The primary purpose of this paper is to state a set of postulates for Boolean algebra and show the characteristic theorems derivable from them, and to unify in one paper the more important methods of representing Boolean algebra and show their equivalence.

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Thesis--Karnatak University. / Bibliography: p. [77]-78.

Boolean valued structures are defined and some of their properties are studied. Completeness and compactness theorems are proved and Lowenheim-Skolem theorems are looked at. It is seen that for any consistent theory T and cardinal number KT there is a model N of T a "universal" model) such that for any model M of T with M <, K, M can be written as a quotient of N. A theory T is shown to be open if and only if given structures M c N, if N is a model of T, then M is a model of T, T is shown to be existential if and only if the union of every chain of models of T is a model of T. The prefix problem and obstructions to elementary extensions are examined. Various forms of completeness are compared and, finally, an example is given where Boolean valued models are used to prove a theorem of Mathematics (Hilbert's 17-th Problem) without using the Axiom of Choice. Throughout, it is seen that good Boolean valued structures (for all Φ [(Ev[sub j])Φ][sub M] = I(Φ,a][sub M] for some a Є U[sub M]) behave very much like relational structures and much of the theory depends upon their existence. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis obtains information about Boolean algebras by means of the radical concept. One group of results revolves about the concept, theorems, and constructions of general radical theory. We obtain some subdirect product representations by methods suggested by the theory. A large number of specific radicals are defined, and their properties and inter-relationships are examined. This provides a natural frame-work for results describing what epimorphs an algebra can have. Some new results of this nature are obtained in the process. Finally, a contribution is made to the structure theory of complete Boolean algebras. Product decomposition theorems are obtained, some of which make use of chains of radical classes. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this section is to define a boolean algebra and to determine some of the important properties of it. A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complement ation, denoted by ', which satisfy the following axioms: (1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws), (2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) and a (b + c) = a b + a c, (the distributive laws), (3) there exists 0 ∑B such that for each a ∑B, a + 0 = a, and there exists 1 ∑B such that for each a ∑ B, a 1 = a, (4) for each a∑B, a + a' = 1 and a a' = 0. If a + e = a for all a in B then 0 = 0 + e = e + 0 = e, so that there is exactly one element in B which satisfies the first half of axiom 3, namely 0. Similarly there is exactly one element in B which satisfies the second half of axiom 3, namely 1. The O and 1 as defined above will be called the distinguished elements.

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The purpose of this chapter is to develop a form of a "free" Boolean algebra with Σ as a base, by imposing the usual Boolean operations on the set Σ and thus generating new elements freely within explicitly prescribed restrictions.

Mak, Yan Kei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 59-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Historical background --- p.6 / Chapter 1.2 --- Why study communication complexity? --- p.7 / Chapter 1.3 --- Main ideas and results --- p.8 / Chapter 1.4 --- Recent development --- p.12 / Chapter 1.5 --- Structure of the thesis --- p.12 / Chapter 2 --- Deterministic Communication Complexity --- p.13 / Chapter 2.1 --- Definitions --- p.13 / Chapter 2.2 --- Tiling lower bound --- p.16 / Chapter 2.3 --- Fooling set lower bound --- p.21 / Chapter 2.4 --- Rank lower bound --- p.24 / Chapter 2.5 --- Comparison of the bounds --- p.27 / Chapter 3 --- Nondeterministic Communication Complexity --- p.29 / Chapter 3.1 --- Definitions --- p.29 / Chapter 3.2 --- "Gaps between N0(f), N1(f) and D(f)" --- p.31 / Chapter 3.3 --- Aho-Ullman-Yannakakis Theorem --- p.33 / Chapter 4 --- Randomized Communication Complexity --- p.38 / Chapter 4.1 --- Preliminaries --- p.38 / Chapter 4.2 --- Definitions --- p.39 / Chapter 4.3 --- Error reduction --- p.41 / Chapter 4.4 --- Exponential gap with D(f) --- p.42 / Chapter 4.5 --- The public coin model --- p.44 / Chapter 4.6 --- Distributional complexity --- p.46 / Chapter 5 --- Communication Complexity Classes --- p.51 / Chapter 5.1 --- Basic classes --- p.51 / Chapter 5.2 --- Polynomial-time hierarchy --- p.52 / Chapter 5.3 --- Reducibility and completeness --- p.53 / Chapter 6 --- Further topics --- p.56 / Chapter 6.1 --- Quantum communication complexity --- p.56 / Chapter 6.2 --- More techniques for bounds --- p.57 / Chapter 6.3 --- Complexity of communication complexity --- p.57 / Bibliography --- p.59

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