderedRing, some of +the attributes do not hold. In particular associative("+") +does not hold. Algorithms defined over a field need special +considerations when the field is a floating point system. + See Also: o )show FloatingPointSystem @@ -70192,13 +70101,6 @@ These exports come from \refto{RealNumberSystem}(): \begin{chunk}{category FPS FloatingPointSystem} )abbrev category FPS FloatingPointSystem -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: approximate, base, bits, digits, exponent, float, -++ mantissa, order, precision, round? -++ Related Constructors: -++ Keywords: float, floating point ++ Description: ++ This category is intended as a model for floating point systems. ++ A floating point system is a model for the real numbers. In fact, @@ -70392,6 +70294,8 @@ digraph pic { FramedAlgebra examples ==================================================================== +A FramedAlgebra is a FiniteRankAlgebra together with a fixed R-module basis. + See Also: o )show FramedAlgebra @@ -70515,14 +70419,6 @@ where R:CommutativeRing and UP:UnivariatePolynomialCategory R): \begin{chunk}{category FRAMALG FramedAlgebra} )abbrev category FRAMALG FramedAlgebra ++ Author: Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together ++ with a fixed R-module basis. @@ -70757,6 +70653,28 @@ digraph pic { PseudoAlgebraicClosureOfFiniteFieldCategory examples ==================================================================== +This category exports the function for the domain +PseudoAlgebraicClosureOfFiniteField which implement dynamic extension +using the simple notion of tower extensions. + +A tower extension T of the ground field K is any sequence of field extension + (T : K_0, K_1, ..., K_i...,K_n) +where K_0 = K and for i =1,2,...,n, K_i is an extension +of K_{i-1} of degree > 1 and defined by an irreducible polynomial +p(Z) in K_{i-1}. + +Two towers + (T_1: K_01, K_11,...,K_i1,...,K_n1) +and + (T_2: K_02, K_12,...,K_i2,...,K_n2) +are said to be related if + T_1 <= T_2 (or T_1 >= T_2), +that is if + K_i1 = K_i2 for i=1,2,...,n1 +(or i=1,2,...,n2). Any algebraic operations defined for several elements +are only defined if all of the concerned elements are comming from +a set of related tour extensions. + See Also: o )show PseudoAlgebraicClosureOfFiniteFieldCategory @@ -70980,9 +70898,8 @@ These exports come from \refto{FiniteFieldCategory}(): -- PseudoAlgebraicClosureOfFiniteFieldCategory ++ Authors: Gaetan Hache ++ Date Created: june 1996 -++ Date Last Updated: -++ References: -++ Description: This category exports the function for the domain +++ Description: +++ This category exports the function for the domain ++ PseudoAlgebraicClosureOfFiniteField which implement dynamic extension ++ using the simple notion of tower extensions. ++ A tower extension T of the ground @@ -71245,6 +71162,11 @@ digraph pic { UnivariateLaurentSeriesConstructorCategory examples ==================================================================== +This is a category of univariate Laurent series constructed from +univariate Taylor series. A Laurent series is represented by a pair +[n,f(x)], where n is an arbitrary integer and f(x) is a Taylor series. +This pair represents the Laurent series x**n * f(x). + See Also: o )show UnivariateLaurentSeriesConstructorCategory @@ -71767,13 +71689,6 @@ where UTS:UnivariateLaurentSeriesCategory(Coef:Ring) ++ Author: Clifton J. Williamson ++ Date Created: 6 February 1990 ++ Date Last Updated: 10 May 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Laurent, Taylor -++ Examples: -++ References: ++ Description: ++ This is a category of univariate Laurent series constructed from ++ univariate Taylor series. A Laurent series is represented by a pair @@ -72056,6 +71971,11 @@ digraph pic { UnivariatePuiseuxSeriesConstructorCategory examples ==================================================================== +This is a category of univariate Puiseux series constructed from +univariate Laurent series. A Puiseux series is represented by a pair +[r,f(x)], where r is a positive rational number and f(x) is a Laurent +series. This pair represents the Puiseux series f(x^r). + See Also: o )show UnivariatePuiseuxSeriesConstructorCategory @@ -72400,13 +72320,6 @@ These exports come from \refto{UnivariatePuiseuxSeriesCategory}(Coef:Ring): ++ Author: Clifton J. Williamson ++ Date Created: 6 February 1990 ++ Date Last Updated: 22 March 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Puiseux, Laurent -++ Examples: -++ References: ++ Description: ++ This is a category of univariate Puiseux series constructed ++ from univariate Laurent series. A Puiseux series is represented @@ -72643,6 +72556,31 @@ digraph pic { FiniteAlgebraicExtensionField examples ==================================================================== +FiniteAlgebraicExtensionField F is the category of fields +which are finite algebraic extensions of the field F. + +If F is finite then any finite algebraic extension of F is finite, too. +Let K be a finite algebraic extension of the finite field F. The +exponentiation of elements of K defines a Z-module structure on the +multiplicative group of K. + +The additive group of K becomes a module over the ring of polynomials +over F via the operation + linearAssociatedExp(a:K,f:SparseUnivariatePolynomial F) +which is linear over F, i.e. for elements a from K, c,d from F and +f,g univariate polynomials over F we have linearAssociatedExp}(a,cf+dg) +equals c times linearAssociatedExp}(a,f) plus d times +linearAssociatedExp}(a,g). + +Therefore linearAssociatedExp is defined completely by its action on +monomials from F[X]: linearAssociatedExp(a,monomial(1,k)\$SUP(F)) is +defined to be Frobenius(a,k) which is a**(q**k) where q=size()\$F. + +The operations order and discreteLog associated with the multiplicative +exponentiation have additive analogues associated to the operation +linearAssociatedExp. These are the functions linearAssociatedOrder +and linearAssociatedLog, respectively. + See Also: o )show FiniteAlgebraicExtensionField @@ -72934,11 +72872,6 @@ These exports come from \refto{FiniteFieldCategory}(): ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 11 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: _+, _*, extensionDegree, -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: field, extension field, algebraic extension, finite extension ++ References: ++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and ++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, @@ -73437,6 +73370,9 @@ digraph pic { MonogenicAlgebra examples ==================================================================== +A MonogenicAlgebra is an algebra of finite rank which can be +generated by a single element. + See Also: o )show MonogenicAlgebra @@ -73786,14 +73722,6 @@ These exports come from \refto{FiniteFieldCategory}(): \begin{chunk}{category MONOGEN MonogenicAlgebra} )abbrev category MONOGEN MonogenicAlgebra ++ Author: Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A \spadtype{MonogenicAlgebra} is an algebra of finite rank which ++ can be generated by a single element. @@ -74033,6 +73961,27 @@ digraph pic { PseudoAlgebraicClosureOfRationalNumberCategory examples ==================================================================== +This category exports the function for the domain +PseudoAlgebraicClosureOfRationalNumber which implement dynamic extension +using the simple notion of tower extensions. A tower extension T of the +ground field K is any sequence of field extension + (T : K_0, K_1, ..., K_i...,K_n) +where K_0 = K and for i =1,2,...,n, K_i is an extension +of K_{i-1} of degree > 1 and defined by an irreducible polynomial +p(Z) in K_{i-1}. + +Two towers + (T_1: K_01, K_11,...,K_i1,...,K_n1) +and + (T_2: K_02, K_12,...,K_i2,...,K_n2) +are said to be related if + T_1 <= T_2 (or T_1 >= T_2), +that is if + K_i1 = K_i2 for i=1,2,...,n1 +(or i=1,2,...,n2). Any algebraic operations defined for several elements +are only defined if all of the concerned elements are comming from +a set of related tour extensions. + See Also: o )show PseudoAlgebraicClosureOfRationalNumberCategory @@ -74264,8 +74213,8 @@ These exports come from \refto{ExtensionField}(Fraction(Integer)): )abbrev category PACRATC PseudoAlgebraicClosureOfRationalNumberCategory ++ Authors: Gaetan Hache ++ Date Created: feb 1997 -++ Date Last Updated: -++ Description: This category exports the function for the domain +++ Description: +++ This category exports the function for the domain ++ PseudoAlgebraicClosureOfRationalNumber ++ which implement dynamic extension using the simple notion of tower ++ extensions. A tower extension T of the ground @@ -74524,6 +74473,8 @@ digraph pic { ComplexCategory examples ==================================================================== +This category represents the extension of a ring by a square root of -1. + See Also: o )show ComplexCategory @@ -74960,15 +74911,7 @@ These exports come from \refto{PolynomialFactorizationExplicit}(): \begin{chunk}{category COMPCAT ComplexCategory} )abbrev category COMPCAT ComplexCategory -++ Author: -++ Date Created: ++ Date Last Updated: 18 March 1994 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: complex, gaussian -++ References: ++ Description: ++ This category represents the extension of a ring by a square root of -1. @@ -75642,6 +75585,8 @@ digraph pic { FunctionFieldCategory examples ==================================================================== +This category is a model for the function field of a plane algebraic curve. + See Also: o )show FunctionFieldCategory @@ -76162,12 +76107,11 @@ UPUP:UnivariatePolynomialCategory Fraction UP \begin{chunk}{category FFCAT FunctionFieldCategory} )abbrev category FFCAT FunctionFieldCategory -++ Function field of a curve ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 19 Mai 1993 -++ Keywords: algebraic, curve, function, field. ++ Description: +++ Function field of a curve ++ This category is a model for the function field of a ++ plane algebraic curve. @@ -76736,6 +76680,27 @@ digraph pic { PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory examples ==================================================================== +This category exports the function for the domain +PseudoAlgebraicClosureOfAlgExtOfRationalNumber which implement dynamic +extension using the simple notion of tower extensions. A tower extension +T of the ground field K is any sequence of field extension + (T : K_0, K_1, ..., K_i...,K_n) +where K_0 = K and for i =1,2,...,n, + K_i is an extension of K_{i-1} of degree > 1 +and defined by an irreducible polynomial p(Z) in K_{i-1}. + +Two towers + (T_1: K_01, K_11,...,K_i1,...,K_n1) +and + (T_2: K_02, K_12,...,K_i2,...,K_n2) +are said to be related if + T_1 <= T_2 (or T_1 >= T_2), +that is if + K_i1 = K_i2 for i=1,2,...,n1 (or i=1,2,...,n2). +Any algebraic operations defined for several elements +are only defined if all of the concerned elements are comming from +a set of related tour extensions. + See Also: o )show PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory @@ -92092,33 +92057,6 @@ digraph dotfull { \end{chunk} \eject \begin{thebibliography}{99} -\bibitem{1} N. Jacobson: Structure and Representations of Jordan Algebras -AMS, Providence, 1968 -\bibitem{2} MacLane and Birkhoff, Algebra 2d Edition, MacMillan 1979 -\bibitem{3} Encyclopedic Dictionary of Mathematics, MIT Press, 1977 -\bibitem{4} R.D. Schafer: An Introduction to Nonassociative Algebras -Academic Press, New York, 1966 -\bibitem{5} R. Wisbauer: Bimodule Structure of Algebra -Lecture Notes Univ. Duesseldorf 1991 -\bibitem{6} J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. -AXIOM Technical Report Series, ATR/5 NP2522. -\bibitem{7} R. Rioboo, -{\sl Real Algebraic Closure of an ordered Field : Implementation in Axiom.}, -In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215. -\bibitem{8} Z. Ligatsikas, R. Rioboo, M. F. Roy -{\sl Generic computation of the real closure of an ordered field.}, -In Mathematics and Computers in Simulation Volume 42, Issue 4-6, -November 1996. -\bibitem{9} D. LAZARD ``A new method for solving algebraic systems of -positive dimension'' Discr. App. Math. 33:147-160,1991 -\bibitem{10} P. AUBRY, D. LAZARD and M. MORENO MAZA ``On the Theories -of Triangular Sets'' Journal of Symbol. Comp. (to appear) -\bibitem{11} M. MORENO MAZA and R. RIOBOO ``Computations of gcd over -algebraic towers of simple extensions'' In proceedings of AAECC11 -Paris, 1995. -\bibitem{12} M. MORENO MAZA ``Calculs de pgcd au-dessus des tours -d'extensions simples et resolution des systemes d'equations -algebriques'' These, Universite P.etM. Curie, Paris, 1997. \end{thebibliography} \printindex \end{document} diff --git a/changelog b/changelog index b8bfaaa..328dc18 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20130228 tpd src/axiom-website/patches.html 20130228.02.tpd.patch +20130228 tpd books/bookvol10.2 write help documentation for all categories 20130228 tpd src/axiom-website/patches.html 20130228.01.tpd.patch 20130228 tpd books/bookvolbib add references 20130227 tpd src/axiom-website/patches.html 20130227.02.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 96222a4..e3fb43b 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -3997,5 +3997,7 @@ books/bookvol10.3 add U8Matrix books/bookvol10.4 add U32VectorPolynomialOperations 20130228.01.tpd.patch books/bookvolbib add references +20130228.02.tpd.patch +books/bookvol10.2 write help documentation for all categories