Minimality over free monoid presentations

As a continues study of the paper [4], in here, we first state and prove the p-Cockcroft property insert ignore into journalissuearticles values(or, equivalently, efficiency); 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 sufficient conditions on a special case for PE to be minimal whilst it is inefficient.