Minimality of Group and Monoid Presentations

[摘要] In Chapter 1 of this thesis we review existing theory concerning group and monoid presentations, and the concept of pictures over these. We also recall aspherical, combi-natorial aspherical, n-Cockcroft (n ∈ Z+), efficient and inefficient presentations. Minimality is the final concept introduced in this chapter: we present an important theorem, due to Lustig in the case of groups and to Pride for monoids. In Chapter 2 we prove necessary and sufficient conditions for the presentation of the central extension to be p-Cockcroft (p a prime or 0). The starting point of this result is the joint paper of Baik-Harlander-Pride. We end the chapter by giving some examples. In Chapter 3 we prove a theorem on the efficiency of standard wreath products of two finite groups. We also present some applications of the theorem and end by giving examples. Chapter 4 sees discussion on the semi-direct product of any two monoids. In particular we prove necessary and sufficient conditions for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft (p a prime or 0). We end by giving some applications of this theorem to the direct product of two monoids and the semi-direct product of two finite cyclic monoids. We begin Chapter 5 with an application of the main theorem of Chapter 4, namely we give necessary and sufficient conditions for a presentation of the semi-direct product of a one-relator monoid by an infinite cyclic monoid to be p-Cockcroft (p a prime or 0), and give some examples of this. Following this we present the main theorem of this chapter, which is sufficient conditions for the presentation of a semi-direct product of a one-relator monoid by an infinite cyclic monoid to be minimal but inefficient. We end by giving some examples.

[发布机构] University:University of Glasgow

[关键词] Mathematics