XFreeAlgebra(vl, R)ΒΆ

xpoly.spad line 28 [edit on github]

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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, 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 CoercibleFrom FreeMonoid vl

coerce: Integer -> %

from NonAssociativeRing

coerce: R -> %

from XAlgebra R

coerce: vl -> %

coerce(v) returns v.

commutator: (%, %) -> %

from NonAssociativeRng

constant?: % -> Boolean

constant?(x) returns true if x is constant.

constant: % -> R

constant(x) returns the constant term of x.

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?: % -> Boolean

quasiRegular?(x) return true if constant(x) is zero.

quasiRegular: % -> %

quasiRegular(x) return x minus its constant term.

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

CoercibleFrom FreeMonoid vl

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