Minimality over free monoid presentations

As a continues study of the paper [4], in here, we first state and prove thep-Cockcroft property (or, equivalently, eciency) for a presentation, say PE, of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Then, in a separate section, we present sucient conditions on a special case for PE to be minimal whilst it is inecient. 2000 AMS Classification: 20L05, 20M05, 20M15, 20M50.