Formal description of multigraded lex

To specify a multigraded lex order on monic monomials in the variables x, ..., x such that if 1 i j n, then one needs to specify a subset of . If M and N are monic monomials in the variables , then M is less than N with respect to the multigraded lex order if one of the following two conditions hold:

(1)

(2) and M is less than N with respect to the graded lex order

where

(a) (we set and ) for

(b) occ(M,V) is the sum of occ(M,v) for all .

To denote the above order, we write down the sequence of characters

where is ``«'' if j is one of the 's and is ``<'' otherwise.

Note that if the collection of subsets is the empty set, then the multigraded lex order is graded lex and if the collection of subsets is {1, 2, ..., n - 1}, then the multigraded lex order is a pure lex order.

As another example, a multigraded lex order for monic monomials in a, b, c, d and e such that a<b<c<d<e can be specified by specifying the set {2, 4}. This multigraded lex order would be denoted a < b « c < d « e

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