Efficiency for self semi-direct 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 D = K x (theta) A be the corresponding moncid semi-direct product. We give necessary and sufficient conditions for the efficiency of a standard presentation for D in terms of the matrix representation for theta. Let p be a prime or 0. In [4], necessary and sufficient conditions were given for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft. We use that result to give more explicit conditions in the special case here.