Minimal but inefficient presentations for self semidirect products of the free abelian monoid on two generators

Let A and K both be copies of the free Abelian monoid on two generators. For any connecting monoid homomorphism theta:A -> End (K), let M = K x(theta) A be the corresponding monoid semidirect product. In Cevik ( 2005), it has been proved necessary and sufficient conditions for the efficiency of a standard presentation for M in terms of the matrix representation for theta. As a main result of this article, we give sufficient conditions for this presentation of M to be minimal but not efficient. To do that we will use the same method as given in Cevik ( 2003b).