Covers for regular semigroups and an application to complexity

AbstractA major result of D.B. McAlister for inverse semigroups is generalised in the paper to classes of regular semigroups, including the class of all regular semigroups. It is shown that any regular semigroup is a homomorphic image of a regular semigroup whose least full self-conjugate subsemigroup is unitary; the homomorphism is injective on the subsemigroup. As an application, the group complexity of any finite E-solid regular semigroup is shown to be the same as, or one more than that of its least full self-conjugate subsemigroup (the subsemigroup is completely regular and is the type II subsemigroup). In an addition to the paper, by P.R. Jones, it is shown that any finite locally orthodox semigroup has group complexity 0 or 1

Weakly free regular semigroups I. The general case

We prove the existence of a regular semigroup F(X) weakly generated by X such that all other regular semigroups weakly generated by X are homomorphic images of F(X). The semigroup F(X) is introduced by a presentation and the word problem for that presentation is solved.Comment: 23 pages; 4 figure

The lattice of varieties of monoids

We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated

On a family of semigroup congruences

We introduce in this thesis a new family of semigroup congruences, and we set out to prove that it is worth studying them for the following very important reasons: (a) that it provides an alternative way of studying algebraic structures of semigroups, thus shedding new light over semigroup structures already known, and it also provides new information about other structures not formerly understood; (b) that it is useful for constructing new semigroups, hence producing new and interesting classes of semigroups from known classes; and (c) that it is useful for classifying semigroups, particularly in describing lattices formed by semigroup species such as varieties, pseudovarieties, existence varieties etc. This interesting family of congruences is described as follows: for any semigroup S, and any ordered pair (n,m) of non-negative integers, define ⦵(n,m) = {(a,b): uav = ubv, for all ⋿Sn and v ⋿Sm}, and we make the convention that S¹ = S and that S0 denotes the set containing only the empty word. The particular cases ⦵(0,1), ⦵(1,0) and ⦵(0,0) were considered by the author in his M.Sc. thesis (1991). In fact, one can recognise ⦵(1,0) to be the well known kernel of the right regular representation of S. It turns out that if S is reductive (for example, if S is a monoid), then ⦵(i,j) is equal to ⦵(0,0) - the identity relation on S, for every (i,j). After developing the tools required for the latter part of the thesis in Chapters 0-2, in Chapter 3 we introduce a new class of semigroups - the class of all structurally regular semigroups. Making use of a new Mal'tsev-type product, in Chapters 4,5,6 and 7, we describe the lattices formed by certain varieties of structurally regular semigroups. Many interesting open problems are posed throughout the thesis, and brief literature reviews are inserted in the text where appropriate

Complete reducibility of pseudovarieties

The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specific pseudovarieties.FCT through the Centro de Matemática da Universidade do Minho and Centro de Matemática da Universidade do Port