MonoidRingCategory(R, M)ΒΆ

mring.spad line 1

MonoidRingCategory(R, M) defines the algebra of all maps from the monoid M to the commutative ring R with finite support.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, R) -> % if M has Comparable or R has CommutativeRing
from RightModule R
*: (Integer, %) -> %
from AbelianGroup
*: (M, R) -> % if M has Comparable
from FreeModuleCategory(R, M)
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> % if M has Comparable or R has CommutativeRing
from LeftModule R
*: (R, M) -> % if M has Comparable
from FreeModuleCategory(R, M)
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
<: (%, %) -> Boolean if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from PartialOrder
<=: (%, %) -> Boolean if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from PartialOrder
>=: (%, %) -> Boolean if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from PartialOrder
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associator: (%, %, %) -> %
from NonAssociativeRng
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, M) -> R
coefficient(f, m) extracts the coefficient of m in f with respect to the canonical basis M.
coefficients: % -> List R
coefficients(f) lists all non-zero coefficients.
coerce: % -> % if R has CommutativeRing and M has CommutativeStar
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Integer -> %
from NonAssociativeRing
coerce: List Record(k: M, c: R) -> %
coerce(lt) converts a list of terms and coefficients to a member of the domain.
coerce: M -> %
from RetractableTo M
coerce: R -> %
from RetractableTo R
commutator: (%, %) -> %
from NonAssociativeRng
construct: List Record(k: M, c: R) -> % if M has Comparable
from IndexedDirectProductCategory(R, M)
constructOrdered: List Record(k: M, c: R) -> % if M has Comparable
from IndexedDirectProductCategory(R, M)
convert: % -> InputForm if R has Finite and M has Finite
from ConvertibleTo InputForm
enumerate: () -> List % if R has Finite and M has Finite
from Finite
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if R has Finite and M has Finite
from Finite
latex: % -> String
from SetCategory
leadingCoefficient: % -> R if M has Comparable
from IndexedDirectProductCategory(R, M)
leadingMonomial: % -> % if M has Comparable
from IndexedDirectProductCategory(R, M)
leadingSupport: % -> M if M has Comparable
from IndexedDirectProductCategory(R, M)
leadingTerm: % -> Record(k: M, c: R) if M has Comparable
from IndexedDirectProductCategory(R, M)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
linearExtend: (M -> R, %) -> R if R has CommutativeRing and M has Comparable
from FreeModuleCategory(R, M)
listOfTerms: % -> List Record(k: M, c: R) if M has Comparable
from IndexedDirectProductCategory(R, M)
lookup: % -> PositiveInteger if R has Finite and M has Finite
from Finite
map: (R -> R, %) -> %
map(fn, u) maps function fn onto the coefficients of the non-zero monomials of u.
max: (%, %) -> % if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from OrderedSet
min: (%, %) -> % if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable
from OrderedSet
monomial: (R, M) -> %
monomial(r, m) creates a scalar multiple of the basis element m.
monomial?: % -> Boolean
monomial?(f) tests if f is a single monomial.
monomials: % -> List %
monomials(f) gives the list of all monomials whose sum is f.
numberOfMonomials: % -> NonNegativeInteger
numberOfMonomials(f) is the number of non-zero coefficients with respect to the canonical basis.
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
random: () -> % if R has Finite and M has Finite
from Finite
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> % if M has Comparable
from IndexedDirectProductCategory(R, M)
retract: % -> M
from RetractableTo M
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(M, failed)
from RetractableTo M
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
size: () -> NonNegativeInteger if R has Finite and M has Finite
from Finite
smaller?: (%, %) -> Boolean if R has Comparable and M has Comparable or R has Finite and M has Finite
from Comparable
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
sup: (%, %) -> % if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from OrderedAbelianMonoidSup
support: % -> List M if M has Comparable
from FreeModuleCategory(R, M)
terms: % -> List Record(k: M, c: R)
terms(f) gives the list of non-zero coefficients combined with their corresponding basis element as records. This is the internal representation.
totalDegree: % -> NonNegativeInteger if M has coerce: M -> Vector NonNegativeInteger and M has Comparable
from FreeModuleCategory(R, M)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianProductCategory R if M has Comparable

AbelianSemiGroup

Algebra % if R has CommutativeRing and M has CommutativeStar

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R) if M has Comparable or R has CommutativeRing

CancellationAbelianMonoid

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing and M has CommutativeStar

CommutativeStar if R has CommutativeRing and M has CommutativeStar

Comparable if R has Comparable and M has Comparable or R has Finite and M has Finite

ConvertibleTo InputForm if R has Finite and M has Finite

Finite if R has Finite and M has Finite

FreeModuleCategory(R, M) if M has Comparable

IndexedDirectProductCategory(R, M) if M has Comparable

LeftModule %

LeftModule R if M has Comparable or R has CommutativeRing

Magma

MagmaWithUnit

Module % if R has CommutativeRing and M has CommutativeStar

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

OrderedAbelianMonoid if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable

OrderedAbelianMonoidSup if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable

OrderedCancellationAbelianMonoid if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup

OrderedSet if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable

PartialOrder if M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup or R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable

RetractableTo M

RetractableTo R

RightModule %

RightModule R if M has Comparable or R has CommutativeRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown