The Word Problem for $(\ddot{I}‰-1)$-Terms over $\mathrm{DAb}$

Abstract

We give a ranker-based description using finite-index congruences for the variety $\mathrm{DAb}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over $\mathrm{DAb}$. For $(\ddot{I}‰-1)$-terms, this normal form is computable, which yields an algorithm for the word problem for $(\ddot{I}‰-1)$-terms of $\mathrm{DAb}$.