From monomials to words to graphs

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.