The idea of a seminear-ring was introduced in [8], as an algebraic system that can be constructed from a set <I>S</I> with two binary operations : addition + and multiplication ., such that (<I>S</I>, +) and (<I>S</I>, .) are semigroups and one distributive law is satisfied. A seminear-ring <I>S</I> is called distributively generated (d.g.) if <I>S</I> contains a multiplicative subsemigroup (<I>T</I>, .) of distributive elements which generates (<I>S</I>, +). 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 thesis 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 <I>M</I>(<I>S</I>). Fröhlich [1], [2] and Meldrum [5] have given some results concerning free d.g. near-rings in a variety <I>V</I>. In chapter 2, we generalize some of these results to free d.g. seminear-rings and we can prove the existence of free (<I>S,T</I>)-semigroups on a set <I>X</I> in a variety <I>V</I>. 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 [5] 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 near-ring. 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:661551 |
| Date | January 1998 |
| Creators | Samman, M. S. |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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