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B.2.7 Product orderings

Let $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_m)$be two ordered sets of variables, $<_1$ a monomial ordering on $K[x]$ and $<_2$ a monomial ordering on $K[y]$. The product ordering (or block ordering) $<\ := (<_1,<_2)$ on $K[x,y]$ is the following:
         $x^a y^b < x^A y^B \Leftrightarrow x^a <_1 x^A $ or ($x^a = x^A$ and $y^b <_2 y^B$).

Inductively one defines the product ordering of more than two monomial orderings.

In SINGULAR, any of the above global orderings, local orderings or matrix orderings may be combined (in an arbitrary manner and length) to a product ordering. E.g., (lp(3), M(1, 2, 3, 1, 1, 1, 1, 0, 0), ds(4), ws(1,2,3)) defines: lp on the first 3 variables, the matrix ordering M(1, 2, 3, 1, 1, 1, 1, 0, 0) on the next 3 variables, ds on the next 4 variables and ws(1,2,3) on the last 3 variables.

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