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

Summary

Abstract

Let A and K be arbitrary two monoids. For any connecting monoid homomorphism θ: A → End(K), let M = K ⋊_θA 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.