On finite complete rewriting systems and large subsemigroups

TL;DR

This paper proves that if a semigroup has a finite complete rewriting system, then any large subsemigroup of it also has one, establishing a two-way relationship between the properties of the semigroup and its large subsemigroups.

Contribution

It demonstrates the converse of a known result, showing that finite complete rewriting systems are preserved when passing to large subsemigroups, with a purely combinatorial and constructive proof.

Findings

If $S$ has a finite complete rewriting system, then so does any large subsemigroup $T$.

The proof is purely combinatorial and constructive.

Establishes a two-way relationship between semigroups and their large subsemigroups regarding rewriting systems.

Abstract

Let $S$ be a semigroup and $T$ be a subsemigroup of finite index in $S$ (that is, the set $S\setminus T$ is finite). The subsemigroup $T$ is also called a large subsemigroup of $S$. It is well known that if $T$ has a finite complete rewriting system then so does $S$. In this paper, we will prove the converse, that is, if $S$ has a finite complete rewriting system then so does $T$. Our proof is purely combinatorial and also constructive.