AbelianMonoidRing(R, E)ΒΆ

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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 Magma
*: (%, 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 % has AbelianGroup or R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> % if % has AbelianGroup or R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if % has AbelianGroup or R 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
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
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
leadingCoefficient(p) returns the coefficient highest degree term of p.
leadingMonomial: % -> %
leadingMonomial(p) returns the monomial of p with the highest degree.
leftPower: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
map: (R -> R, %) -> %
map(fn, u) maps function fn onto the coefficients of the non-zero monomials of u.
monomial: (R, E) -> %
monomial(r, e) makes a term from a coefficient r and an exponent e.
monomial?: % -> Boolean
monomial?(p) tests if p is a single monomial.
one?: % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
recip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
reductum: % -> %
reductum(u) returns u minus its leading monomial returns zero if handed the zero element.
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

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

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

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

unitsKnown if R has Ring