On the word problem for weakly compressible monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-AFL , as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.