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Restriction Semigroups: Structure, Varieties and Presentations

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Restriction Semigroups: Structure, Varieties and Presentations Restriction Semigroups: Structure, Varieties and Presentations Claire Cornock PhD University of York Mathematics May 2011 Abstract Classes of (left) restriction semigroups arise from partial transformation monoids and form a wider class than inverse semigroups. Firstly, we produce a presentation of the Szendrei expansion of a monoid, which is a left restriction monoid, using a similar approach to Exel’s presentation for the Szendrei expansion of a group. Presentations for the Szendrei expansion of an arbitrary left restriction semigroup and of an inverse semigroup are also found. For our second set of results we look at structure theorems, or P-theorems, for proper restriction semigroups and produce results in a number of ways. Initially, we generalise Lawson’s approach for the proper ample case, in which he adapted the one-sided result for proper left ample semi- groups. The awkwardness of this approach illustrates the need for a sym- metrical two-sided result. Creating a construction from partial actions, based on the idea of a double action, we produce structure theorems for proper restriction semigroups. We also consider another construction based on double actions which yields a structure theorem for a particular class of restriction semigroups. In fact, this was our first idea, but the class of proper restriction semigroups it produces is not the whole class. For our final topic we consider varieties of left restriction semigroups. Specifically, we shall show that the class of (left) restriction semigroups having a cover over a variety of monoids is a variety of (left) restriction semigroups. We do this in two ways. Generalising results by Gomes and Gould on graph expansions, we consider the graph expansion of a monoid and obtain our result for the class of left restriction monoids. Following the same approach as Petrich and Reilly we produce the result for the class of left restriction semigroups and for the class of restriction semigroups. i Contents 1 Universal Algebra 1 1.1 Inversesemigroups ..................... 1 1.2 TypesofAlgebras...................... 3 1.3 Varieties........................... 8 1.4 Freeobjects ......................... 10 1.5 Categories.......................... 12 2 Restriction, Weakly Ample and Ample semigroups 15 2.1 Background ......................... 15 2.2 Restriction and weakly ample semigroups . 18 2.3 Partialtransformationmonoids . 25 2.4 Amplesemigroups ..................... 26 2.5 Examples .......................... 29 2.6 Thenaturalpartialorder. 38 2.7 The least congruence identifying E ............ 40 2.8 Properrestrictionsemigroups . 44 3 The Szendrei expansion 50 3.1 TheSzendreiexpansionofamonoid. 50 3.1.1 Definitionsandbackground . 50 3.1.2 Premorphisms. .. .. .. .. ... .. .. .. .. 51 3.1.3 Presentations via generators and relations . 53 3.2 The Szendrei expansion of a left restriction semigroup . 58 ii 3.2.1 Definitionsandbackground . 58 3.2.2 Premorphisms. .. .. .. .. ... .. .. .. .. 58 3.2.3 Presentations via generators and relations . 60 3.3 The Szendrei expansion of an inverse semigroup . 63 3.3.1 Definitionsandbackground . 63 3.3.2 Premorphisms. .. .. .. .. ... .. .. .. .. 64 3.3.3 Presentations via generators and relations . 65 4 Background:McAlister’sP-Theorem 69 4.1 E-unitary inverse semigroups and McAlister’s covering the- orem............................. 69 4.2 P-semigroups and McAlister’s P-theorem . 70 5 Background:One-sidedP-theorems 72 5.1 Definitionsandcoveringtheorems . 72 5.2 M-semigroups and P-theorem for proper left ample semi- groups ............................ 73 5.3 Strong M-semigroups and P-theorems for proper left re- striction and proper weakly left ample semigroups . 74 6 Two-sided P-theorems 81 6.1 Definitionsandcoveringtheorems . 81 6.2 Structure theorem for proper ample semigroups . 82 6.3 Structure theorem for proper restriction and proper weakly amplesemigroups...................... 82 7 Constructionbasedondoubleactions 87 7.1 Doubleactions ....................... 87 7.2 Construction ........................ 88 7.3 Conversetothestructuretheorem. 93 iii 8 Constructionbasedonpartialactions 96 8.1 Partialactions........................ 96 8.2 Constructionbasedonpartialactions . 96 8.3 Symmetrical two-sided structure theorem for proper re- strictionsemigroups. 98 8.4 Structure theorems independent of one-sided results . 110 8.5 Symmetrical two-sided structure theorems for proper weakly ample, proper ample and proper inverse semigroups . 112 8.6 Acoveringtheorem . 115 9 Graph expansions 118 9.1 Definitions.......................... 118 9.2 The categories PLR(X) and PLR(X,f,S)........ 123 e σ e σ 9.3 The Functors F , F , FX and FX ............. 136 9.4 A construction of M(X,f,S) ............... 155 9.5 Propercoversandvarieties. 163 9.6 A class of left restriction semigroups having a cover over avarietyofmonoids . 164 10Varietiesofrestrictionsemigroups 172 10.1 Propercoversandvarieties. 172 10.2 Subhomomorphisms. 178 10.3 Propercoversandvarieties. 194 Bibliography 209 iv Preface Left restriction semigroups have appeared in the literature under various names including function semigroups in [57] in the work of Trokhimenko, type SL2 γ-semigroups in the work of Batbedat in [4] and [5], twisted LC- semigroups in the work of Jackson and Stokes in [33], guarded semigroups in the work of Manes in [39] and more recently as weakly left E-ample semigroups. Restriction semigroups are believed to have first appeared as function systems in the work of Schweizer and Sklar [54] in the 1960s. We shall look at these appearances in more detail in Chapter 2. We shall provide an abstract definition of left restriction semigroups and look at how they are precisely the (2, 1)-subalgebras of partial transformation monoids, along with examples. We shall provide another definition for left restriction semigroups as a class of algebras defined by identities. In Chapter 2 we shall also give an introduction to weakly (left) ample and (left) ample semigroups. We shall look at the natural partial order on (left) restriction semigroups and the least congruence identifying the distingished semilattice of idempotents associated with the (left) restric- tion semigroup. We shall look at proper (left) restriction semigroups and proper covers in subsequent chapters, but provide a brief introduction in Chapter 2. In Chapters 1 and 2 we provide background definitions and results from universal algebra. In particular, we look at different types of algebras, generating sets, morphisms and congruences. We also look at free objects, categories and varieties. We present some results, which are generalisa- tions from the weakly ample case, for which the proofs are essentially the same. After the introductory chapters we look at three different, but related, topics. Our first, presentations of Szendrei expansions, will be looked at in Chapter 3. Across Chapters 4, 5, 6, 7 and 8 we look at structure theorems, our second topic. Our final topic, varieties, shall be considered in Chapters 9 and 10 where we use two different approaches to prove the same result. The Szendrei expansion is one of two types of expansions we consider. Expansions are used to produce a global action from a partial action, but we shall not study this directly. As well as Szendrei expansions, we also consider graph expansions. We use graph expansions in Chapter 9 as a tool to obtain the result that the class of left restriction monoids v having a proper cover over a variety of monoids is itself a variety of left restriction monoids. In Chapter 3 we look at presentations of the Szendrei expansion of various algebras. Looking first at the Szendrei ex- pansion of a group, which coincides with the Birget-Rhodes expansion (as pointed out in [56]), we consider the “expansion” of a group which Exel described, via generators and relations, in [11]. Kellendonk and Lawson later proved in [35] that Exel’s expansion is isomorphic to the Szendrei expansion. We therefore have a presentation of the Szendrei expansion of a group, which involves factoring a free semigroup by the congruence generated by certain relations inspired by the definition of premorphism for groups. By looking at the relevant definition of a premorphism, we obtain a presentation for the Szendrei expansion of a monoid by factoring the free left restriction semigroup by a congruence generated by certain relations. Similarly we produce presentations of the Szendrei expansion of a left restriction semigroup and inverse semigroup by factoring the free left restriction semigroup and free inverse semigroup respectively by congruences determined by premorphisms. Looking at our second topic, we provide mainly background material in Chapters 4 and 5. In Chapter 4 we present McAlister’s covering theorem from [42] which states that every inverse semigroup has an E- unitary cover, which is the important point behind his P-theorem. The P-theorem from [43] is a structure theorem which states that every E- unitary inverse semigroup is isomorphic to a P-semigroup, a structure consisting of the ingredients of a group, a semilattice and a partially or- dered set, and conversely that every such P-semigroup is an E-unitary inverse semigroup. In Chapter 5 we present covering theorems and structure theorems that were prompted by McAlister’s work, for proper left ample, proper weakly left ample and proper left restriction semigroups. In particular, we look at a structure theorem from [12] for proper left ample semigroups based on a structure M (T, X , Y ), where X is a partially ordered set, Y is a subsemilattice
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