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Let $S$ be a semigroup and $T$ be a subsemigroup of finite index in $S$ (thatis, the set $S\setminus T$ is finite). The subsemigroup $T$ is also called alarge subsemigroup of $S$. It is well known that if $T$ has a finite completerewriting 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 $T$. Ourproof is purely combinatorial and also constructive.

On finite complete rewriting systems and large subsemigroups