Tarek Sayed-Ahmed

Topics in algebraic logic (on neat reducts of algebras of logics)

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 preface, and is self-contained. In particular, every chapter can be read independently of the other chapters. 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 representations and the metalogical one of omitting types. We also solve a long-standing open problem of Tarski and his co-athours Andréka, Henkin, Monk, and Németi on neat reducts but only for the finite dimensional case. In chapter six, we extend this result to the infinite dimensional case.

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