\documentclass{article} \usepackage{open-axiom} \title{\$SPAD/src/algebra catdef.spad} \author{James Davenport, Lalo Gonzalez-Vega, Gabriel Dos~Reis} \begin{document} \maketitle \begin{abstract} \end{abstract} \tableofcontents \eject \section{Linear sets} <>= )abbrev category LLINSET LeftLinearSet ++ Author: Gabriel Dos Reis ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: ++ A set is an \spad{S}-left linear set if it is stable by left-dilation ++ by elements in the semigroup \spad{S}. ++ See Also: RightLinearSet. LeftLinearSet(S: SemiGroup): Category == SetCategory with 0: % ++ \spad{0} represents the origin of the linear set zero?: % -> Boolean ++ \spad{zero? x} holds if \spad{x} is the origin. *: (S,%) -> % ++ \spad{s*x} is the left-dilation of \spad{x} by \spad{s}. @ <>= )abbrev category RLINSET RightLinearSet ++ Author: Gabriel Dos Reis ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: ++ A set is an \spad{S}-right linear set if it is stable by right-dilation ++ by elements in the semigroup \spad{S}. ++ See Also: LeftLinearSet. RightLinearSet(S: SemiGroup): Category == SetCategory with 0: % ++ \spad{0} represents the origin of the linear set zero?: % -> Boolean ++ \spad{zero? x} holds if \spad{x} is the origin. *: (%,S) -> % ++ \spad{x*s} is the right-dilation of \spad{x} by \spad{s}. @ <>= )abbrev category LINSET LinearSet ++ Author: Gabriel Dos Reis ++ Date Created: May 31, 2009 ++ Date Last Modified: May 31, 2009 ++ Description: ++ A set is an \spad{S}-linear set if it is stable by dilation ++ by elements in the semigroup \spad{S}. ++ See Also: LeftLinearSet, RightLinearSet. LinearSet(S: SemiGroup): Category == Join(LeftLinearSet S, RightLinearSet S) @ \section{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. ++ ++ Axioms: ++ \spad{-(-x) = x} ++ \spad{x+(-x) = 0} -- following domain must be compiled with subsumption disabled AbelianGroup(): Category == Join(CancellationAbelianMonoid, LeftLinearSet Integer) with -: % -> % ++ \spad{-x} is the additive inverse of \spad{x} "-": (%,%) -> % ++ \spad{x-y} is the difference of \spad{x} ++ and \spad{y} i.e. \spad{x + (-y)}. add (x:% - y:%):% == x+(-y) subtractIfCan(x:%, y:%):Union(%, "failed") == (x-y) :: Union(%,"failed") n:NonNegativeInteger * x:% == (n::Integer) * x import RepeatedDoubling(%) if not (% has Ring) then n:Integer * x:% == zero? n => 0 n>0 => double(n pretend PositiveInteger,x) double((-n) pretend PositiveInteger,-x) @ \section{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. ++ ++ Axioms: ++ \spad{leftIdentity("+":(%,%)->%,0)}\tab{30}\spad{ 0+x=x } ++ \spad{rightIdentity("+":(%,%)->%,0)}\tab{30}\spad{ x+0=x } -- following domain must be compiled with subsumption disabled -- define SourceLevelSubset to be EQUAL AbelianMonoid(): Category == AbelianSemiGroup with --operations 0: % ++ 0 is the additive identity element. sample: % ++ sample yields a value of type % zero?: % -> Boolean ++ zero?(x) tests if x is equal to 0. *: (NonNegativeInteger,%) -> % ++ n * x is left-multiplication by a non negative integer add import RepeatedDoubling(%) zero? x == x = 0 n:PositiveInteger * x:% == (n::NonNegativeInteger) * x sample() == 0 if not (% has Ring) then n:NonNegativeInteger * x:% == zero? n => 0 double(n pretend PositiveInteger,x) @ \section{category ABELSG AbelianSemiGroup} <>= import PositiveInteger )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{+}. ++ ++ Axioms: ++ \spad{associative("+":(%,%)->%)}\tab{30}\spad{ (x+y)+z = x+(y+z) } ++ \spad{commutative("+":(%,%)->%)}\tab{30}\spad{ x+y = y+x } AbelianSemiGroup(): Category == SetCategory with --operations +: (%,%) -> % ++ x+y computes the sum of x and y. *: (PositiveInteger,%) -> % ++ n*x computes the left-multiplication of x by the positive integer n. ++ This is equivalent to adding x to itself n times. add import RepeatedDoubling(%) if not (% has Ring) then n:PositiveInteger * x:% == double(n,x) @ \section{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). ++ ++ Axioms: ++ \spad{(b+c)::% = (b::%) + (c::%)} ++ \spad{(b*c)::% = (b::%) * (c::%)} ++ \spad{(1::R)::% = 1::%} ++ \spad{b*x = (b::%)*x} ++ \spad{r*(a*b) = (r*a)*b = a*(r*b)} Algebra(R:CommutativeRing): Category == Join(Ring, Module R, CoercibleFrom R) add coerce(x:R):% == x * 1$% @ \section{category BASTYPE BasicType} <>= import Boolean )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(): Category == with =: (%,%) -> Boolean ++ x=y tests if x and y are equal. ~=: (%,%) -> Boolean ++ x~=y tests if x and y are not equal. add x:% ~= y:% == not(x=y) @ \section{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. ++ ++ Axiom: ++ \spad{ r*(x*s) = (r*x)*s } BiModule(R:Ring,S:Ring):Category == Join(LeftModule(R),RightModule(S)) with leftUnitary ++ \spad{1 * x = x} rightUnitary ++ \spad{x * 1 = x} @ \section{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. ++ \spad{ a+b = a+c => b=c }. ++ This is formalised by the partial subtraction operator, ++ which satisfies the axioms listed below: ++ ++ Axioms: ++ \spad{c = a+b <=> c-b = a} CancellationAbelianMonoid(): Category == AbelianMonoid with --operations subtractIfCan: (%,%) -> Union(%,"failed") ++ subtractIfCan(x, y) returns an element z such that \spad{z+y=x} ++ or "failed" if no such element exists. @ \section{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 CharacteristicNonZero():Category == Ring with charthRoot: % -> Union(%,"failed") ++ charthRoot(x) returns the pth root of x ++ where p is the characteristic of the ring. @ \section{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. CharacteristicZero():Category == Ring @ \section{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. ++ element. --CommutativeRing():Category == Join(Ring,BiModule(%:Ring,%:Ring)) with CommutativeRing():Category == Join(Ring,BiModule(%,%)) with commutative("*") ++ multiplication is commutative. @ \section{Differential Domain} <>= )abbrev category DIFFDOM DifferentialDomain ++ Author: Gabriel Dos Reis ++ Date Created: June 13, 2010 ++ Date Last Modified: June 13, 2010 ++ Description: ++ This category captures the interface of domains with a distinguished ++ operation named \spad{differentiate}. Usually, additional properties ++ are wanted. For example, that it obeys the usual Leibniz identity ++ of differentiation of product, in case of differential rings. One ++ could also want \spad{differentiate} to obey the chain rule when ++ considering differential manifolds. ++ The lack of specific requirement in this category is an implicit ++ admission that currently \Language{} is not expressive enough to ++ express the most general notion of differentiation in an adequate ++ manner, suitable for computational purposes. DifferentialDomain(T: Type): Category == Type with differentiate: % -> T ++ \spad{differentiate x} compute the derivative of \spad{x}. D: % -> T ++ \spad{D x} is a shorthand for \spad{differentiate x} add D x == differentiate x @ \section{Differential Space} <>= )abbrev category DIFFSPC DifferentialSpace ++ Author: Gabriel Dos Reis ++ Date Created: June 13, 2010 ++ Date Last Modified: June 15, 2010 ++ Description: ++ This category is like \spadtype{DifferentialDomain} where the ++ target of the differentiation operator is the same as its source. DifferentialSpace(): Category == DifferentialDomain % with differentiate: (%, NonNegativeInteger) -> % ++ \spad{differentiate(x,n)} returns the \spad{n}-th ++ derivative of \spad{x}. D: (%, NonNegativeInteger) -> % ++ \spad{D(x, n)} returns the \spad{n}-th derivative of \spad{x}. add differentiate(r, n) == for i in 1..n repeat r := differentiate r r D(r,n) == differentiate(r,n) @ \section{Differential Space Extension} <>= )abbrev category DSEXT DifferentialSpaceExtension ++ Author: Gabriel Dos Reis ++ Date Created: June 16, 2010 ++ Date Last Updated: Jun 16, 2010 ++ Related Constructors: Module, DifferentialSpace ++ Also See: ++ Description: ++ Extension of a base differential space with a derivation. ++ DifferentialSpaceExtension(R: Type): Category == Type with differentiate: (%,R -> R) -> % ++ \spad{differentiate(x,d)} computes the derivative of ++ \spad{x}, extending differentiation \spad{d} on \spad{R}. differentiate: (%,R -> R,NonNegativeInteger) -> % ++ \spad{differentiate(x,d,n)} computes the \spad{n}-th derivative ++ of \spad{x} using a derivation extending \spad{d} on \spad{R}. D: (%,R -> R) -> % ++ \spad{D(x,d)} is a shorthand for \spad{differentiate(x,d)}. D: (%,R -> R,NonNegativeInteger) -> % ++ \spad{D(x,d,n)} is a shorthand for \spad{differentiate(x,d,n)}. if R has DifferentialSpace then DifferentialSpace if R has PartialDifferentialSpace Symbol then PartialDifferentialSpace Symbol add differentiate(x: %, d: R -> R, n: NonNegativeInteger):% == for i in 1..n repeat x := differentiate(x,d) x D(x: %, d: R -> R) == differentiate(x, d) D(x: %, d: R -> R, n: NonNegativeInteger) == differentiate(x,d,n) if R has DifferentialSpace then differentiate x == differentiate(x, differentiate$R) if R has PartialDifferentialSpace Symbol then differentiate(x:%, v: Symbol):% == differentiate(x, differentiate(#1, v)$R) @ \section{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}. ++ ++ Axioms: ++ \spad{differentiate(x+y) = differentiate(x)+differentiate(y)} ++ \spad{differentiate(x*y) = x*differentiate(y) + differentiate(x)*y} DifferentialRing(): Category == Join(Ring,DifferentialSpace) @ \section{Differential Module} <>= )abbrev category DIFFMOD DifferentialModule ++ Author: Gabriel Dos Reis ++ Date Created: June 14, 2010 ++ Date Last Updated: Jun 16, 2010 ++ Related Constructors: Module, DifferentialSpace ++ Also See: ++ Description: ++ An R-module equipped with a distinguised differential operator. ++ If R is a differential ring, then differentiation on the module ++ should extend differentiation on the differential ring R. The ++ latter can be the null operator. In that case, the differentiation ++ operator on the module is just an R-linear operator. For that ++ reason, we do not require that the ring R be a DifferentialRing; ++ ++ Axioms: ++ \spad{differentiate(x + y) = differentiate(x) + differentiate(x)} ++ \spad{differentiate(r*y) = r*differentiate(y) + differentiate(r)*y} DifferentialModule(R: Ring): Category == Join(BiModule(R,R), DifferentialSpace) with if R has CommutativeRing then Module R @ \section{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 %. DifferentialExtension(R:Ring): Category == Join(Ring,DifferentialSpaceExtension R) with if R has DifferentialRing then DifferentialRing if R has PartialDifferentialRing(Symbol) then PartialDifferentialRing(Symbol) @ \section{Differential Module Extension} <>= )abbrev category DMEXT DifferentialModuleExtension ++ Author: Gabriel Dos Reis ++ Date Created: June 16, 2010 ++ Date Last Updated: Jun 16, 2010 ++ Related Constructors: Module, DifferentialSpaceExtension ++ Also See: ++ DifferentialExtension ++ Description: ++ Category of modules that extend differential rings. ++ DifferentialModuleExtension(R: Ring): Category == Join(BiModule(R,R),DifferentialSpaceExtension R) with if R has DifferentialSpace then DifferentialModule R if R has PartialDifferentialSpace Symbol then PartialDifferentialModule(R,Symbol) @ \section{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 ++ all non-zero elements have multiplicative inverses. DivisionRing(): Category == Join(EntireRing, Algebra Fraction Integer) with **: (%,Integer) -> % ++ x**n returns x raised to the integer power n. inv : % -> % ++ inv x returns the multiplicative inverse of x. ++ Error: if x is 0. -- Q-algebra is a lie, should be conditional on characteristic 0, -- but knownInfo cannot handle the following commented -- if % has CharacteristicZero then Algebra Fraction Integer add n: Integer x: % import RepeatedSquaring(%) x ** n: Integer == zero? n => 1 zero? x => n<0 => error "division by zero" x n<0 => expt(inv x,(-n) pretend PositiveInteger) expt(x,n pretend PositiveInteger) -- if % has CharacteristicZero() then q:Fraction(Integer) * x:% == numer(q) * inv(denom(q)::%) * x @ \section{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. ++ ++ Axioms: ++ \spad{ab=0 => a=0 or b=0} -- known as noZeroDivisors ++ \spad{not(1=0)} --EntireRing():Category == Join(Ring,BiModule(%:Ring,%:Ring)) with EntireRing():Category == Join(Ring,BiModule(%,%)) with noZeroDivisors ++ if a product is zero then one of the factors ++ must be zero. @ \section{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 ++ or is smaller (\spadfun{euclideanSize}) than the divisor. ++ ++ Conditional attributes: ++ multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)} ++ additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)} EuclideanDomain(): Category == PrincipalIdealDomain with --operations sizeLess?: (%,%) -> Boolean ++ sizeLess?(x,y) tests whether x is strictly ++ smaller than y with respect to the \spadfunFrom{euclideanSize}{EuclideanDomain}. euclideanSize: % -> NonNegativeInteger ++ euclideanSize(x) returns the euclidean size of the element x. ++ Error: if x is zero. divide: (%,%) -> Record(quotient:%,remainder:%) ++ divide(x,y) divides x by y producing a record containing a ++ \spad{quotient} and \spad{remainder}, ++ where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain}) ++ than the divisor y. quo : (%,%) -> % ++ x quo y is the same as \spad{divide(x,y).quotient}. ++ See \spadfunFrom{divide}{EuclideanDomain}. rem: (%,%) -> % ++ x rem y is the same as \spad{divide(x,y).remainder}. ++ See \spadfunFrom{divide}{EuclideanDomain}. extendedEuclidean: (%,%) -> Record(coef1:%,coef2:%,generator:%) -- formerly called princIdeal ++ extendedEuclidean(x,y) returns a record rec where ++ \spad{rec.coef1*x+rec.coef2*y = rec.generator} and ++ rec.generator is a gcd of x and y. ++ The gcd is unique only ++ up to associates if \spadatt{canonicalUnitNormal} is not asserted. ++ \spadfun{principalIdeal} provides a version of this operation ++ which accepts an arbitrary length list of arguments. extendedEuclidean: (%,%,%) -> Union(Record(coef1:%,coef2:%),"failed") -- formerly called expressIdealElt ++ extendedEuclidean(x,y,z) either returns a record rec ++ where \spad{rec.coef1*x+rec.coef2*y=z} or returns "failed" ++ if z cannot be expressed as a linear combination of x and y. multiEuclidean: (List %,%) -> Union(List %,"failed") ++ multiEuclidean([f1,...,fn],z) returns a list of coefficients ++ \spad{[a1, ..., an]} such that ++ \spad{ z / prod fi = sum aj/fj}. ++ If no such list of coefficients exists, "failed" is returned. add -- declarations x,y,z: % l: List % -- definitions sizeLess?(x,y) == zero? y => false zero? x => true euclideanSize(x) 0 zero? y => "failed" qr:=divide(x,y) zero?(qr.remainder) => qr.quotient "failed" gcd(x,y) == --Euclidean Algorithm x:=unitCanonical x y:=unitCanonical y while not zero? y repeat (x,y):= (y,x rem y) y:=unitCanonical y -- this doesn't affect the -- correctness of Euclid's algorithm, -- but -- a) may improve performance -- b) ensures gcd(x,y)=gcd(y,x) -- if canonicalUnitNormal x IdealElt ==> Record(coef1:%,coef2:%,generator:%) unitNormalizeIdealElt(s:IdealElt):IdealElt == (u,c,a):=unitNormal(s.generator) one? a => s [a*s.coef1,a*s.coef2,c]$IdealElt extendedEuclidean(x,y) == --Extended Euclidean Algorithm s1:=unitNormalizeIdealElt([1$%,0$%,x]$IdealElt) s2:=unitNormalizeIdealElt([0$%,1$%,y]$IdealElt) zero? y => s1 zero? x => s2 while not zero?(s2.generator) repeat qr:= divide(s1.generator, s2.generator) s3:=[s1.coef1 - qr.quotient * s2.coef1, s1.coef2 - qr.quotient * s2.coef2, qr.remainder]$IdealElt s1:=s2 s2:=unitNormalizeIdealElt s3 if not(zero?(s1.coef1)) and not sizeLess?(s1.coef1,y) then qr:= divide(s1.coef1,y) s1.coef1:= qr.remainder s1.coef2:= s1.coef2 + qr.quotient * x s1 := unitNormalizeIdealElt s1 s1 TwoCoefs ==> Record(coef1:%,coef2:%) extendedEuclidean(x,y,z) == zero? z => [0,0]$TwoCoefs s:= extendedEuclidean(x,y) (w:= z exquo s.generator) case "failed" => "failed" zero? y => [s.coef1 * w, s.coef2 * w]$TwoCoefs qr:= divide((s.coef1 * w), y) [qr.remainder, s.coef2 * w + qr.quotient * x]$TwoCoefs principalIdeal l == l = [] => error "empty list passed to principalIdeal" rest l = [] => uca:=unitNormal(first l) [[uca.unit],uca.canonical] rest rest l = [] => u:= extendedEuclidean(first l,second l) [[u.coef1, u.coef2], u.generator] v:=principalIdeal rest l u:= extendedEuclidean(first l,v.generator) [[u.coef1,:[u.coef2*vv for vv in v.coef]],u.generator] expressIdealMember(l,z) == z = 0 => [0 for v in l] pid := principalIdeal l (q := z exquo (pid.generator)) case "failed" => "failed" [q*v for v in pid.coef] multiEuclidean(l,z) == n := #l zero? n => error "empty list passed to multiEuclidean" n = 1 => [z] l1 := copy l l2 := split!(l1, n quo 2) u:= extendedEuclidean(*/l1, */l2, z) u case "failed" => "failed" v1 := multiEuclidean(l1,u.coef2) v1 case "failed" => "failed" v2 := multiEuclidean(l2,u.coef1) v2 case "failed" => "failed" concat(v1,v2) @ \section{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. ++ The \spadfun{factor} operation while trivial is useful to have defined. ++ ++ Axioms: ++ \spad{a*(b/a) = b} ++ \spad{inv(a) = 1/a} Field(): Category == Join(EuclideanDomain,UniqueFactorizationDomain, DivisionRing) with --operations /: (%,%) -> % ++ x/y divides the element x by the element y. ++ Error: if y is 0. canonicalUnitNormal ++ either 0 or 1. canonicalsClosed ++ since \spad{0*0=0}, \spad{1*1=1} add --declarations x,y: % n: Integer -- definitions UCA ==> Record(unit:%,canonical:%,associate:%) unitNormal(x) == if zero? x then [1$%,0$%,1$%]$UCA else [x,1$%,inv(x)]$UCA unitCanonical(x) == if zero? x then x else 1 associates?(x,y) == if zero? x then zero? y else not(zero? y) inv x ==((u:=recip x) case "failed" => error "not invertible"; u) x exquo y == (y=0 => "failed"; x / y) gcd(x,y) == 1 euclideanSize(x) == 0 prime? x == false squareFree x == x::Factored(%) factor x == x::Factored(%) x / y == (zero? y => error "catdef: division by zero"; x * inv(y)) divide(x,y) == [x / y,0] @ \section{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} to give a bijection ++ between the finite set and an initial segment of positive integers. ++ ++ Axioms: ++ \spad{lookup(index(n)) = n} ++ \spad{index(lookup(s)) = s} Finite(): Category == SetCategory with --operations size: () -> NonNegativeInteger ++ size() returns the number of elements in the set. index: PositiveInteger -> % ++ index(i) takes a positive integer i less than or equal ++ to \spad{size()} and ++ returns the \spad{i}-th element of the set. This operation establishs a bijection ++ between the elements of the finite set and \spad{1..size()}. lookup: % -> PositiveInteger ++ lookup(x) returns a positive integer such that ++ \spad{x = index lookup x}. random: () -> % ++ random() returns a random element from the set. @ \section{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 ++ \spadtype{LinearlyExplicitRingOver Integer}, then so is S FullyLinearlyExplicitRingOver(R:Ring):Category == LinearlyExplicitRingOver R with if (R has LinearlyExplicitRingOver Integer) then LinearlyExplicitRingOver Integer add if not(R is Integer) then if (R has LinearlyExplicitRingOver Integer) then reducedSystem(m:Matrix %):Matrix(Integer) == reducedSystem(reducedSystem(m)@Matrix(R)) reducedSystem(m:Matrix %, v:Vector %): Record(mat:Matrix(Integer), vec:Vector(Integer)) == rec := reducedSystem(m, v)@Record(mat:Matrix R, vec:Vector R) reducedSystem(rec.mat, rec.vec) @ \section{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 ++ of the existence of \spadfun{factor} operation for factorisation into irreducibles. ++ However, if such a \spadfun{factor} operation exist, factorization will be ++ unique up to order and units. GcdDomain(): Category == IntegralDomain with --operations gcd: (%,%) -> % ++ gcd(x,y) returns the greatest common divisor of x and y. -- gcd(x,y) = gcd(y,x) in the presence of canonicalUnitNormal, -- but not necessarily elsewhere gcd: List(%) -> % ++ gcd(l) returns the common gcd of the elements in the list l. lcm: (%,%) -> % ++ lcm(x,y) returns the least common multiple of x and y. -- lcm(x,y) = lcm(y,x) in the presence of canonicalUnitNormal, -- but not necessarily elsewhere lcm: List(%) -> % ++ lcm(l) returns the least common multiple of the elements of the list l. gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % ++ gcdPolynomial(p,q) returns the greatest common divisor (gcd) of ++ univariate polynomials over the domain add lcm(x: %,y: %) == y = 0 => 0 x = 0 => 0 LCM : Union(%,"failed") := y exquo gcd(x,y) LCM case % => x * LCM error "bad gcd in lcm computation" lcm(l:List %) == reduce(lcm,l,1,0) gcd(l:List %) == reduce(gcd,l,0,1) SUP ==> SparseUnivariatePolynomial gcdPolynomial(p1,p2) == zero? p1 => unitCanonical p2 zero? p2 => unitCanonical p1 c1:= content(p1); c2:= content(p2) p1:= (p1 exquo c1)::SUP % p2:= (p2 exquo c2)::SUP % if (e1:=minimumDegree p1) > 0 then p1:=(p1 exquo monomial(1,e1))::SUP % if (e2:=minimumDegree p2) > 0 then p2:=(p2 exquo monomial(1,e2))::SUP % e1:=min(e1,e2); c1:=gcd(c1,c2) p1:= degree p1 = 0 or degree p2 = 0 => monomial(c1,0) p:= subResultantGcd(p1,p2) degree p = 0 => monomial(c1,0) c2:= gcd(leadingCoefficient p1,leadingCoefficient p2) unitCanonical(c1 * primitivePart(((c2*p) exquo leadingCoefficient p)::SUP %)) zero? e1 => p1 monomial(1,e1)*p1 @ \section{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. ++ ++ Axioms: ++ \spad{leftInverse("*":(%,%)->%,inv)}\tab{30}\spad{ inv(x)*x = 1 } ++ \spad{rightInverse("*":(%,%)->%,inv)}\tab{30}\spad{ x*inv(x) = 1 } Group(): Category == Monoid with --operations inv: % -> % ++ inv(x) returns the inverse of x. /: (%,%) -> % ++ x/y is the same as x times the inverse of y. **: (%,Integer) -> % ++ x**n returns x raised to the integer power n. unitsKnown ++ unitsKnown asserts that recip only returns ++ "failed" for non-units. conjugate: (%,%) -> % ++ conjugate(p,q) computes \spad{inv(q) * p * q}; this is 'right action ++ by conjugation'. commutator: (%,%) -> % ++ commutator(p,q) computes \spad{inv(p) * inv(q) * p * q}. add import RepeatedSquaring(%) x:% / y:% == x*inv(y) recip(x:%) == inv(x) x:% ** n:Integer == zero? n => 1 n<0 => expt(inv(x),(-n) pretend PositiveInteger) expt(x,n pretend PositiveInteger) conjugate(p,q) == inv(q) * p * q commutator(p,q) == inv(p) * inv(q) * p * q @ \section{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. ++ ++ Conditional attributes: ++ canonicalUnitNormal\tab{20}the canonical field is the same for all associates ++ canonicalsClosed\tab{20}the product of two canonicals is itself canonical IntegralDomain(): Category == -- Join(CommutativeRing, Algebra(%:CommutativeRing), EntireRing) with Join(CommutativeRing, Algebra(%), EntireRing) with --operations exquo: (%,%) -> Union(%,"failed") ++ exquo(a,b) either returns an element c such that ++ \spad{c*b=a} or "failed" if no such element can be found. unitNormal: % -> Record(unit:%,canonical:%,associate:%) ++ unitNormal(x) tries to choose a canonical element ++ from the associate class of x. ++ The attribute canonicalUnitNormal, if asserted, means that ++ the "canonical" element is the same across all associates of x ++ if \spad{unitNormal(x) = [u,c,a]} then ++ \spad{u*c = x}, \spad{a*u = 1}. unitCanonical: % -> % ++ \spad{unitCanonical(x)} returns \spad{unitNormal(x).canonical}. associates?: (%,%) -> Boolean ++ associates?(x,y) tests whether x and y are associates, i.e. ++ differ by a unit factor. unit?: % -> Boolean ++ unit?(x) tests whether x is a unit, i.e. is invertible. add -- declaration x,y: % -- definitions UCA ==> Record(unit:%,canonical:%,associate:%) if not (% has Field) then unitNormal(x) == [1$%,x,1$%]$UCA -- the non-canonical definition unitCanonical(x) == unitNormal(x).canonical -- always true recip(x) == if zero? x then "failed" else 1$% exquo x unit?(x) == (recip x case "failed" => false; true) if % has canonicalUnitNormal then associates?(x,y) == (unitNormal x).canonical = (unitNormal y).canonical else associates?(x,y) == zero? x => zero? y zero? y => false x exquo y case "failed" => false y exquo x case "failed" => false true @ \section{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 ++ the rng. ++ ++ Axioms: ++ \spad{ (a*b)*x = a*(b*x) } ++ \spad{ (a+b)*x = (a*x)+(b*x) } ++ \spad{ a*(x+y) = (a*x)+(a*y) } LeftModule(R:Rng):Category == Join(AbelianGroup, LeftLinearSet R) @ \section{category LINEXP LinearlyExplicitRingOver} <>= )abbrev category LINEXP LinearlyExplicitRingOver ++ Author: ++ Date Created: ++ Date Last Updated: June 14, 2010 ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An extension of left-module with an explicit linear dependence test. LinearlyExplicitRingOver(R:Ring): Category == LeftModule R with reducedSystem: Vector % -> Matrix R ++ \spad{reducedSystem [v1,...,vn]} returns a matrix \spad{M} ++ with coefficients in \spad{R} such that the system of equations ++ \spad{c1*v1 + ... + cn*vn = 0$%} has the same solution as ++ \spad{c * M = 0} where \spad{c} is the row vector \spad{[c1,...cn]}. reducedSystem: Matrix % -> Matrix R ++ reducedSystem(A) returns a matrix B such that \spad{A x = 0} and \spad{B x = 0} ++ have the same solutions in R. reducedSystem: (Matrix %,Vector %) -> Record(mat:Matrix R,vec:Vector R) ++ reducedSystem(A, v) returns a matrix B and a vector w such that ++ \spad{A x = v} and \spad{B x = w} have the same solutions in R. @ \section{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. ++ ++ Axioms: ++ \spad{1*x = x} ++ \spad{(a*b)*x = a*(b*x)} ++ \spad{(a+b)*x = (a*x)+(b*x)} ++ \spad{a*(x+y) = (a*x)+(a*y)} Module(R:CommutativeRing): Category == Join(BiModule(R,R), LinearSet R) add if not(R is %) then x:%*r:R == r*x @ \section{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. ++ ++ Axioms: ++ \spad{leftIdentity("*":(%,%)->%,1)}\tab{30}\spad{1*x=x} ++ \spad{rightIdentity("*":(%,%)->%,1)}\tab{30}\spad{x*1=x} ++ ++ Conditional attributes: ++ unitsKnown\tab{15}\spadfun{recip} only returns "failed" on non-units Monoid(): Category == SemiGroup with --operations 1: % ++ 1 is the multiplicative identity. sample: % ++ sample yields a value of type % one?: % -> Boolean ++ one?(x) tests if x is equal to 1. **: (%,NonNegativeInteger) -> % ++ x**n returns the repeated product ++ of x n times, i.e. exponentiation. recip: % -> Union(%,"failed") ++ recip(x) tries to compute the multiplicative inverse for x ++ or "failed" if it cannot find the inverse (see unitsKnown). add import RepeatedSquaring(%) one? x == x = 1 sample() == 1 recip x == one? x => x "failed" x:% ** n:NonNegativeInteger == zero? n => 1 expt(x,n pretend PositiveInteger) @ \section{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. OrderedAbelianGroup(): Category == Join(OrderedCancellationAbelianMonoid, AbelianGroup) @ \section{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. OrderedAbelianMonoid(): Category == Join(OrderedAbelianSemiGroup, AbelianMonoid) @ \section{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 \spadfun{sup} operation added. ++ The purpose of the \spadfun{sup} operator in this domain is to act as a supremum ++ with respect to the partial order imposed by \spadop{-}, rather than with respect to ++ the total \spad{>} order (since that is "max"). ++ ++ Axioms: ++ \spad{sup(a,b)-a \~~= "failed"} ++ \spad{sup(a,b)-b \~~= "failed"} ++ \spad{x-a \~~= "failed" and x-b \~~= "failed" => x >= sup(a,b)} OrderedAbelianMonoidSup(): Category == OrderedCancellationAbelianMonoid with --operation sup: (%,%) -> % ++ sup(x,y) returns the least element from which both ++ x and y can be subtracted. @ \section{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. ++ \spad{ x < y => x+z < y+z} OrderedAbelianSemiGroup(): Category == Join(OrderedSet, AbelianSemiGroup) @ \section{The Ordered Semigroup Category} <>= )abbrev category OSGROUP OrderedSemiGroup ++ Author: Gabriel Dos Reis ++ Date Create May 25, 2008 ++ Date Last Updated: May 25, 2008 ++ Description: Semigroups with compatible ordering. OrderedSemiGroup(): Category == Join(OrderedSet, SemiGroup) @ \section{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. OrderedCancellationAbelianMonoid(): Category == Join(OrderedAbelianMonoid, CancellationAbelianMonoid) @ \section{category ORDFIN OrderedFinite} <>= )abbrev category ORDFIN OrderedFinite ++ Author: ++ Date Created: ++ Date Last Updated: December 27, 2008 ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ Ordered finite sets. OrderedFinite(): Category == Join(OrderedSet, Finite) with min: % ++ \spad{min} is the minimum value of %. max: % ++ \spad{max} is the maximum value of %. @ \section{category OINTDOM OrderedIntegralDomain} <>= )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 ++ OrderedIntegralDomain(): Category == Join(IntegralDomain, OrderedRing) @ \section{category ORDMON OrderedMonoid} <>= )abbrev category ORDMON OrderedMonoid ++ Author: ++ Date Created: ++ Date Last Updated: May 28, 2008 ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ Ordered sets which are also monoids, such that multiplication ++ preserves the ordering. ++ ++ Axioms: ++ \spad{x < y => x*z < y*z} ++ \spad{x < y => z*x < z*y} OrderedMonoid(): Category == Join(OrderedSemiGroup, Monoid) @ \section{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. ++ ++ Axiom: ++ \spad{0 ab< ac} OrderedRing(): Category == Join(OrderedAbelianGroup,Ring,Monoid) with positive?: % -> Boolean ++ positive?(x) tests whether x is strictly greater than 0. negative?: % -> Boolean ++ negative?(x) tests whether x is strictly less than 0. sign : % -> Integer ++ sign(x) is 1 if x is positive, -1 if x is negative, 0 if x equals 0. abs : % -> % ++ abs(x) returns the absolute value of x. add positive? x == x>0 negative? x == x<0 sign x == positive? x => 1 negative? x => -1 zero? x => 0 error "x satisfies neither positive?, negative? or zero?" abs x == positive? x => x negative? x => -x zero? x => 0 error "x satisfies neither positive?, negative? or zero?" @ \section{category ORDSET OrderedSet} <>= import Boolean )abbrev category ORDSET OrderedSet ++ Author: ++ Date Created: ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ The class of totally ordered sets, that is, sets such that for each pair of elements \spad{(a,b)} ++ exactly one of the following relations holds \spad{a a Boolean ++ x < y is a strict total ordering on the elements of the set. >: (%, %) -> Boolean ++ x > y is a greater than test. >=: (%, %) -> Boolean ++ x >= y is a greater than or equal test. <=: (%, %) -> Boolean ++ x <= y is a less than or equal test. max: (%,%) -> % ++ max(x,y) returns the maximum of x and y relative to "<". min: (%,%) -> % ++ min(x,y) returns the minimum of x and y relative to "<". add before?(x,y) == x < y -- These really ought to become some sort of macro max(x,y) == x > y => x y min(x,y) == x > y => y x ((x: %) > (y: %)) : Boolean == y < x ((x: %) >= (y: %)) : Boolean == not (x < y) ((x: %) <= (y: %)) : Boolean == not (y < x) @ \section{Partial Differential Domain} <>= )abbrev category PDDOM PartialDifferentialDomain ++ Author: Gabriel Dos Reis ++ Date Created: June 16, 2010 ++ Date Last Modified: June 16, 2010 ++ Description: ++ This category captures the interface of domains with a distinguished ++ operation named \spad{differentiate} for partial differentiation with ++ respect to some domain of variables. ++ See Also: ++ DifferentialDomain, PartialDifferentialSpace PartialDifferentialDomain(T: Type, S: Type): Category == Type with differentiate: (%,S) -> T ++ \spad{differentiate(x,v)} computes the partial derivative ++ of \spad{x} with respect to \spad{v}. D: (%,S) -> T ++ \spad{D(x,v)} is a shorthand for \spad{differentiate(x,v)} add D(x,v) == differentiate(x,v) @ \section{Partial Differential Space} <>= )abbrev category PDSPC PartialDifferentialSpace ++ Author: Gabriel Dos Reis ++ Date Created: June 16, 2010 ++ Date Last Modified: June 16, 2010 ++ Description: ++ This category captures the interface of domains stable by partial ++ differentiation with respect to variables from some domain. ++ See Also: ++ PartialDifferentialDomain PartialDifferentialSpace(S: SetCategory): Category == PartialDifferentialDomain(%,S) with differentiate: (%,List S) -> % ++ \spad{differentiate(x,[s1,...sn])} computes successive ++ partial derivatives, i.e. ++ \spad{differentiate(...differentiate(x, s1)..., sn)}. differentiate: (%,S,NonNegativeInteger) -> % ++ \spad{differentiate(x,s,n)} computes multiple partial ++ derivatives, i.e. \spad{n}-th derivative of \spad{x} ++ with respect to \spad{s}. differentiate: (%,List S,List NonNegativeInteger) -> % ++ \spad{differentiate(x,[s1,...,sn],[n1,...,nn])} computes ++ multiple partial derivatives, i.e. D: (%,List S) -> % ++ \spad{D(x,[s1,...sn])} is a shorthand for ++ \spad{differentiate(x,[s1,...sn])}. D: (%,S,NonNegativeInteger) -> % ++ \spad{D(x,s,n)} is a shorthand for \spad{differentiate(x,s,n)}. D: (%,List S,List NonNegativeInteger) -> % ++ \spad{D(x,[s1,...,sn],[n1,...,nn])} is a shorthand for ++ \spad{differentiate(x,[s1,...,sn],[n1,...,nn])}. add differentiate(r: %,l: List S) == for s in l repeat r := differentiate(r, s) r differentiate(r: %,s: S,n: NonNegativeInteger) == for i in 1..n repeat r := differentiate(r, s) r differentiate(r: %,ls: List S,ln: List NonNegativeInteger) == for s in ls for n in ln repeat r := differentiate(r, s, n) r D(r: %,v: S) == differentiate(r,v) D(r: %,lv: List S) == differentiate(r,lv) D(r: %,v: S,n: NonNegativeInteger) == differentiate(r,v,n) D(r: %,lv: List S,ln: List NonNegativeInteger) == differentiate(r, lv, ln) @ \section{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. ++ ++ Axioms: ++ \spad{differentiate(x+y,e) = differentiate(x,e)+differentiate(y,e)} ++ \spad{differentiate(x*y,e) = x*differentiate(y,e) + differentiate(x,e)*y} PartialDifferentialRing(S:SetCategory): Category == Join(Ring,PartialDifferentialSpace S) @ \section{Partial Differential Module} <>= )abbrev category PDMOD PartialDifferentialModule ++ Author: Gabriel Dos Reis ++ Date Created: June 16, 2010 ++ Date Last Updated: June 18, 2010 ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A partial differential R-module with differentiations ++ indexed by a parameter type S. ++ ++ Axioms: ++ \spad{differentiate(x+y,e) = differentiate(x,e)+differentiate(y,e)} ++ \spad{differentiate(r*x,e) = r*differentiate(x,e) + differentiate(r,e)*x} ++ \spad{differentiate(x*r,e) = x*differentiate(r,e) + differentiate(x,e)*r} PartialDifferentialModule(R: Ring,S: SetCategory): Category == Join(BiModule(R,R),PartialDifferentialSpace S) with if R has CommutativeRing then Module R @ \section{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. ++ 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. PolynomialFactorizationExplicit(): Category == Definition where P ==> SparseUnivariatePolynomial % Definition ==> UniqueFactorizationDomain with -- operations squareFreePolynomial: P -> Factored(P) ++ squareFreePolynomial(p) returns the ++ square-free factorization of the ++ univariate polynomial p. factorPolynomial: P -> Factored(P) ++ factorPolynomial(p) returns the factorization ++ into irreducibles of the univariate polynomial p. factorSquareFreePolynomial: P -> Factored(P) ++ factorSquareFreePolynomial(p) factors the ++ univariate polynomial p into irreducibles ++ where p is known to be square free ++ and primitive with respect to its main variable. gcdPolynomial: (P, P) -> P ++ gcdPolynomial(p,q) returns the gcd of the univariate ++ polynomials p qnd q. -- defaults to Euclidean, but should be implemented via -- modular or p-adic methods. solveLinearPolynomialEquation: (List P, P) -> Union(List P,"failed") ++ solveLinearPolynomialEquation([f1, ..., fn], g) ++ (where the fi are relatively prime to each other) ++ returns a list of ai such that ++ \spad{g/prod fi = sum ai/fi} ++ or returns "failed" if no such list of ai's exists. if % has CharacteristicNonZero then conditionP: Matrix % -> Union(Vector %,"failed") ++ conditionP(m) returns a vector of elements, not all zero, ++ whose \spad{p}-th powers (p is the characteristic of the domain) ++ are a solution of the homogenous linear system represented ++ by m, or "failed" is there is no such vector. charthRoot: % -> Union(%,"failed") ++ charthRoot(r) returns the \spad{p}-th root of r, or "failed" ++ if none exists in the domain. -- this is a special case of conditionP, but often the one we want add gcdPolynomial(f,g) == zero? f => g zero? g => f cf:=content f if not one? cf then f:=(f exquo cf)::P cg:=content g if not one? cg then g:=(g exquo cg)::P ans:=subResultantGcd(f,g)$P gcd(cf,cg)*(ans exquo content ans)::P if % has CharacteristicNonZero then charthRoot f == -- to take p'th root of f, solve the system X-fY=0, -- so solution is [x,y] -- with x^p=X and y^p=Y, then (x/y)^p = f zero? f => 0 m:Matrix % := matrix [[1,-f]] ans:= conditionP m ans case "failed" => "failed" (ans.1) exquo (ans.2) if % has Field then solveLinearPolynomialEquation(lf,g) == multiEuclidean(lf,g)$P else solveLinearPolynomialEquation(lf,g) == LPE ==> LinearPolynomialEquationByFractions % solveLinearPolynomialEquationByFractions(lf,g)$LPE @ \section{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 ++ 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. PrincipalIdealDomain(): Category == GcdDomain with --operations principalIdeal: List % -> Record(coef:List %,generator:%) ++ principalIdeal([f1,...,fn]) returns a record whose ++ generator component is a generator of the ideal ++ generated by \spad{[f1,...,fn]} whose coef component satisfies ++ \spad{generator = sum (input.i * coef.i)} expressIdealMember: (List %,%) -> Union(List %,"failed") ++ expressIdealMember([f1,...,fn],h) returns a representation ++ of h as a linear combination of the fi or "failed" if h ++ is not in the ideal generated by the fi. @ \section{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 multiplation by elements of ++ the rng. ++ ++ Axioms: ++ \spad{ x*(a*b) = (x*a)*b } ++ \spad{ x*(a+b) = (x*a)+(x*b) } ++ \spad{ (x+y)*x = (x*a)+(y*a) } RightModule(R:Rng):Category == Join(AbelianGroup, RightLinearSet R) @ \section{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. Ring(): Category == Join(Rng,Monoid,LeftModule(%),CoercibleFrom Integer) with --operations characteristic: NonNegativeInteger ++ characteristic() returns the characteristic of the ring ++ this is the smallest positive integer n such that ++ \spad{n*x=0} for all x in the ring, or zero if no such n ++ exists. --We can not make this a constant, since some domains are mutable unitsKnown ++ recip truly yields ++ reciprocal or "failed" if not a unit. ++ Note: \spad{recip(0) = "failed"}. add n:Integer coerce(n) == n * 1$% @ \section{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 ++ and a semigroup, with multiplication distributing over addition. ++ ++ Axioms: ++ \spad{ x*(y+z) = x*y + x*z} ++ \spad{ (x+y)*z = x*z + y*z } ++ ++ Conditional attributes: ++ \spadnoZeroDivisors\tab{25}\spad{ ab = 0 => a=0 or b=0} Rng(): Category == Join(AbelianGroup,SemiGroup) @ \section{category SGROUP SemiGroup} <>= import PositiveInteger )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{*}. ++ ++ Axioms: ++ \spad{associative("*":(%,%)->%)}\tab{30}\spad{ (x*y)*z = x*(y*z)} ++ ++ Conditional attributes: ++ \spad{commutative("*":(%,%)->%)}\tab{30}\spad{ x*y = y*x } SemiGroup(): Category == SetCategory with --operations *: (%,%) -> % ++ x*y returns the product of x and y. **: (%,PositiveInteger) -> % ++ x**n returns the repeated product ++ of x n times, i.e. exponentiation. add import RepeatedSquaring(%) x:% ** n:PositiveInteger == expt(x,n) @ \section{category SETCAT SetCategory} <>= )abbrev category SETCAT SetCategory ++ Author: ++ Date Created: ++ Date Last Updated: ++ 09/09/92 RSS added latex and hash ++ May 21, 2009: added before? -- gdr ++ 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 output form. ++ ++ Conditional Attributes: ++ canonical\tab{15}data structure equality is the same as \spadop{=} SetCategory(): Category == Join(BasicType,CoercibleTo OutputForm) with --operations hash: % -> SingleInteger ++ hash(s) calculates a hash code for s. latex: % -> String ++ latex(s) returns a LaTeX-printable output ++ representation of s. before?: (%,%) -> Boolean ++ spad{before?(x,y)} holds if \spad{x} comes before \spad{y} ++ in the internal total ordering used by OpenAxiom. add hash(s : %): SingleInteger == SXHASH(s)$Lisp latex(s : %): String == "\mbox{\bf Unimplemented}" before?(x,y) == GGREATERP(y,x)$Foreign(Builtin) @ \section{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 ++ return duplicate items and only return "failed" after exhausting ++ all elements of the domain. ++ This assumes that the sequence starts with \spad{init()}. ++ For infinite domains, repeated application ++ of \spadfun{nextItem} is not required to reach all possible domain elements ++ starting from any initial element. ++ ++ Conditional attributes: ++ infinite\tab{15}repeated \spad{nextItem}'s are never "failed". StepThrough(): Category == SetCategory with --operations init: % ++ init() chooses an initial object for stepping. nextItem: % -> Union(%,"failed") ++ nextItem(x) returns the next item, or "failed" if domain is exhausted. @ \section{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 ++ a finite number of irreducible elements. UniqueFactorizationDomain(): Category == GcdDomain with --operations prime?: % -> Boolean ++ prime?(x) tests if x can never be written as the product of two ++ non-units of the ring, ++ i.e., x is an irreducible element. squareFree : % -> Factored(%) ++ squareFree(x) returns the square-free factorization of x ++ i.e. such that the factors are pairwise relatively prime ++ and each has multiple prime factors. squareFreePart: % -> % ++ squareFreePart(x) returns a product of prime factors of ++ x each taken with multiplicity one. factor: % -> Factored(%) ++ factor(x) returns the factorization of x into irreducibles. add squareFreePart x == unit(s := squareFree x) * _*/[f.factor for f in factors s] prime? x == # factorList factor x = 1 @ \section{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. VectorSpace(S:Field): Category == Module(S) with / : (%, S) -> % ++ x/y divides the vector x by the scalar y. dimension: () -> CardinalNumber ++ dimension() returns the dimensionality of the vector space. add (v:% / s:S):% == inv(s) * v @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2009, Gabriel Dos Reis. --All rights reversed. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}