View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ROS: The Research Output Service. Heriot-Watt University Edinburgh Cosets in Inverse Semigroups and Inverse Subsemigroups of Finite Index Amal AlAli SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY HERIOT-WATT UNIVERSITY DEPARTMENT OF MATHEMATICS, SCHOOL OF MATHEMATICAL AND COMPUTER SCIENCES. July 2016 The copyright in this thesis is owned by the author. Any quotation from the thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. Abstract The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the exis- tence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. Although a developed theory of cosets in inverse semigroups exists, it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finte index are much weaker than in the group case. Nevertheless, many aspects of this theory remain of interest, and some of them are addressed in this thesis. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall’s Theorem on counting subgroups of finite index in finitely generated groups. We then look at specific examples, classifying the finite index inverse subsemigroups in polycyclic monoids and in graph inverse semigroups. Finally, we look at the connection between the properties of finite generation and having finte index: these were shown to be equivalent for free inverse monoids by Margolis and Meakin. i Acknowledgements First and foremost, I would like to express my deep thanks and gratitude to my supervisor Prof. Nick Gilbert, whose expertise, guidance, understanding, and patience, added con- siderably to this thesis. This work could not have been completed without his unwavering support and encouragement. I would also like to thank my parents, to whom I owe my life for their constant love, encouragement, and blessings. I also acknowledge with deep thanks my husband, Waleed. Without his love, patience, support and belief in me, I would not have finished this thesis. Thank you to my sister and brothers for all their support and encouragement they provided me through my entire life. Very special thanks to my children for their complete and unconditional love that carries me through always. Finally, I am grateful for Princess Norah University, and the Ministry of Education; King- dom of Saudi Arabia for their financial support. ii ACADEMIC REGISTRY Research Thesis Submission Name: AMAL ALALI School/PGI: MACS Version: (i.e. First, FIRST Degree Sought PhD Mathematics Resubmission, Final) (Award and Subject area) Declaration In accordance with the appropriate regulations I hereby submit my thesis and I declare that: 1) the thesis embodies the results of my own work and has been composed by myself 2) where appropriate, I have made acknowledgement of the work of others and have made reference to work carried out in collaboration with other persons 3) the thesis is the correct version of the thesis for submission and is the same version as any electronic versions submitted*. 4) my thesis for the award referred to, deposited in the Heriot-Watt University Library, should be made available for loan or photocopying and be available via the Institutional Repository, subject to such conditions as the Librarian may require 5) I understand that as a student of the University I am required to abide by the Regulations of the University and to conform to its discipline. * Please note that it is the responsibility of the candidate to ensure that the correct version of the thesis is submitted. Signature of Date: Candidate: Submission Submitted By (name in capitals): Signature of Individual Submitting: Date Submitted: For Completion in the Student Service Centre (SSC) Received in the SSC by (name in capitals): Method of Submission (Handed in to SSC; posted through internal/external mail): E-thesis Submitted (mandatory for final theses) Signature: Date: Please note this form should bound into the submitted thesis. Updated February 2008, November 2008, February 2009, January 2011 iii Contents 1 Introduction 1 1.1 Regular and Inverse Semigroups . 3 1.2 Properties of Inverses . 4 1.3 The Natural Partial Order . 5 1.4 Compatibility Relations . 11 1.5 Congruence . 12 1.6 The Minimum Group Congruence . 13 1.7 Green’s Relations . 14 1.8 Clifford Semigroups . 17 1.9 Symmetric Inverse Monoid . 19 1.10 Inverse Semigroups and Inductive Groupoids . 20 1.11 Free Monoids and Free Inverse Monoids . 22 2 Cosets in Inverse Semigroups 25 2.1 Group Cosets and Actions . 25 2.2 Closed Inverse Subsemigroups and Cosets . 27 2.3 The Index Formula for Inverse Semigroups . 36 2.4 Actions on Cosets in Inverse Semigroups . 42 2.5 An Inverse Semigroup Version of M. Hall’s Theorem . 45 iv CONTENTS 3 Cosets in the Bicyclic Monoid 48 3.1 Definitions and Preliminaries . 48 3.2 Basic Properties . 49 3.3 Closed Inverse Subsemigroups of B ..................... 53 3.4 Generating Sets for Finite Index Inverse Subsemigroups of The Bicyclic Monoid . 62 3.5 Bruck-Reilly Semigroups . 68 4 Polycyclic Monoids and Graph Inverse Semigroups 75 4.1 Closed Inverse Submonoids of the Polycyclic Monoids . 75 4.2 Closed Inverse Subsemigroups of Graph Inverse Semigroups . 88 5 Finite Index and Finite Generation 103 5.1 Preliminaries . 104 5.2 Finite Index implies Finite Generation . 109 5.3 Recognizable Closed Inverse Submonoids . 114 5.4 Rational Generation . 115 5.5 Concordance . 118 Bibliography 122 v Chapter 1 Introduction It has long been recognized that the origin and development of the theory of inverse semi- groups are due to the substantial contributions of two prime foci: the Soviet and Western schools, led by Wagner and Schein in the Soviet school, and Preston and Munn in the school of the West. The concept of inverse semigroups was defined first by Wagner [31] in 1952 and called ”generalized groups” and he and some of his followers have used this term ever since. The term ”inverse semi-groups” was introduced by Preston [21] who independently discovered this class of semigroups in 1954. Inverse semigroups are the mathematical structures which can provide a full description of the concept of partial symmetries. In this sense, inverse semigroups are the generalizations of the idea of transforming symmetries in natural forms into the mathematical precise no- tion of groups. There has been a large body of literature concerning the structure of inverse semigroups. The reason for this is probably due to the simple and beautifully constructed theorems for different classes of inverse semigroups. Therefore, it is not surprising that the subject has attracted the attention of many researchers. Many structure theorems and concepts for various classes of semigroups have been formu- lated through the years which give the subject a broad spectrum of various applications and connections with other branches of mathematics. In fact, inverse semigroups recently have wide applications in many aspects of modern sciences including: geometric group theory, combinatorial group theory, model theory, linear logic, solid state physics and C∗- algebras. 1 Chapter 1: Introduction It is worth noting that the similarities between groups and inverse semigroups are not as substantial as it might appear at first sight. However, there is a significant analogy between the two theories. Inverse semigroups represent the concept of sets of partial symmetries just as groups play the same role for groups of symmetries. Consequently, there has been a tendency among many researchers in the field of semigroups to model structural theorems on those in group theory. Even though this approach has had a limited success, groups still play a significant role in many important theorems for different classes of inverse semigroups. In the present research, the process of generalizing some notions and group theoretical approaches to semigroups, in particular inverse semigroups, is considered in a way that is similar in spirit to the group-theoretic formalization. Nevertheless, there are some significant differences between the two theories. Among them is the natural partial order that exists in each inverse semigroup. Moreover, the basic theo- retical structure of semigroups is very different from that of groups: for example, a congru- ence on a semigroup is not generally determined by one congruence class, as is the case in groups. For further background information on the general theory of inverse semigroups, see [11, 14, 20] 2 Chapter 1: Introduction 1.1 Regular and Inverse Semigroups A semigroup is defined as a non-empty set S on which an associative binary operation is defined, by which we mean a map f : S × S −! S written (x; y) 7! xy such that for all x; y; z 2 S; we have (xy)z = x(yz). If a semigroup has an identity element 1 2 S; then S is called a monoid. A semigroup has at most one identity element. If a semigroup S has no identity element, then we can adjoin an extra element 1 to S to form a monoid. We define 1s = s1 = s for all s 2 S; and 11 = 1; and then S [ f1g becomes a monoid. Thus we now can define 8 > <>S if S has an identity element ; S1 = > :>S [ f1g otherwise . If a semigroup S has the property that, for all x; y 2 S; xy = yx; we shall say that S is a commutative semigroup. An element x of a semigroup S is called regular, if there exists y 2 S such that x = x y x: The semigroup S is called a regular semigroup if all its elements are regular.