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

Abstract

Let 𝑀 be a monoid and 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, an introductory lemma is proposed, proved and a relevant example is given. Then, it is shown that if 𝜌 can be presented by a finite complete rewriting system, then so can 𝑀. As the final part of the main result, it is proved that if 𝜌 can be presented by a finite complete rewriting system, then so can 𝑀/𝜌.