Minimal but inefficient presentations of the semi-direct products of some monoids

Abstract

In recent papers of Ruskuc, Saito and J. Wang, the semi-direct product of two arbitrary monoids and a standard presentation, say P, for this product have received considerable attention. Wang defined a trivialiser set of the Squier complex associated with P and after that necessary and sufficient conditions for P to be efficient have been given by Cevik. As a main result of this paper, we give sufficient conditions for a presentation of the semi-direct product of a one-relator monoid by an infinite cyclic monoid to be minimal but not efficient. In the final part of this paper we give some applications of this result.