MathAction NonCommutativeLaurentPolynomials

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	NoncommutativePolynomials
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NonCommutativeLaurentPolynomials

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Consider a noncommutative group ring over commuting polynomials

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(1) -> NPOLY:=MonoidRing(Polynomial(Fraction(Integer)),FreeGroup(Symbol))

$\label{eq1}\hbox{\axiomType{MonoidRing}\ } \left({{\hbox{\axiomType{Polynomial}\ } \left({\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}\right)}, \:{\hbox{\axiomType{FreeGroup}\ } \left({\hbox{\axiomType{Symbol}\ }}\right)}}\right)$

Type: Type

Generators

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x:NPOLY:='x::FreeGroup Symbol

$\label{eq2}x$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))

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y:NPOLY:='y::FreeGroup Symbol

$\label{eq3}y$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))

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x':NPOLY:=('x::FreeGroup Symbol)^(-1)

$\label{eq4}{x}^{- 1}$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))

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y':NPOLY:=('y::FreeGroup Symbol)^(-1)

$\label{eq5}{y}^{- 1}$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))

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a:POLY FRAC INT := 'a::Symbol

$\label{eq6}a$

Type: Polynomial(Fraction(Integer))

Example

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p:=x*y*x+y

$\label{eq7}{x \ y \ x}+ y$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))

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(p+3*a+1)*x'

$\label{eq8}{y \ {{x}^{- 1}}}+{x \ y}+{{\left({3 \ a}+ 1 \right)}\ {{x}^{- 1}}}$

Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))