No description available.

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

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

No description available.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

M.Sc.

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.

In this thesis we are concerned with arithmetic in a certain partially ordered, commutative semigroup D. The first chapter investigates the class of countable Boolean algebras from which this semigroup arises. The elements of D correspond to the isomorphism classes of the Boolean algebras under consideration. In Chapter 2 we begin the study of the semigroup structure of D. D is axiomatically described by three groups of axioms. It is proved that these axioms are categorical. The ordering of D is used to investigate the multiplication. The set of T of torsion elements of D (elements with only finite many distinct powers), form a subsemigroup whose structure is studied. There is a natural torsion free quotient D/T whose structure is also investigated. In Chapter 3, the axioms are used to characterize elements s of T in terms of the arithmetic in the subsemigroup generated by the elements that are smaller than s. The characterization is used to determine elements of T that cover a single element. In the last part of Chapter 3, we obtain some sufficient, purely combinatorial conditions for an element to have infinite order.