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

TL;DR

This paper proves that a class of monoids defined by specific relations admits finite complete rewriting systems and provides an example of a finitely presented non-residually finite monoid with a linear Dehn function.

Contribution

It establishes the existence of finite complete rewriting systems for a family of monoids and presents a non-hopfian example with particular properties.

Findings

Monoids $ ext{Mon} angle a,b:a^{ ext{alpha}}b^{ ext{beta}}a^{ ext{gamma}}b^{ ext{delta}}=b$ admit finite complete rewriting systems.

The monoid $ ext{Mon} angle a,b:ab^2a^2b^2=b$ is non-hopfian.

The monoid with the given relation has a linear Dehn function.

Abstract

We prove that every monoid $\mathrm{Mon}\langle a,b:a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}=b\rangle$ admits a finite complete rewriting system. Furthermore we prove that $\mathrm{Mon}\langle a,b:ab^2a^2b^2=b\rangle$ is non-hopfian, providing an example of a finitely presented non-residually finite monoid with linear Dehn function.