Abstract:
The main purpose of this dissertation consists on studying the class of principally ordered regular semigroups. This thesis is made up of three chapters: the first one deals with general purpose considerations, the second is on principally ordered semigroups and the third on residuated regular semigroups. On the first chapter we present the basic concepts and results that we understand to be basic in the understanding of the remaining chapters of this thesis. We opted not to present the proofs of the results as they can be found in any basic level textbook of the Theory of Semigroups. On chapter two we study an ordered regular semigroups subclass: the principally ordered regular semigroups. A regular ordered semigroup S is said to be principally ordered if, for every x 2 S, there exists the biggest element of the set {y 2 S : XVX _ X}. This biggest element is represented by X_. We describe and prove several properties satisfied by this class of semigroups. We also present some examples of principally ordered regular semigroups. A concept connected to principally ordered regular semigroups is the concept of antitone mapping. The principally ordered regular semigroups in which the mapping x 7! x_ is antitone, that is, x_ _ y_ whenever x _ y, satisfy interesting properties. We present these properties together with their proofs. Another concept explored in this work is the concept of biggest inverse. Given a regular semigroup S, we say that x 2 S is an inverse of x0 2 S if x = xx0x and x0 = x0xx0. If the regular semigroup is ordered, it may exist the biggest inverse of x_, which we represent by x_. In a principally ordered semigroup S, not always x_ = x, for all x 2 S. If this is the case, we say that the semigroup is compact. We end the second chapter by presenting a characterization of a compact regular principally and naturally ordered semigroup. In the last chapter the residuated semigroups are studied. This chapter starts with a definition of residuated semigroup and follows with the presentation of the properties of these semigroups. In particular, we study one class of residuated semigroups: the residuated regular semigroups. Linking this with the second chapter, we study the sub-semigroup S_ = {x_ : x 2 S} of a regular residuated semigroup. This chapter follows with the definition of a concise element and the characterization of the set of concise elements in a residuated regular semigroup. Then, we define the concise semigroup and we present the theorems that characterize it. We finish this chapter by characterizing the semigroups that are simultaneously concise and compact.

Semigrupos regulares principalmente ordenados (pdf) »»