The type CommAlgebra checks if the algebra A is an algebra of commutative polynomials, as declared by Ore_algebra[poly_algebra] (or Ore_algebra[skew_algebra] with no commutation and commutative parameters only).

The type SkewAlgebra checks if the algebra A is built by using Ore_algebra[skew_algebra] with commutations of the form

for constants p, r, and s only.  This is the case for the commutation types delta, diff, euler, shift, and their dual forms.

The type SkewParamAlgebra checks if the algebra A is built by using Ore_algebra[skew_algebra] with commutations of the form

for constants p, q, r, and s with at least one commutation with $q\ne 1$.  This is the case for the commutation types qdelta, qdiff, qdilat, qshift, `shift+qshift`, and their dual forms.

The type OreAlgebra checks if the algebra A is any of the above.

The type SkewPolynomial checks if the membership of the polynomial P in the algebra A.  When this algebra allows rational function coefficients, a polynomial with rational function coefficients is a member of the algebra.

$\mathrm{type}\left(1\,\mathrm{OreAlgebra}\right)$

$\mathrm{false}$

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$A≔\mathrm{poly\_algebra}\left(a\,b\,c\right)\:$

$\mathrm{type}\left(A\,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A\,\mathrm{OreAlgebra}\right)$

$\mathrm{true},\mathrm{true}$

$\mathrm{type}\left(a^2+b^2+c^2-1\,\mathrm{SkewPolynomial}\left(A\right)\right)$

$\mathrm{true}$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\right)\:$

$\mathrm{type}\left(A\,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewParamAlgebra}\right)$

$\mathrm{false},\mathrm{true},\mathrm{false}$

$\mathrm{type}\left(x\mathrm{Dx}+1\,\mathrm{SkewPolynomial}\left(A\right)\right),\mathrm{type}\left(\mathrm{Dx}+\frac{1}{x}\,\mathrm{SkewPolynomial}\left(A\right)\right)$

$\mathrm{true},\mathrm{true}$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\,\mathrm{polynom}=x\right)\:$

$\mathrm{type}\left(A\,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewParamAlgebra}\right)$

$\mathrm{false},\mathrm{true},\mathrm{false}$

$\mathrm{type}\left(x\mathrm{Dx}+1\,\mathrm{SkewPolynomial}\left(A\right)\right),\mathrm{type}\left(\mathrm{Dx}+\frac{1}{x}\,\mathrm{SkewPolynomial}\left(A\right)\right)$

$\mathrm{true},\mathrm{false}$

$A≔\mathrm{qshift\_algebra}\left(\left[\mathrm{Sn}\,q^n\right]\right)\:$

$\mathrm{type}\left(A\,\mathrm{CommAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewAlgebra}\right),\mathrm{type}\left(A\,\mathrm{SkewParamAlgebra}\right)$

$\mathrm{false},\mathrm{false},\mathrm{true}$

$\mathrm{type}\left(\frac{q^n}{1-q^n}\mathrm{Sn}+1\,\mathrm{SkewPolynomial}\left(A\right)\right)$

$\mathrm{true}$