On a family of semigroup congruences

Abstract

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.