‘A countable family of congruence-free finitely presented monoids’
[with F. Al-Kharousi, V. Maltcev & A. Umar]
Acta Universitatis Szegediensis: Acta Scientiarum Mathematicarum, 81, no. 3–4 (2015), pp. 437–445.
DOI: 10.14232/actasm-013-028-z. MR: 3443762. ZBL: 1363.20044.

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 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.