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Nine Chapters on the Semigroup Art Alan J. Cain Nine Chapters on the Semigroup Art Lecture notes for a tour through semigroups Porto & Lisbon 2020 version 0.66.62 (2020-06-13) [A4 / One-sided / Colour] 3b9596955641bc1af1e88645064dd8dc6c2cf041 : 000000005ee533ff To download the most recent version, and files suitable for colour or greyscale printing, or for viewing on tablets and ebook readers, visit http://www-groups.mcs.st-andrews.ac.uk/~alanc/pub/c_ semigroups/ • © 2012–20 Alan J. Cain ([email protected]) This work is licensed under the Creative Commons Attribu- tion–Non-Commercial–NoDerivs 4.0 International Licence. To view a copy of this licence, visit https://creativecommons.org/licenses/by-nc-nd/4.0/ Contents Preface v Prerequisites vii ◆ Acknowledgements vii Chapter 1 | Elementary semigroup theory 1 Basic concepts and examples 1 ◆ Generators and subsemi- groups 8 ◆ Binary relations 11 ◆ Orders and lattices 15 ◆ Homomorphisms 19 ◆ Congruences and quotients 20 ◆ Generating equivalences and congruences 22 ◆ Subdirect products 28 ◆ Actions 29 ◆ Cayley graphs 30 ◆ Exercises 32 ◆ Notes 34 Chapter 2 | Free semigroups & presentations 36 Alphabets and words 36 ◆ Universal property 37 ◆ Proper- ties of free semigroups 40 ◆ Semigroup presentations 41 ◆ Exercises 50 ◆ Notes 52 Chapter 3 | Structure of semigroups 53 Green’s relations 53 ◆ Simple and 0-simple semigroups 56 ◆ D-class structure 58 ◆ Inverses and D-classes 61 ◆ Schützenberger groups 63 ◆ Exercises 66 ◆ Notes 68 Chapter 4 | Regular semigroups 70 Completely 0-simple semigroups 72 ◆ Ideals and completely 0-simple semigroups 78 ◆ Completely simple semigroups 79 ◆ Completely regular semigroups 81 ◆ Left and right groups 83 ◆ Homomorphisms 85 ◆ Exercises 86 ◆ Notes 88 Chapter 5 | Inverse semigroups 90 Equivalent characterizations 90 ◆ Vagner–Preston theo- rem 94 ◆ The natural partial order 97 ◆ Clifford semi- groups 99 ◆ Free inverse semigroups 103 ◆ Exercises 113 ◆ Notes 116 Chapter 6 | Commutative semigroups 117 Cancellative commutative semigroups 117 ◆ Free commut- ative semigroups 119 ◆ Rédei’s theorem 121 ◆ Exercises 124 ◆ Notes 125 • iii Chapter 7 | Finite semigroups 126 Green’s relations and ideals 126 ◆ Semidirect and wreath products 128 ◆ Division 130 ◆ Krohn–Rhodes decomposition theorem 135 ◆ Exercises 142 ◆ Notes 143 Chapter 8 | Varieties & pseudovarieties 144 Varieties 144 ◆ Pseudovarieties 152 ◆ Pseudovarieties of semigroups and monoids 154 ◆ Free objects for pseudovari- eties 156 ◆ Projective limits 157 ◆ Pro-V semigroups 159 ◆ Pseudoidentities 162 ◆ Semidirect products of pseudovarie- ties 167 ◆ Exercises 168 ◆ Notes 169 Chapter 9 | Automata & finite semigroups 170 Finite automata and rational languages 170 ◆ Syntactic sem- igroups and monoids 179 ◆ Eilenberg correspondence 183 ◆ Schützenberger’s theorem 190 ◆ Exercises 196 ◆ Notes 197 Solutions to exercises 198 Bibliography 246 Index 250 List of Tables Table 8.1 Varieties of semigroups 151 Table 8.2 Varieties of monoids 151 Table 8.3 Varieties of semigroups with a unary operation −1 152 Table 8.4 S-pseudovarieties of semigroups 164 Table 8.5 M-pseudovarieties of monoids 164 Table 9.3 Varieties of rational ∗-languages 188 Table 9.4 Varieties of rational +-languages 188 • • iv Preface A preface is frequently a superior composition to the work itself ‘ — Isaac D’Israeli,’ ‘Prefaces’. In: Curiosities of Literature. • This course is a tour through selected areas of semi- group theory. There are essentially three parts: ◆ Chapters 1–3 study general semigroups, including presentations for semigroups and basic structure theory. ◆ Chapters 4–6 examine special classes: namely regular, inverse, and commutative semigroups. ◆ Chapters 7–9 study finite semigroups, their classification using pseu- dovarieties, and connections with the theory of automata and regular languages. The course is broad rather than deep. It is not intended to be comprehens- ive: it does not try to study (for instance) structure theory as deeply as Howie, Fundamentals of Semigroup Theory, pseudovarieties as deeply as Almeida, Finite Semigroups and Universal Algebra, or languages as deeply as Pin, Varieties of Formal Languages; rather, it samples highlights from each area. It should be emphasized that there is very little that is original in this course. It is heavily based on the treatments in these and other standard textbooks, as the bibliographic notes in each chapter make clear. The main novelty is in the selection and arrangement of material, the slightly slower pace, and the general policy of avoiding leaving proofs to the reader when the corresponding results are required for later proofs. Figure P.1 shows the dependencies between the chapters. At the end of each chapter, there are a number of exercises, intended to reinforce concepts introduced in the chapter, and also to explore some related topics that are not covered in the main text. The most important exercises are marked with a star ✴ . Solutions are supplied for all exercises. At the end of each chapter are bibliographic notes, which give sources and suggestions for further reading. Warnings against potential misunderstandings are marked (like this) with a ‘dangerous bend’ symbol, as per Bourbaki or Knuth. Important observations that are not potential misunderstandings per se are marked with an ‘exclamation’ symbol (like this). This course was originally delivered to master’s students at the Uni- • v Chapter 1 Elementary semigroup theory Chapter 2 Free semigroups & presentations Chapter 3 Structure of semigroups Chapter 4 Regular semigroups Chapter 5 Inverse semigroups Chapter 6 Commutative semigroups Chapter 7 Finite semigroups Chapter 8 FIGURE P.1 Varieties & pseudovarieties Chart of the dependencies between the chapters. Dotted arrows indicate that the Chapter 9 dependency is only in the Automata & finite semigroups exercises, not in the main text. versities of Porto and Santiago de Compostella. The course was covered during 56 hours of classes, which included lectures and discussions of the exercises. Revisions have increased the length of the notes, and about 70 hours of class time would now be required to cover them fully. These notes were heavily revised in 2013–15. Most of the main text is now stable, but Chapter 8 will be further revised, and further exercises will be added. At present, the index is limited to names and ‘named results’ only. There may be minor typesetting problems that arise from the ‘in-development’ status of the LuaLaTEX software and many of the required packages. The author welcomes any corrections, observations, or constructive criticisms; please send them to the email address on the copyright page. Preface • vi Prerequisites There are few formal prerequisites: general mathematical maturity is the main one. An understanding of the most basic concepts from elementary group theory is assumed, such as the definition of groups, cosets, and factor groups. Some knowledge of linear algebra will help with understanding certain examples, but is not vital. For Chapters 1 and 5, knowledge of the basic definitions of graph theory is assumed. Some basic topology is necessary to appreciate part of Chapter 8 fully (although most of the chapter can be understood without it, and the relevant sections can simply be skipped), and some background in universal algebra is useful, but not essential. For Chapter 9, some experience with formal language theory and automata is useful, but again not essential. Acknowledgements Attila Egri-Nagy, Darij Grinberg, Akihiko Koga, and Guil- herme Rito made valuable suggestions and indicated various errors. Some exercises were suggested by Victor Maltcev. Typos were pointed out by Nick Ham, Samuel Herman, José Manuel dos Santos dos Santos, and Alexandre Trocado. Many improvements are due to the students who took the first version of this course: Miguel Couto, Xabier García, and Jorge Soares. The imperfections that remain are my responsibility. The title alludes to 九章算術 (Jiuzhāngˇ Suànshù), Nine Chapters on the Mathematical Art. A. J. C. • Acknowledgements • vii Elementary semigroup theory 1 I use the word “elementary” in the sense ‘ in which professional mathematicians use it — G.H. Hardy, A Mathematician’s’ Apology, § 21. • A binary operation ∘ on a set 푆 is a map ∘ ∶ 푆 × 푆 → 푆. Binary operation This operation is associative if 푥 ∘ (푦 ∘ 푧) = (푥 ∘ 푦) ∘ 푧 for all elements 푥, 푦, 푧 ∈ 푆.A semigroup is a non-empty set equipped with an associative Semigroup binary operation. Semigroups are therefore one of the most basic types of algebraic structure. We could weaken the definition further by removing the as- sociativity condition and requiring only a binary operation on a set. A structure that satisfies this weaker condition is called a magma or group- oid. (These ‘groupoids’ are different from the category-theoretic notion of groupoid.) On the other hand, we can strengthen the definition by requiring an identity and inverses. Structures satisfying this stronger condition are of course groups. However, there are many more semigroups than groups. For instance, there are 5 essentially different groups with 8 elements (the cyclic group 퐶8, the direct products 퐶4 × 퐶2 and 퐶2 × 퐶2 × 퐶2, the dihed- ral group 퐷4, and the quaternion group 푄8), but there are 3 684 030 417 different (non-isomorphic) semigroups with 8 elements. Some authors define a semigroup as a (possibly empty) set equipped with ‘Empty semigroup’ an associative binary operation. That is, the empty set forms the ‘empty semigroup’. This has advantages from a category-theoretic viewpoint. Note, however, that other definitions must be adjusted if a semigroup can be empty. In these notes, semigroups are required to be non-empty. Basic concepts and examples Throughout this chapter, 푆 will denote a semigroup with operation ∘. Formally, we write (푆, ∘) to indicate that we are considering the set 푆 with the operation ∘, but we will only do this when we need to distinguish a particular operation. Unless we need to distinguish between • 1 different operations, we will often write 푥푦 instead of 푥∘푦 (where 푥, 푦 ∈ 푆) and we will call the operation multiplication and the element 푥푦 (i.e. the Multiplication, product result of applying the operation to 푥 and 푦) the product of the elements 푥 and 푦.
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