A countable family of finitely presented infinite congruence-free monoids

Abstract

We prove that monoids $\mathrm{Mon}⟨a,b,c,d:a^{n}b=0,ac=1,db=1,dc=1,dab=1,d{a}^{2}b=1,\dots ,d{a}^{n-1}b=1⟩$ are congruence-free for all $n\ge 1$. 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.

7 pages