Title:

Topics in seminearring theory

The idea of a seminearring was introduced in [8], 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 seminearring S is called distributively generated (d.g.) if S contains a multiplicative subsemigroup (T, .) of distributive elements which generates (S, +). Unlike the nearrings case for which a rich theory has already been developed, very little seems to be known about seminearrings. The aim of this dissertation consists mainly of two goals. The first is to generalize some results which are known in the theory of nearrings. The second goal of this thesis appears mainly in the last 6 chapters in which we obtain some results about seminearrings of endomorphisms. In chapter 1, the definitions and basic concepts about seminearrings are given; e.g. an arbitrary seminearring can be embedded in a seminearring of the form M(S). Fröhlich [1], [2] and Meldrum [5] have given some results concerning free d.g. nearrings in a variety V. In chapter 2, we generalize some of these results to free d.g. seminearrings and we can prove the existence of free (S,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. seminearring has a faithful representation. This would generalize the result which was given by Meldrum [5] for the nearring case. Chapter 3 gives an overview of strong semilattices of nearrings and of rings. In this context we show that a strong semilattice of nearrings is a seminearring while a strong semilattice of rings is a nearring. 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.
