Topics in seminear-ring theory

Abstract

The idea of a seminear-ring was introduced in [9], as an algebraic system that can be constructed from a set S with two binary operations : addition + and multiplication such that (S, +) and (S,.) are semigroups and one distributive law is satisfied. A seminear-ring S is called distributively generated (d.g.) if S contains a multiplicative subsemigroup (T,.) of distributive elements which generates (S, +). Unlike the near-rings case for which a rich theory has already been developed, very little seems to be known about seminear-rings. The aim of this dissertation consists mainly of two goals. The first is to generalize some results which are known in the theory of near-rings. The second goal of this the¬ sis appears mainly in the last 6 chapters in which we obtain some results about seminear-rings of endomorphisms.

In chapter 1, the definitions and basic concepts about seminear-rings are given; e.g. an arbitrary seminear-ring can be embedded in a seminear-ring of the form M(S).

Frohlich [1], [2] andMeldrum [6] have given some results concerning free d.g. near-rings in a variety V. In chapter 2, we generalize some of these results to free d.g. seminear-rings and we can prove the existence of free (3, T)-semigroups on a set X in a variety V. In section 2.4, we prove a theorem which asserts that not every d.g. seminear-ring has a faithful representation. This would generalize the result which was given by Meldrum [6] for the near-ring case.

Chapter 3 gives an overview of strong semilattices of near-rings and of rings. In this context we show that a strong semilattice of near-rings is a seminear-ring while a strong semilattice of rings is a semiring.

Chapter 4 is designed to be a preparatory chapter for the remaining part of the thesis. It explains the main plan which will be followed in all the last 6 chapters. It also includes some basic ideas and results which are of great use in the remaining work.

Throughout chapter 5 to chapter 10 we will be considering seminear-rings of endomorphisms of some special types. In each chapter we consider some groups Ga, where each a belongs to the semilattice Y, then we study the structure of the corresponding strong semilattice S of groups Ga,a E Y. For each group Ga there will be a d.g. near-ring E(Ga) generated by End(Cxa), the set of all endomorphisms of Ga. On the other hand, considering S, the semilattice of the groups Ga, then End(S), the set of all endomorphisms of S, will generate a d.g. seminear-ring E(S). So we study the structure of E(S) with its connection to {E(Ga);a & Y} which asserts that E(S), indeed, forms a Clifford semigroup. For the background in semigroup theory we refer to Howie [4] which is the standard book in that subject.