Rewriting systems for some semigroup constructions

Whether some important algebraic constructions in semigroup theory has a finite complete rewriting system in which conditions is investigated. Let S be a semigroup and let T be a a large subsemigroup of S. It is shown that T has a finite complete rewriting system if and only if S has a finite complete rewriting system. Let S be an ideal extension of T by U, it is shown that if T and U are presented by finite complete rewriting systems then S is presented by a finite complete rewriting system. Let M be a monoid and let ρ be congruece on M as a submonoid of M×M, finally it is shown that if ρ has a finite complete rewriting system then so do M and M/ρ.