AbelianMonoidRing(R, E)

polycat.spad line 1 [edit on github]

• R : Join(SemiRng, AbelianMonoid)
• E : OrderedAbelianMonoid

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 coefficients 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.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

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

from RightModule(Fraction(Integer))

* : (R, %) -> %

from LeftModule(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

+ : (%, %) -> %

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.

0 : () -> %

from AbelianMonoid

1 : () -> % if R has SemiRing

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

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

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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

from Algebra(R)

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

from Algebra(Fraction(Integer))

coerce : Integer -> % if R has Ring

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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

from IndexedProductCategory(R, E)

monomial : (R, E) -> %

from IndexedProductCategory(R, E)

monomial? : % -> Boolean

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

~= : (%, %) -> Boolean

from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

noZeroDivisors

LeftModule(Fraction(Integer))

RightModule(%)

Monoid

Algebra(%)

Algebra(R)

AbelianMonoid

BiModule(R, R)

NonAssociativeAlgebra(Fraction(Integer))

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

RightModule(Fraction(Integer))

CommutativeStar

LeftModule(%)

LeftModule(R)

Module(%)

SetCategory

Algebra(Fraction(Integer))

Rng

CommutativeRing

IntegralDomain

SemiGroup

TwoSidedRecip

Magma

BiModule(%, %)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

IndexedProductCategory(R, E)

AbelianProductCategory(R)

Module(R)

RightModule(R)

BiModule(Fraction(Integer), Fraction(Integer))

NonAssociativeRng

Ring

SemiRng

EntireRing

NonAssociativeSemiRng

CharacteristicZero

BasicType

SemiRing