The TestOrder command compares monomials with respect to a monomial order. It returns true if and only if u is less than or equal to v.

The third argument T can be either a MonomialOrder or a ShortMonomialOrder description. See the Monomial Orders help page for a list of the available monomial orders.

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

M := [op(randpoly([x,y,z],degree=3,dense,coeffs=proc() 1 end))];

$M≔\left[x^3\,x^{2}y\,x^{2}z\,x{y}^2\,xyz\,x{z}^2\,y^3\,y^{2}z\,y{z}^2\,z^3\,x^2\,xy\,xz\,y^2\,yz\,z^2\,x\,y\,z\,1\right]$

$\mathrm{sort}\left(M\,\left(a\,b\right)\mapsto \mathrm{TestOrder}\left(a\,b\,\mathrm{plex}\left(x\,y\,z\right)\right)\right)$

$\left[1\,z\,z^2\,z^3\,y\,yz\,y{z}^2\,y^2\,y^{2}z\,y^3\,x\,xz\,x{z}^2\,xy\,xyz\,x{y}^2\,x^2\,x^{2}z\,x^{2}y\,x^3\right]$

$\mathrm{sort}\left(M\,\left(a\,b\right)\mapsto \mathrm{TestOrder}\left(a\,b\,\mathrm{grlex}\left(x\,y\,z\right)\right)\right)$

$\left[1\,z\,y\,x\,z^2\,yz\,y^2\,xz\,xy\,x^2\,z^3\,y{z}^2\,y^{2}z\,y^3\,x{z}^2\,xyz\,x{y}^2\,x^{2}z\,x^{2}y\,x^3\right]$

$\mathrm{sort}\left(M\,\left(a\,b\right)\mapsto \mathrm{TestOrder}\left(a\,b\,\mathrm{tdeg}\left(x\,y\,z\right)\right)\right)$

$\left[1\,z\,y\,x\,z^2\,yz\,xz\,y^2\,xy\,x^2\,z^3\,y{z}^2\,x{z}^2\,y^{2}z\,xyz\,x^{2}z\,y^3\,x{y}^2\,x^{2}y\,x^3\right]$