Reflectivity: A Generalization Of Commutativity, Cancellation And Separativity In Semigroups

The concept of reflectivity is interesting in that it is a simultaneous generalization of commutativity, cancellation and separativity. This thesis studies reflective semigroups as well as a related notion called reflexive subset of a semigroup.;In Chapter 2, a new notion called strongly viable semigroups is introduced. This notion is weaker than reflectivity. Some previously known results for reflective semigroups are shown to hold for strongly viable semigroups. The maximal separative homomorphic image of any given periodic strongly viable semigroup is described. Every homomorphism into a group of a reflective semigroup with a minimal one-sided ideal is shown to have a group image. Chapter 3 shows that a semigroup is reflective, (eta)-simple and contains an idempotent if and only if it is an ideal extension of a group by a reflective nil semigroup. Properties of simple, idempotent-free reflective semigroups are listed. In Chapter 4, an effective construction of finite reflective nil semigroups of order n is given as ideal extensions of reflective nil semigroups of order n-1. Chapter 5 is a study of reflexive and completely reflexive subsets of semigroups. A characterization is given of the reflexive and completely reflexive ideals of (SIGMA)*. Semigroups in which every one-sided ideal is reflexive are shown to share many properties with reflective semigroups. A new notion called power-commutativity is introduced and investigated. In Chapter 6, an effective construction of a finite automaton recognizing the reflexive closure of a given regular language is given. Chapter 7 introduces three notions, related to reflectivity for future study