The MonomialOrder command constructs a term order on a polynomial algebra A. This object (called a MonomialOrder) may be used with other commands in the Groebner package to do computations with modules and (left) ideals of non-commutative skew polynomials.

For commutative problems you typically do not need to construct a MonomialOrder, you can use a short monomial order description (such as plex(x,y,z)) instead. For a list of these short monomial orders, see the Monomial Orders help page.

The first argument is an algebra A defined by the Ore_algebra package.  It can be one of the following:

a commutative polynomial algebra defined by Ore_algebra[poly_algebra]

a Weyl algebra defined by Ore_algebra[diff_algebra]

a difference algebra defined by Ore_algebra[shift_algebra]

a q-difference algebra defined by Ore_algebra[qshift_algebra]

a general Ore algebra defined by Ore_algebra[skew_algebra]

The second argument should be a ShortMonomialOrder, which describes the order imposed on variables of the algebra. For example, plex(x,y,z) describes lexicographic order with x > y > z. All of the variables of the algebra A must be ordered.  For a list of short monomial orders, see the Monomial Orders help page.

The second argument can also describe a user-defined monomial order, which can have the form user(P, L) or user(P, Q, L). In both cases P is a procedure taking two monomials as arguments and returning true if and only if they are in increasing order, L is the list of indeterminates with respect to which the monomial order is defined, and (optionally) Q is a procedure for computing the leading term of a polynomial (see LeadingTerm).  The procedure Q should take a polynomial as input and return the sequence (leading coefficient, leading monomial).

The optional third argument M is a list or set of module placeholder variables. These variables must be included in the algebra A and ordered by the monomial order tord. Multiplication by elements of M is not allowed, so distinct monomials in the variables of M indicate different module components.  See the Groebner[Basis_details] help page for more information.

Due to technical limitations in the Ore_algebra package, you can not use algebras with pairs of variables that depend on each other, such as d[x] and x, to construct a MonomialOrder. In general, depends(u,v) must return false for each pair of variables $u\ne v$, otherwise MonomialOrder will return an error. Use variable names such as dx and x or d[1] and x[1] instead. This limitation does not apply to ordinary commutative computations using ShortMonomialOrders.

Note: the termorder command is deprecated.  It may not be supported in a future Maple release.

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

$\mathrm{with}\left(\mathrm{Groebner}\right)\:$

$A≔\mathrm{poly\_algebra}\left(x\,y\,z\right)$

$A≔\mathrm{Ore\_algebra}$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{plex}\left(x\,y\,z\right)\right)$

$T≔\mathrm{monomial\_order}$

$f≔-4yx-3z^2+y{z}^2-5x^4+x^3{z}^2$

$f≔x^3{z}^2-5x^4+y{z}^2-4yx-3z^2$

$\mathrm{LeadingTerm}\left(f\,T\right)$

$\mathrm{-5},x^4$

$\mathrm{LeadingTerm}\left(f\,\mathrm{plex}\left(x\,y\,z\right)\right)$

$\mathrm{-5},x^4$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{D}\left[1\right]\,x\left[1\right]\right]\,\left[\mathrm{D}\left[2\right]\,x\left[2\right]\right]\right)$

$A≔\mathrm{Ore\_algebra}$

$A\left[''polynomial\_indets''\right]$

$\left\{{\mathrm{D}}_1\,{\mathrm{D}}_2\right\}$

$A\left[''rational\_indets''\right]$

$\left\{x_1\,x_2\right\}$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(\mathrm{D}\left[1\right]\,\mathrm{D}\left[2\right]\right)\right)$

$T≔\mathrm{monomial\_order}$

$\mathrm{SPolynomial}\left(\mathrm{D}\left[1\right]x\left[2\right]-x\left[2\right]\,\mathrm{D}\left[2\right]x\left[1\right]-x\left[1\right]\,T\right)$

$\left(x_2{x}_1+x_1\right){\mathrm{D}}_1+\left(-x_2{x}_1-x_2\right){\mathrm{D}}_2-x_1+x_2$

$A≔\mathrm{poly\_algebra}\left(x\,y\,z\right)\:$

P := proc(t1, t2) global x,y,z; (degree(t1,x) < degree(t2,x)) or (degree(t1,x) = degree(t2,x) and degree(t1,y) < degree(t2,y)) or (degree(t1,x) = degree(t2,x) and degree(t1,y) = degree(t2,y) and degree(t1,z) <= degree(t2,z)) end proc:

$T≔\mathrm{MonomialOrder}\left(A\&comma;'\mathrm{user}'\left(P\&comma;\left[x\&comma;y\&comma;z\right]\right)\right)$

$T≔\mathrm{monomial\_order}$

$\mathrm{LeadingTerm}\left(f\&comma;T\right)$

$\mathrm{-5},x^4$