On Finite Semigroup Cross-Sections and Complete Rewriting Systems

Abstract

In this paper we obtain a [finite] complete rewriting system defining a semigroup/monoid S, from a given finite right cross-section of a subsemigroup/submonoid defined by a [finite] complete presentation. In the semigroup case the subsemigroup must have a right identity element which must also be part of the cross-section. In the monoid case the submonoid and the cross-section must include the identity of the semigroup. The result on semigroups allow us to show that if G is a group defined by a [finite] complete rewriting system then the completely simple semigroup M[G;I,J;P] is also defined by a [finite] complete rewriting system.