SolvableSkewPolynomialCategory(R, Expon)

skpol.spad line 6 [edit on github]

• R : Ring
• Expon : OrderedAbelianMonoidSup

This is the category of polynomials in noncommutative variables over noncommutative rings. We do not assume that variables and elements of the base ring commute. We assume that the polynomial ring is of solvable type, so noncommutative version of Buchberger algorithm works.

* : (%, %) -> %

from Magma

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

degree : % -> Expon

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

latex : % -> String

from SetCategory

leadingCoefficient : % -> R

leadingCoefficient(p) returns the coefficient of the highest degree term of p.

leadingMonomial : % -> %

leadingMonomial(p) returns the monomial of p with the highest degree.

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

monomial : (R, Expon) -> %

monomial(r, e) makes a term from a coefficient r and an exponent e.

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

recip : % -> Union(%, "failed")

from MagmaWithUnit

reductum : % -> %

reductum(u) returns u minus its leading monomial returns zero if handed the zero element.

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

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

from CancellationAbelianMonoid

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

RightModule(%)

Rng

Monoid

Ring

SemiGroup

CancellationAbelianMonoid

LeftModule(%)

NonAssociativeRing

BasicType

unitsKnown

Magma

NonAssociativeSemiRng

SemiRing

AbelianGroup

NonAssociativeSemiRing

SetCategory

AbelianSemiGroup

AbelianMonoid

BiModule(%, %)

NonAssociativeRng

LeftModule(R)

MagmaWithUnit

CoercibleTo(OutputForm)

SemiRng