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

DOI: 10.14232/actasm-013-028-z

AMS Subject Classification (1991): 20M05, 20M10

Keyword(s): Boone-Higman Conjecture, congruence-free, finitely presented, rewriting systems

Received May 1, 2013, and in revised form May 21, 2015. (Registered under 28/2013.)