Minimal but inefficient presentations for semi-direct products of finite cyclic monoids

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Let A and K be arbitrary two monoids. For any connecting monoid homomorphism θ: A → End(K), let M = K ⋊ θ A be the corresponding monoid semi-direct product. In [2], Cevik discussed necessary and sufficient conditions for the standard presentation of M to be efficient (or, equivalently, p -Cockcroft for any prime p or 0), and then, as an application of this, he showed the efficiency for the presentation, say P M , of the semi-direct product of any two finite cyclic monoids. As a main result of this paper, we give sufficient conditions for P M to be minimal but not efficient. To do that we will use the same method as given in [3]. AMS Classification : 20L05, 20M05, 20M15, 20M50, 20M99. Keywords: Minimality, Efficiency, p -Cockcroft property, Finite cyclic monoids. Introduction Let P = [X ; r] be a monoid presentation where a typical element R ∈ r has the form R + = R − . Here R + , R − are words on X (that is, elements of the free monoid F(X) on X ). The monoid defined by [ X ; r] is the quotient of F(X) by the smallest congruence generated by r. We have a (Squier) graph Γ = Γ ( X ; r) associated with [ X ; r], where the vertices are the elements of F(X) and the edges are the 4-tuples e = ( U,R, ε, V ) where U, V ∈ F(X) , R ∈ r and ε = ±1.