Abstract
Given a finite alphabet X and an ordering ≺ on the letters, the map σ ≺ sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Gröbner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal 〈 σ ≺( I)〉 generated by σ ≺( I) in the free monoid is finitely generated. Whether there exists an ≺ such that 〈 σ ≺( I)〉 is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.
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