Topics in Algebraic Logic

Abstract

This dissertation revolves around the notion of neat reducts and the related notion of neat embeddings. It has six chapters. Chapter one is a broad introduction to algebraic logic with emphasis on the significance of the notion of neat reducts. Every other chapter is preceded by an abstract and a more detailed technical prdan•. and is self-contained. In particular, every chapter can be read independently. In chapters two and three we relate results on neat embeddings to results on amalgamation. In chapter four we address amalgamation proper. In• chapter five we relate results on neat embeddings to the algebraic notion of complete representatiuns and to the met:alogical one of omitting types. We also solve a long-standing open problem of Tarski and his co-athours Andreka, Henkin, Monk, and Nemeti on neat reducts but only for the finite dimensional case. In chapter six, we extend this result to the infinite dimensional case. In more detail, in chapter two we show that the class of o:-dimensionalneat reduct.s of /3-dimensional algebras in several cylindric-like algebras of relations is not closed under forming subalgebras for any pair of ordinals 1 < o < ,6. As a coroll<ny we settle a conjecture of Tarski for cylindric algebras in the affirmative. The algebras we deal \Yith in chapter two are Tarski's cylindric algebras, Halmos' quasipolyadic algebras with and \Yithout. equality and Pinter's substitution algebras. In particular. all algebras considered are reducts of Halmos' polyadic algebras. We give a constrast.­ ing result for Halmos' polyadic algebras of infinite dimension; for these we show that t.he class of neat reducts forms a variety.