FreeModule(R, S)ΒΆ

poly.spad line 90

A bi-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored. old domain FreeModule1 was merged to it in May 2009 The description of the latter: This domain implements linear combinations of elements from the domain S with coefficients in the domain R where S is an ordered set and R is a ring (which may be non-commutative). This domain is used by domains of non-commutative algebra such as: XDistributedPolynomial, XRecursivePolynomial. Author: Michel Petitot (petitot@lifl.fr)

0: %
from AbelianMonoid
*: (%, R) -> %
from RightModule R
*: (Integer, %) -> % if R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
*: (R, S) -> %
from FreeModuleCategory(R, S)
*: (S, R) -> %
from FreeModuleCategory(R, S)
+: (%, %) -> %
from AbelianSemiGroup
-: % -> % if R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if R has AbelianGroup
from AbelianGroup
<: (%, %) -> Boolean if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from PartialOrder
<=: (%, %) -> Boolean if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from PartialOrder
>=: (%, %) -> Boolean if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from PartialOrder
~=: (%, %) -> Boolean
from BasicType
coefficient: (%, S) -> R
from FreeModuleCategory(R, S)
coefficients: % -> List R
from FreeModuleCategory(R, S)
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: S -> % if R has SemiRing
from RetractableTo S
construct: List Record(k: S, c: R) -> %
from IndexedDirectProductCategory(R, S)
constructOrdered: List Record(k: S, c: R) -> %
from IndexedDirectProductCategory(R, S)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
from IndexedDirectProductCategory(R, S)
leadingMonomial: % -> %
from IndexedDirectProductCategory(R, S)
leadingSupport: % -> S
from IndexedDirectProductCategory(R, S)
leadingTerm: % -> Record(k: S, c: R)
from IndexedDirectProductCategory(R, S)
linearExtend: (S -> R, %) -> R if R has CommutativeRing
from FreeModuleCategory(R, S)
listOfTerms: % -> List Record(k: S, c: R)
from IndexedDirectProductCategory(R, S)
map: (R -> R, %) -> %
from IndexedDirectProductCategory(R, S)
max: (%, %) -> % if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from OrderedSet
min: (%, %) -> % if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet
from OrderedSet
monomial: (R, S) -> %
from IndexedDirectProductCategory(R, S)
monomial?: % -> Boolean
from IndexedDirectProductCategory(R, S)
monomials: % -> List %
from FreeModuleCategory(R, S)
numberOfMonomials: % -> NonNegativeInteger
from IndexedDirectProductCategory(R, S)
opposite?: (%, %) -> Boolean
from AbelianMonoid
reductum: % -> %
from IndexedDirectProductCategory(R, S)
retract: % -> S if R has SemiRing
from RetractableTo S
retractIfCan: % -> Union(S, failed) if R has SemiRing
from RetractableTo S
sample: %
from AbelianMonoid
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed) if R has CancellationAbelianMonoid
from CancellationAbelianMonoid
sup: (%, %) -> % if R has OrderedAbelianMonoidSup and S has OrderedSet
from OrderedAbelianMonoidSup
support: % -> List S
from FreeModuleCategory(R, S)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid

AbelianProductCategory R

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid if R has CancellationAbelianMonoid

CoercibleTo OutputForm

Comparable if R has Comparable

FreeModuleCategory(R, S)

IndexedDirectProductCategory(R, S)

LeftModule R

Module R if R has CommutativeRing

OrderedAbelianMonoid if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet

OrderedAbelianMonoidSup if R has OrderedAbelianMonoidSup and S has OrderedSet

OrderedAbelianSemiGroup if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet

OrderedCancellationAbelianMonoid if R has OrderedAbelianMonoidSup and S has OrderedSet

OrderedSet if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet

PartialOrder if R has OrderedAbelianMonoidSup and S has OrderedSet or R has OrderedAbelianMonoid and S has OrderedSet

RetractableTo S if R has SemiRing

RightModule R

SetCategory