MonoidRing(R, M)ΒΆ
mring.spad line 33 [edit on github]
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 LeftModule %
- *: (%, R) -> %
from RightModule R
- *: (Integer, %) -> %
from AbelianGroup
- *: (NonNegativeInteger, %) -> %
from AbelianMonoid
- *: (PositiveInteger, %) -> %
from AbelianSemiGroup
- *: (R, %) -> %
from LeftModule R
- +: (%, %) -> %
from AbelianSemiGroup
- -: % -> %
from AbelianGroup
- -: (%, %) -> %
from AbelianGroup
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
- coefficient: (%, M) -> R
from FreeModuleCategory(R, M)
- coefficients: % -> List R
from FreeModuleCategory(R, M)
- coerce: % -> % if M has CommutativeStar and R has CommutativeRing
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: Integer -> %
from NonAssociativeRing
- coerce: List Record(k: M, c: R) -> %
from MonoidRingCategory(R, M)
- coerce: M -> %
from CoercibleFrom M
- coerce: R -> %
from Algebra R
- commutator: (%, %) -> %
from NonAssociativeRng
- construct: List Record(k: M, c: R) -> %
from IndexedProductCategory(R, M)
- constructOrdered: List Record(k: M, c: R) -> % if M has Comparable
from IndexedProductCategory(R, M)
- hash: % -> SingleInteger if M has Finite and R has Finite
from Hashable
- index: PositiveInteger -> % if M has Finite and R has Finite
from Finite
- latex: % -> String
from SetCategory
- leadingCoefficient: % -> R if M has Comparable
from IndexedProductCategory(R, M)
- leadingMonomial: % -> % if M has Comparable
from IndexedProductCategory(R, M)
- leadingSupport: % -> M if M has Comparable
from IndexedProductCategory(R, M)
- leadingTerm: % -> Record(k: M, c: R) if M has Comparable
from IndexedProductCategory(R, M)
- leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
from Magma
- leftRecip: % -> Union(%, failed)
from MagmaWithUnit
- linearExtend: (M -> R, %) -> R if R has CommutativeRing
from FreeModuleCategory(R, M)
- listOfTerms: % -> List Record(k: M, c: R)
from IndexedDirectProductCategory(R, M)
- lookup: % -> PositiveInteger if M has Finite and R has Finite
from Finite
- map: (R -> R, %) -> %
from IndexedProductCategory(R, M)
- monomial?: % -> Boolean
from IndexedProductCategory(R, M)
- monomial: (R, M) -> %
from IndexedProductCategory(R, M)
- monomials: % -> List %
from FreeModuleCategory(R, M)
- numberOfMonomials: % -> NonNegativeInteger
from IndexedDirectProductCategory(R, M)
- one?: % -> Boolean
from MagmaWithUnit
- opposite?: (%, %) -> Boolean
from AbelianMonoid
- plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing
from NonAssociativeAlgebra R
- recip: % -> Union(%, failed)
from MagmaWithUnit
- reductum: % -> % if M has Comparable
from IndexedProductCategory(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 M has Finite and R has Finite
from Finite
- smaller?: (%, %) -> Boolean if R has Comparable and M has Comparable or M has Finite and R has Finite
from Comparable
- subtractIfCan: (%, %) -> Union(%, failed)
- support: % -> List M
from FreeModuleCategory(R, M)
- terms: % -> List Record(k: M, c: R)
from MonoidRingCategory(R, M)
- zero?: % -> Boolean
from AbelianMonoid
Algebra % if M has CommutativeStar and R has CommutativeRing
Algebra R if R has CommutativeRing
BiModule(%, %)
BiModule(R, R)
CharacteristicNonZero if R has CharacteristicNonZero
CharacteristicZero if R has CharacteristicZero
CommutativeRing if M has CommutativeStar and R has CommutativeRing
CommutativeStar if M has CommutativeStar and R has CommutativeRing
Comparable if R has Comparable and M has Comparable or M has Finite and R has Finite
ConvertibleTo InputForm if M has Finite and R has Finite
Finite if M has Finite and R has Finite
FreeModuleCategory(R, M)
Hashable if M has Finite and R has Finite
IndexedDirectProductCategory(R, M)
IndexedProductCategory(R, M)
Module % if M has CommutativeStar and R has CommutativeRing
Module R if R has CommutativeRing
MonoidRingCategory(R, M)
NonAssociativeAlgebra % if M has CommutativeStar and R has CommutativeRing
NonAssociativeAlgebra R if R has CommutativeRing
TwoSidedRecip if M has CommutativeStar and R has CommutativeRing