type/MonomialOrder - Maple Programming Help

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type/MonomialOrder

check for monomial order data structures

type/ShortMonomialOrder

check for short monomial order descriptions

 Calling Sequence type(T, MonomialOrder) type(ST, ShortMonomialOrder)

Parameters

 T - table that denotes a monomial ordering on an algebra SP - short monomial order description

Description

 • The type ShortMonomialOrder checks if ST is a short monomial order description.
 • The type MonomialOrder checks if T is a monomial order, as declared by the command Groebner[MonomialOrder]. This representation is used to denote general monomial orders over general skew algebras, possibly with parameters, and possibly in positive characteristic.

Examples

A short monomial order description.

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $\mathrm{type}\left(\mathrm{plex}\left(x,y,z\right),\mathrm{MonomialOrder}\right),\mathrm{type}\left(\mathrm{plex}\left(x,y,z\right),\mathrm{ShortMonomialOrder}\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}$ (1)
 > $\mathrm{LeadingMonomial}\left(a{x}^{2}+bxy+cxz,\mathrm{plex}\left(x,y,z\right)\right)$
 ${{x}}^{{2}}$ (2)

The previous order can equivalently be declared as a (general) monomial order:

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{poly_algebra}\left(x,y,z,\mathrm{rational}=\left\{a,b,c\right\}\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{plex}\left(x,y,z\right)\right):$
 > $\mathrm{LeadingMonomial}\left(a{x}^{2}+bxy+cxz,T\right)$
 ${{x}}^{{2}}$ (3)
 > $\mathrm{type}\left(T,\mathrm{MonomialOrder}\right),\mathrm{type}\left(T,\mathrm{ShortMonomialOrder}\right)$
 ${\mathrm{true}}{,}{\mathrm{false}}$ (4)

A monomial order for a skew polynomial ring.

 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\mathrm{comm}=\left\{\mathrm{μ},s\right\},\mathrm{polynom}=s\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dx},s\right)\right):$
 > $\mathrm{type}\left(T,\mathrm{MonomialOrder}\right),\mathrm{type}\left(T,\mathrm{ShortMonomialOrder}\right)$
 ${\mathrm{true}}{,}{\mathrm{false}}$ (5)

Neither a monomial order nor a short monomial order description.

 > $\mathrm{type}\left(1,\mathrm{MonomialOrder}\right),\mathrm{type}\left(1,\mathrm{ShortMonomialOrder}\right)$
 ${\mathrm{false}}{,}{\mathrm{false}}$ (6)