MonoidRing(R, M)ΒΆ

mring.spad line 52

MonoidRing(R, M), implements the algebra of all maps from the monoid M to the commutative ring R with finite support. Multiplication of two maps f and g is defined to map an element c of M to the (convolution) sum over f(a)g(b) such that ab = c. Thus M can be identified with a canonical basis and the maps can also be considered as formal linear combinations of the elements in M. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When M is FreeMonoid Symbol, one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups G, where modules over MonoidRing(R, G) are studied.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, R) -> % if R has CommutativeRing or M has Comparable
from RightModule R
*: (Integer, %) -> %
from AbelianGroup
*: (M, R) -> % if M has Comparable
from FreeModuleCategory(R, M)
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> % if R has CommutativeRing or M has Comparable
from LeftModule R
*: (R, M) -> % if M has Comparable
from FreeModuleCategory(R, M)
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
<: (%, %) -> Boolean if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from PartialOrder
<=: (%, %) -> Boolean if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from PartialOrder
>=: (%, %) -> Boolean if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
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
from MonoidRingCategory(R, M)
coefficients: % -> List R
from MonoidRingCategory(R, M)
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Integer -> %
from NonAssociativeRing
coerce: List Record(k: M, c: R) -> %
from MonoidRingCategory(R, M)
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, %) -> %
from MonoidRingCategory(R, M)
max: (%, %) -> % if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from OrderedSet
min: (%, %) -> % if R has OrderedAbelianMonoid and M has OrderedSet and M has Comparable or M has Comparable and M has OrderedSet and R has OrderedAbelianMonoidSup
from OrderedSet
monomial: (R, M) -> %
from MonoidRingCategory(R, M)
monomial?: % -> Boolean
from MonoidRingCategory(R, M)
monomials: % -> List %
from MonoidRingCategory(R, M)
numberOfMonomials: % -> NonNegativeInteger
from MonoidRingCategory(R, M)
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 Finite and M has Finite or R has Comparable and M has Comparable
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)
from MonoidRingCategory(R, M)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianProductCategory R if M has Comparable

AbelianSemiGroup

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

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

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

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 R has CommutativeRing or M has Comparable

Magma

MagmaWithUnit

Module R if R has CommutativeRing

Monoid

MonoidRingCategory(R, M)

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

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

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

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

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

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

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

RetractableTo M

RetractableTo R

RightModule %

RightModule R if R has CommutativeRing or M has Comparable

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown