On finite complete rewriting systems and large subsemigroups

Let S be a semigroup and T be a subsemigroup of finite index in S (that is, the set S∖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.