XFreeAlgebra(vl, R)ΒΆ

xpoly.spad line 28

This category specifies opeations for polynomials and formal series with non-commutative variables.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, R) -> %
x * r returns the product of x by r. Usefull if R is a non-commutative Ring.
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
*: (vl, %) -> %
v * x returns the product of a variable x by x.
+: (%, %) -> %
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
coef: (%, %) -> R
coef(x, y) returns scalar product of x by y, the set of words being regarded as an orthogonal basis.
coef: (%, FreeMonoid vl) -> R
coef(x, w) returns the coefficient of the word w in x.
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: FreeMonoid vl -> %
from RetractableTo FreeMonoid vl
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from XAlgebra R
coerce: vl -> %
coerce(v) returns v.
commutator: (%, %) -> %
from NonAssociativeRng
constant: % -> R
constant(x) returns the constant term of x.
constant?: % -> Boolean
constant?(x) returns true if x is constant.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
lquo: (%, %) -> %
lquo(x, y) returns the left simplification of x by y.
lquo: (%, FreeMonoid vl) -> %
lquo(x, w) returns the left simplification of x by the word w.
lquo: (%, vl) -> %
lquo(x, v) returns the left simplification of x by the variable v.
map: (R -> R, %) -> %
map(fn, x) returns Sum(fn(r_i) w_i) if x writes Sum(r_i w_i).
mindeg: % -> FreeMonoid vl
mindeg(x) returns the little word which appears in x. Error if x=0.
mindegTerm: % -> Record(k: FreeMonoid vl, c: R)
mindegTerm(x) returns the term whose word is mindeg(x).
mirror: % -> %
mirror(x) returns Sum(r_i mirror(w_i)) if x writes Sum(r_i w_i).
monom: (FreeMonoid vl, R) -> %
monom(w, r) returns the product of the word w by the coefficient r.
monomial?: % -> Boolean
monomial?(x) returns true if x is a monomial
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
quasiRegular: % -> %
quasiRegular(x) return x minus its constant term.
quasiRegular?: % -> Boolean
quasiRegular?(x) return true if constant(x) is zero.
recip: % -> Union(%, failed)
from MagmaWithUnit
retract: % -> FreeMonoid vl
from RetractableTo FreeMonoid vl
retractIfCan: % -> Union(FreeMonoid vl, failed)
from RetractableTo FreeMonoid vl
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
rquo: (%, %) -> %
rquo(x, y) returns the right simplification of x by y.
rquo: (%, FreeMonoid vl) -> %
rquo(x, w) returns the right simplification of x by w.
rquo: (%, vl) -> %
rquo(x, v) returns the right simplification of x by the variable v.
sample: %
from AbelianMonoid
sh: (%, %) -> % if R has CommutativeRing
sh(x, y) returns the shuffle-product of x by y. This multiplication is associative and commutative.
sh: (%, NonNegativeInteger) -> % if R has CommutativeRing
sh(x, n) returns the shuffle power of x to the n.
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
varList: % -> List vl
varList(x) returns the list of variables which appear in x.
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo OutputForm

LeftModule %

LeftModule R

Magma

MagmaWithUnit

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has noZeroDivisors

RetractableTo FreeMonoid vl

RightModule %

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

XAlgebra R