The word problem for nilpotent inverse monoids (Q1902161)

The author says that an inverse semigroup \(S\) is \(k\)-nilpotent if \(S\) has a zero and \(a^k=0\) for every non-idempotent \(a\in S\). Let \(N_k(X)\) denote the largest \(k\)-nilpotent Rees quotient of the free inverse monoid on the set \(X\). The author proves that, for every \(k\geq 1\) and for every finitely generated congruence \(\tau\) on \(N_k(X)\), the quotient monoid \(N_k(X)/\tau\) has decidable word problem and finite \({\mathcal D}\)-classes.