A countable family of finitely presented infinite congruence-free monoids

TL;DR

This paper introduces a countable family of finitely presented monoids that are congruence-free, advancing understanding of the Boone--Higman Conjecture and illustrating that such monoids can have quadratic Dehn functions.

Contribution

It constructs a new countable family of finitely presented congruence-free monoids, contributing to the study of algebraic structures related to the Boone--Higman Conjecture.

Findings

Monoids are congruence-free for all n ≥ 1.

Provides examples of finitely presented congruence-free monoids with quadratic Dehn function.

Advances understanding of the structure of congruence-free monoids.

Abstract

We prove that monoids $\mathrm{Mon}\langle a,b,c,d : a^nb=0, ac=1, db=1, dc=1, dab=1, da^2b=1, \ldots, da^{n-1}b=1\rangle$ are congruence-free for all $n\geq 1$. This provides a new countable family of finitely presented congruence-free monoids, bringing one step closer to understanding the Boone--Higman Conjecture. We also provide examples which show that finitely presented congruence-free monoids may have quadratic Dehn function.