AbelianMonoidRing(R, E)ΒΆ

polycat.spad line 1 [edit on github]

Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i, elements of the ordered abelian monoid, are thought of as exponents or monomials. The monomials commute with each other, but in general do not commute with the coefficients (which themselves may or may not be commutative). See FiniteAbelianMonoidRing for the case of finite support. A useful common model for polynomials and power series. Conceptually at least, only the non-zero terms are ever operated on.

0: %

from AbelianMonoid

1: % if R has SemiRing

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer

from RightModule Fraction Integer

*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> % if R has Algebra Fraction Integer

from LeftModule Fraction Integer

*: (Integer, %) -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

/: (%, R) -> % if R has Field

p/c divides p by the coefficient c.

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean if R has Ring

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean if R has IntegralDomain and % has VariablesCommuteWithCoefficients

from EntireRing

associator: (%, %, %) -> % if R has Ring

from NonAssociativeRng

characteristic: () -> NonNegativeInteger if R has Ring

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero

from CharacteristicNonZero

coefficient: (%, E) -> R

coefficient(p, e) extracts the coefficient of the monomial with exponent e from polynomial p, or returns zero if exponent is not present.

coerce: % -> % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> % if R has Algebra Fraction Integer

from Algebra Fraction Integer

coerce: Integer -> % if R has Ring

from NonAssociativeRing

coerce: R -> % if R has CommutativeRing and % has VariablesCommuteWithCoefficients

from Algebra R

commutator: (%, %) -> % if R has Ring

from NonAssociativeRng

construct: List Record(k: E, c: R) -> %

from IndexedProductCategory(R, E)

constructOrdered: List Record(k: E, c: R) -> %

from IndexedProductCategory(R, E)

degree: % -> E

degree(p) returns the maximum of the exponents of the terms of p.

exquo: (%, %) -> Union(%, failed) if R has IntegralDomain and % has VariablesCommuteWithCoefficients

from EntireRing

latex: % -> String

from SetCategory

leadingCoefficient: % -> R

from IndexedProductCategory(R, E)

leadingMonomial: % -> %

from IndexedProductCategory(R, E)

leadingSupport: % -> E

from IndexedProductCategory(R, E)

leadingTerm: % -> Record(k: E, c: R)

from IndexedProductCategory(R, E)

leftPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

map: (R -> R, %) -> %

from IndexedProductCategory(R, E)

monomial?: % -> Boolean

from IndexedProductCategory(R, E)

monomial: (R, E) -> %

from IndexedProductCategory(R, E)

one?: % -> Boolean if R has SemiRing

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has Algebra Fraction Integer or R has CommutativeRing and % has VariablesCommuteWithCoefficients

from NonAssociativeAlgebra %

recip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(R, E)

rightPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

unit?: % -> Boolean if R has IntegralDomain and % has VariablesCommuteWithCoefficients

from EntireRing

unitCanonical: % -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has IntegralDomain and % has VariablesCommuteWithCoefficients

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid

AbelianProductCategory R

AbelianSemiGroup

Algebra % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer

BiModule(R, R)

CancellationAbelianMonoid

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

CommutativeStar if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

EntireRing if R has IntegralDomain and % has VariablesCommuteWithCoefficients

IndexedProductCategory(R, E)

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

Magma

MagmaWithUnit if R has SemiRing

Module % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

Monoid if R has SemiRing

NonAssociativeAlgebra % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has SemiRing

NonAssociativeSemiRng

noZeroDivisors if R has IntegralDomain and % has VariablesCommuteWithCoefficients

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule R

Ring if R has Ring

Rng if R has Ring

SemiGroup

SemiRing if R has SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

unitsKnown if R has Ring