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Axiomatic Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk. 2 OXTOBY. Measure and Category. 2nd ed. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 35 ALEXANDERiWERMER. Several Complex 4 HILTON/STAMMBACH. A Course in Variables and Banach Algebras. 3rd ed. Homological Algebra. 2nd ed. 36 KELLEy!NAMIOKA et al. Linear 5 MAC LANE. Categories for the Working Topological Spaces. Mathematician. 2nd ed. 37 MONK. Mathematical Logic. 6 HUGHES/PIPER. Projective Planes. 38 GRAUERT/F'RITZSCHE. Several Complex 7 SERRE. A Course in Arithmetic. Variables. 8 TAKEUTIlZARING. Axiomatic Set Theory. 39 ARVESON. An Invitation to C*-Algebra~. 9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELL/KNAPP. Denumerable and Representation Theory. Markov Chains. 2nd ed. 10 COHEN. A Course in Simple Homotopy 41 ApOSTOL. Modular Functions and Theory. Dirichlet Series in Number Theory. II CONWAY. Functions of One Complex 2nd ed. Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite 12 BEALS. Advanced Mathematical Analysis. Groups. 13 ANDERSON/fuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous of Modules. 2nd ed. Functions. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry. and Their Singularities. 45 LOEVE. Probability Theory I. 4th ed. 15 BERBERIAN. Lectures in Functional 46 LOEVE. Probability Theory II. 4th ed. Analysis and Operator Theory. 47 MOISE. Geometric Topology in 16 WINTER. The Structure of Fields.Dimensions 2 and 3. 17 ROSENBLATT. Random Processes. 2nd ed.

The techniques employed in the semantic analysis of propositional languages fall roughly into two kinds. The algebraic method treats formulae as polynomial symbols by interpreting logical connectives as operators on certain kinds of lattices. In the set-theoretic approach the models, or frames, carry structural features other than finitary operations, such as neighbourhood systems and finitary relations. Formulae are interpreted as subsets of the frame in a manner constrained by its particular structure.The two kinds of model are closely related. Algebras may be constructed as subset lattices of frames, and frames may be obtained from algebras through various lattice representations. The general concern of this thesis is to explore the relationships between these two semantical frameworks and to discuss their relative strengths and limitations. The vehicle chosen for this work is normal modal logic, although the concepts and results developed may be parallelled in other contexts, for example, intuitionist logic. Sections 1 and 2 outline the syntax of modal logic, its algebraic semantics (modal algebras), and the "possible-worlds" model theory due to Krlpke. A discussion of previous work on correspondences between frames and algebras leads in Section 3 to S.K. Thomason's notion of a first-order frame, that incorporates a restriction on the admissible interpretations of formulae. In Sections 4-7 we develop the basic structure theory of firstorder frames, focusing on validity-preserving constructions such as homomorphisms, subframes, disjoint unions, and ultraproducts, each of which corresponds to an identity-preserving construction on modal-algebras.

In his review (Kaplan [1966] A first-order frame is a triple (W,R,P), where W is a non-empty set, R is a binary relation on W, and P is a non-empty collection of subsets of W closed under the Boolean operations and the unary operation M R