A monoid is said to be efficient if it admits an efficient presentation. If some monoid has a presentation that is minimal but not efficient, then this monoid is called inefficient. The results of this paper help to find examples of inefficient monoids.
The paper studies presentations of the semidirect product of one-relator monoids by the infinite cyclic monoid. It gives necessary and sufficient conditions for these presentations to be minimal but not efficient, and shows how this result can be applied in several situations. The author makes extensive use of the \(p\)-Cockcroft property of monoid presentations [see S. J. Pride, Int. J. Algebra Comput. 5, No. 6, 631-649 (1995; Zbl 0838.20075)], and the equivalence between this property and efficiency.
As for the style, the rigor and detail of some definitions contrasts with the slackness shown in others. This fact, combined with a couple of English mistakes, makes the paper a little hard to read, which does not stain its mathematical value.