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

TL;DR

This paper develops a combinatorial normal form and an algorithm to solve the word problem for $(0)$-terms over the variety of finite monoids with Abelian group regular classes, using finite-index congruences.

Contribution

It introduces a ranker-based description and a computable normal form for pseudowords over the variety 0, enabling an effective solution to the word problem for these terms.

Findings

Provides a normal form for pseudowords over 0.

Develops a computable algorithm for the word problem for $(0)$-terms.

Establishes a finite-index congruence characterization for 0.

Abstract

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