The Word Problem for $(\omega -1)$-Terms over $\mathrm{D}\mathrm{A}\mathrm{b}$

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