Finite Complete Rewriting Systems for the Monoids M, ρ, and M/ρ

Abstract

In this study, we take M to be a monoid and we let ρ be an equivalence relation on M such that ρ is a congruence. So, ρ is a submonoid of the direct product of monoids M×M, and M/ρ={xρ:x∈M} is a monoid with the operation (xρ)(yρ)=(xy)ρ. First, we propose and prove an introductory lemma and we give a relevant example. Then, we show that if ρ can be presented by a finite complete rewriting system, then so can M. As the final part of our main result, we prove that if ρ can be presented by a finite complete rewriting system, then so can M/ρ.