Monoids $\mathrm{Mon}\langle a,b:a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}a^{\varepsilon}b^{\varphi}=b\rangle$ admit finite complete rewriting systems

TL;DR

This paper proves that certain one-relator monoids with a specific relator length admit finite complete rewriting systems, advancing the understanding of their word problem solvability.

Contribution

It demonstrates that monoids with relators of length 6 have finite complete rewriting systems, contributing to the open problem of word problem solvability for one-relator monoids.

Findings

Monoids with relator length 6 admit finite complete rewriting systems.

Progress on the open problem of word problem for one-relator monoids.

Enhanced understanding of the structure of specific one-relator monoids.

Abstract

The aim of this note is to prove that monoids $\mathrm{Mon}\langle a,b:aUb=b\rangle$, with $aUb$ of relative length 6, admit finite complete rewriting systems. This is some advance in the understanding the long-standing open problem whether the word problem for one-relator monoids is soluble.