Edit detail for FreeModuleCategory revision 1 of 2

A domain of this category implements formal linear combinations of elements from a domain S with coefficients in a domain R. The coefficient ring may be non-commutative.

See the XDistributedPolynomial? constructor for examples of domains built with the FreeModuleCategory? category constructor.

Author: Michel Petitot (petitot@lifl.fr)

Modified by: Franz Lehner, June 2009

(1) -> )sh FreeModuleCategory

 FreeModuleCategory(R: Join(SemiRng,AbelianMonoid),S: SetCategory) is a category constructor
 Abbreviation for FreeModuleCategory is FMCAT 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

 ?*? : (%, R) -> %                     ?*? : (R, %) -> %
 ?*? : (PositiveInteger, %) -> %       ?+? : (%, %) -> %
 ?=? : (%, %) -> Boolean               coefficient : (%, S) -> R
 coefficients : % -> List(R)           coerce : % -> OutputForm
 latex : % -> String                   map : ((R -> R), %) -> %
 monomial : (R, S) -> %                monomial? : % -> Boolean
 monomials : % -> List(%)              support : % -> List(S)
 ?~=? : (%, %) -> Boolean             
 ?*? : (NonNegativeInteger, %) -> % if and(OR(has(R,AbelianMonoid),has(R,AbelianMonoid)),has(R,CommutativeRing)) or R has ABELMON or R has ABELMON
 ?*? : (Integer, %) -> % if and(OR(has(R,AbelianGroup),has(R,AbelianGroup)),has(R,CommutativeRing)) or R has ABELGRP or R has ABELGRP
 -? : % -> % if and(OR(has(R,AbelianGroup),has(R,AbelianGroup)),has(R,CommutativeRing)) or R has ABELGRP or R has ABELGRP
 ?-? : (%, %) -> % if and(OR(has(R,AbelianGroup),has(R,AbelianGroup)),has(R,CommutativeRing)) or R has ABELGRP or R has ABELGRP
 0 : () -> % if and(OR(has(R,AbelianMonoid),has(R,AbelianMonoid)),has(R,CommutativeRing)) or R has ABELMON or R has ABELMON
 construct : List(Record(k: S,c: R)) -> %
 constructOrdered : List(Record(k: S,c: R)) -> % if S has COMPAR
 leadingCoefficient : % -> R if S has COMPAR
 leadingMonomial : % -> % if S has COMPAR
 leadingSupport : % -> S if S has COMPAR
 leadingTerm : % -> Record(k: S,c: R) if S has COMPAR
 linearExtend : ((S -> R), %) -> R if R has COMRING
 listOfTerms : % -> List(Record(k: S,c: R))
 numberOfMonomials : % -> NonNegativeInteger
 opposite? : (%, %) -> Boolean if and(OR(has(R,AbelianMonoid),has(R,AbelianMonoid)),has(R,CommutativeRing)) or R has ABELMON or R has ABELMON
 reductum : % -> % if S has COMPAR
 sample : () -> % if and(OR(has(R,AbelianMonoid),has(R,AbelianMonoid)),has(R,CommutativeRing)) or R has ABELMON or R has ABELMON
 smaller? : (%, %) -> Boolean if S has COMPAR and R has COMPAR
 subtractIfCan : (%, %) -> Union(%,"failed") if and(OR(has(R,AbelianGroup),has(R,AbelianGroup)),has(R,CommutativeRing)) or R has CABMON
 zero? : % -> Boolean if and(OR(has(R,AbelianMonoid),has(R,AbelianMonoid)),has(R,CommutativeRing)) or R has ABELMON or R has ABELMON