Lattices and topologies on Newman algebras.
thesisposted on 19.11.2015, 08:55 by R. Beazer
In what was almost certainly an attempt to find a new axiom system for Boolean algebra based on distributivity and the existence of complements MHA Newman discovered a remarkable set of independent postulates defining an algebra which may be regarded as a generalization of Boolean algebra and now bears his name. Shortly after publication of his paper Newman extended his discussion to a wider class of relatively complemented algebras which we call Generalized Newman algebra. Recently K.Roy investigated the properties of an algebra closely related to Newman algebra, called Dual Newman algebra, and found that it has similar properties to its progenitor the opening chapters of the thesis are devoted to a discussion of the properties of the lattices of ideals, congruence relations and filters in Newman algebra and the relationships between them. The concepts of inverse and sub-inverse limits of Newman algebras are introduced, some general properties proved, and a sub-inverse limit representation established for a particular class of Newman algebras together with an inverse limit representation for the class of infinite, couplet, Boolean algebras. Furthermore, it is proved that a Newman algebra can be represented as a direct product of simple algebras if and only if its ideal lattice is a finite Boolean algebra. In the following chapter we investigate, within the framework of Newman algebras, the analogues of the auto and ideal topologies on Boolean algebra discovered by P.S. Rema. It is shown that the set of all ideal topologies L1 on a Newman algebra N is a complete, Brouwerian, dually atomic lattice containing the set L0 of all auto topologies as a complete sub-lattice and that L0 is completely isomorphic to the lattice of filters of N. Some important types of filters in N are characterized in terms of properties of the associated auto topologies on N and the auto topology associated with a given filter characterized within the lattice of ideal topologies on N. Amongst the more general properties proved we mention that the property of a topology, compatible with the fundamental operations on N, being an ideal (auto) topology is, in the algebraic and topological sense, hereditary, productive and divisible. The various connectedness properties of ideal topologized Newman algebra N;J are considered in some detail; the components being exhibited as certain congruence classes of N and necessary and sufficient conditions found for N;J to be connected, locally connected and totally disconnected. Some results are obtained concerning complete ideal uniformities and compact ideal uniformities. The properties of a particular class of ideal uniformities, called chain uniformities, are investigated and a clear cut family of metrizable chain uniformities are exhibited. Necessary and sufficient conditions are then established for a Newman algebra endowed with a separated ideal uniformity to the metrizable. In the closing chapters of the thesis we are concerned with the axiomatics of Dual and Generalized Newman algebras. Two new sets of axioms for Dual Newman algebra are exhibited each containing one less axiom than the system due to K. Roy. A new set of axioms for Generalized Boolean algebra is found containing one less axiom than the system discovered by lawmen together with a new set of independent postulates, characterizing the direct product of an arbitrary Generalized Boolean algebra and Boolean ring, which contains two fewer axioms than the system discovered by Newman.