SolvableSkewPolynomialCategory(R, Expon, VarSet)ΒΆ

skpol.spad line 6

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.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
=: (%, %) -> Boolean
from BasicType
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associator: (%, %, %) -> %
from NonAssociativeRng
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
degree: % -> Expon
degree(p) returns the maximum of the exponents of the terms of p.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CoercibleTo OutputForm

LeftModule %

LeftModule R

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

RightModule %

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown