Computing the maximal subsemigroups of a finite semigroup Wilf Wilson 25th April 2014 1 1 * * 1 2 2 1 * * * * * * 2 * * 2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4 4 4 4 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Semigroup diagrams of the four maximal subsemigroups of the partition monoid of degree 3, B3, which do not arise from maximal subgroups of the group of units. See Figure1 and Section 7.3 for more details. For Alan, my little brother. 2 Abstract We characterise the maximal subsemigroups of some special classes of finite semigroup. This then leads us to describe novel algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup. These algorithms have now been implemented in the Semigroups package [8] for GAP [6]. We conclude by detailing the maximal subsemigroups of some specific semigroups, guided by these algorithms. Acknowledgements The original mathematics and implementation of the algorithms discussed in this project was part of a group project. The team consisted, at various points, of Casey Donoven, Robert Hancock, Julius Jonušas, Dr James Mitchell, Dr Yann Péresse, and myself. I enjoyed working with you all. However, the work on this project report is mine alone. I would like to thank Dr Mitchell in particular for his encouraging supervision. 3 Contents 1 Introduction 1 1.1 Intended readership........................................ 1 1.2 Motivation and aims....................................... 1 1.3 Overview of project........................................ 1 1.4 Notation and conventions .................................... 2 1.5 Pre-requisites for understanding................................. 2 2 Graham, Graham and Rhodes6 2.1 A correction............................................ 11 3 Maximal subsemigroups of special classes of finite semigroup 12 3.1 Groups............................................... 12 3.2 Left-zero, right-zero, and other degenerate semigroups .................... 12 3.3 Commutative semigroups .................................... 13 3.4 Commutative bands ....................................... 13 3.5 Monogenic semigroups...................................... 14 3.6 0-simple semigroups ....................................... 15 3.7 Simple semigroups ........................................ 18 3.8 Regular semigroups........................................ 19 4 Description of algorithms for Rees 0-matrix semigroups and Rees matrix semigroups 20 4.1 Algorithm for Rees 0-matrix semigroups............................ 20 4.2 Algorithm for Rees matrix semigroups ............................. 29 5 Description and justification of algorithms for arbitrary semigroups 30 5.1 Compute the maximal subsemigroups which lack a maximal J -class............ 31 5.2 Compute the maximal subsemigroups which lack a non-maximal J -class ......... 33 6 Notes about algorithms 39 6.1 Possible improvements...................................... 39 7 Maximal subsemigroups of specific finite semigroups 40 7.1 Full transformation monoid Tn ................................. 40 7.2 Symmetric inverse monoid In .................................. 41 7.3 Partition monoid Bn ....................................... 41 References 43 Note that many of the diagrams contained in this project convey information through the use of colour. For this reason, it is recommended that this report be read in colour. 4 1 Introduction In this project, we address the question: given a finite semigroup, what are its maximal subsemigroups? Semigroups are renowned for their unrestrictive definition and large number. For example, there are 12,418,001,077,381,302,684 non-equivalent (where equivalence is up to isomorphism or anti-isomorphism) semigroups of order 10 [2]. Only 2 of these are groups. This suggests that answering a question about arbitrary semigroups, especially when considered computationally, can be much more difficult than an- swering an analogous question for groups. Indeed, we will see that in order to answer this question for arbitrary semigroups, we rely on it having already been answered for groups. However, the inherent diversity in semigroup theory does not translate to a complete lack of structure. We find there is much structure present in an arbitrary semigroup, and we will use this to help us answer the question in a practical way. 1.1 Intended readership Many parts of this project are highly technical, and so an understanding of semigroup theory at an advanced undergraduate honours level is required. In particular, we assume that the reader is familiar with all of the topics covered in the module MT5823 Semigroups, which is taught at the University of St Andrews. A familiarity with basic graph theory is also required for several sections of this project. Beyond this level, additional important definitions and results are stated in Section 1.5. We hope that the following pages are both instructive and intriguing to any suitably qualified reader. 1.2 Motivation and aims The maximal subsemigroups of a semigroup are an interesting attribute in their own right. However, being able to know the maximal subsemigroups of a semigroup allows us to answer further questions. For example, it allows us to calculate every subsemigroup of a finite semigroup: we first calculate the maximal subsemigroups, then the maximal subsemigroups of those subsemigroups, and so on, until the process necessarily terminates. Since we are not aware of any previously existing algorithm for maximal subsemigroups, by creating a working algorithm, we are providing a useful tool for other researchers in the field. This is also a demonstration of the power and utility of computational algebra; in particular of the GAP system [6] and the Semigroups package [8] for GAP. We hope the reader will gain a strong understanding of the topic of maximal subsemigroups, and the included algorithms - and perhaps be inspired to explore related topics in computational semigroup theory. 1.3 Overview of project Much of this project is based on the results of Maximal subsemigroups of finite semigroups by Graham, Graham and Rhodes [5]. This paper describes the possible forms of a maximal subsemigroup of a finite semigroup. Section2 is particularly technical, and is entirely devoted entirely to presenting and proving the main results of Graham et al., which we will use throughout the rest of this project. Guided by the results of Section2, we describe the maximal subsemigroups of some special classes of finite semigroup in Section3, such as groups (3.1) and monogenic semigroups (3.5). Next, we describe novel and practical algorithms for calculating maximal subsemigroups, for which we provide pseudocode. Section4 contains algorithms to compute the maximal subsemigroups of a finite regular Rees 0-matrix semigroup over a group, and as a consequence, a finite Rees matrix semigroup over a group. We use these algorithms, along with further results from [5], to describe an algorithm to calculate the maximal subsemigroups of an arbitrary finite semigroup in Section5. We briefly comment on these algorithms in Section6, and suggest future improvements. 1 Finally, in Section7, aided by these algorithms, we calculate the maximal subsemigroups of particularly famous semigroups, such as the full transformation monoid (7.1) and the symmetric inverse monoid (7.2). 1.4 Notation and conventions • S denotes an arbitrary semigroup (S; ∗); we require that S be non-empty. • If the semigroup S is a monoid, then we define S1 to be the semigroup S. If the semigroup S is not a monoid, then we define S1 to be the monoid formed by adjoining the identity 1 to S. • M denotes a maximal subsemigroup of the semigroup S. • For an element a 2 S, Ja denotes the J -class of the element a, i.e. the equivalence class a=J . To S avoid ambiguity when considering multiple semigroups, we write Ja to denote the J -class of the element a in the semigroup S. The other Green’s relations are handled in the same way. • When there is no possibility for confusion, we occasionally omit brackets, for example writing hX; x; yi to denote the semigroup hX [ fx; ygi. • Mappings are composed from left to right. • N denotes the set of natural numbers without zero: f1; 2; 3; :::g. • N0 denotes the set of natural numbers with zero: f0; 1; 2; 3; :::g. 1 2 3 4 5 • When writing transformations we list only the image; for example, the transformation 4 3 5 1 4 is written as (43514). However, when specifically considering a group of permutations, we write in the usual disjoint cycle notation - we shall make clear when this is the case (such as in Example 4.2). 1.5 Pre-requisites for understanding As stated in Section 1.1, it is assumed that the reader is familiar with the basics of semigroup theory, especially with the definitions of subsemigroups, groups, and ideals; with Green’s relations; and with the notion