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Cylindric-like Algebras and Algebraic Logic

Part of the Bolyai Society Mathematical Studies Book Series (volume 22)

Editors: Andréka, Hajnal; Ferenczi, Miklós; Németi, István

DOI: 10.1007/978-3-642-35025-2

ISBN: 978-3-642-35024-5

Publisher: Springer Berlin Heidelberg - Springer Journals

Jan 23, 2014

1 - 10 of 19 Chapters

LitStream

Introduction

Andréka, Hajnal; Ferenczi, Miklós; Németi, István

Pages: 9 –12

[The theory of cylindric algebras (CAs) and Tarskian algebraic logic have been immensely successful in, e.g., applications to such a diversity of fields as logic, mathematics, computer science, artificial intelligence, linguistics and even relativity theory. In some of the applications one has to...
LitStream

Reducing First-order Logic to Df3, Free Algebras

Andréka, Hajnal; Németi, István

Pages: 15 –35

[Alfred Tarski in 1953 formalized set theory in the equational theory of relation algebras [Tar,53a, Tar,53b]. Why did he do so? Because the equational theory of relation algebras (RA) corresponds to a logic without individual variables, in other words, to a propositional logic. This is why the...
LitStream

Varieties of Two-Dimensional Cylindric Algebras

Bezhanishvili, Nick

Pages: 37 –59

[In this chapter we survey recent developments in the theory of twodimensional cylindric algebras. In particular, we will deal with varieties of two-dimensional diagonal-free cylindric algebras and with varieties of two-dimensional cylindric algebras with the diagonal. It is well known that...
LitStream

Completions and Complete Representations

Hirsch, Robin; Hodkinson, Ian

Pages: 61 –89

[The title of this chapter indicates a rather technical topic, but it can also be thought of as a foundational issue in logic. The question to be considered is this: to what extent can we use an abstract mathematical language to express and reason about relations? Going back at least as far as...
LitStream

Amalgamation, Interpolation and Epimorphisms in Algebraic Logic

Madarász, Judit X.; Ahmed, Tarek Sayed

Pages: 91 –103

[The amalgamation property (for classes of models), since its discovery, has played a dominant role in algebra and model theory. Algebraic logic is the natural interface between universal algebra and logic (in our present context a variant of first order logic). Indeed, in algebraic logic...
LitStream

Neat Reducts and Neat Embeddings in Cylindric Algebras

Ahmed, Tarek Sayed

Pages: 105 –131

[An important central concept introduced in [Hen-Mon-Tar,85] is that of neat reducts, and the related one of neat embeddings. The notion of neat reducts is due to Leon Henkin, and one can find that the discussion of this notion is comprehensive and detailed in [Hen-Mon-Tar,85] (closer to the end...
LitStream

A New Representation Theory: Representing Cylindric-like Algebras by Relativized Set Algebras

Ferenczi*, Miklós

Pages: 135 –162

[Representation theorems. As is known, cylindric algebras are not representable in the classical sense (as a subdirect product of cylindric set algebras), in general. But, the Resek-Thompson theorem states that if the system of cylindric axioms is extended by a new axiom, by the merry-goround...
LitStream

Representing all Cylindric Algebras by Twisting on a Problem of Henkin

Simon, András

Pages: 163 –182

[Had the class of cylindric set algebras turned out to be finitely (and/or nicely) axiomatizable, algebraic logic would have evolved along a markedly different path than it did in the past 40 years. Among other things, it would have probably spelled the end of the “abstract” class CAα as a...
LitStream

Representable Cylindric Algebras and Many-Dimensional Modal Logics

Kurucz, Agi

Pages: 185 –203

[The equationally expressible properties of the cylindrifications and the diagonals in finite-dimensional representable cylindric algebras can be divided into two groups: ‘One-dimensional’ properties describing individual cylindrifications. These can be fully characterised by finitely many...
LitStream

Completions, Complete Representations and Omitting Types

Ahmed, Tarek Sayed

Pages: 205 –221

[Algebraic logic arose as a subdiscipline of algebra mirroring constructions and theorems of mathematical logic. It is similar in this respect to such fields as algebraic geometry and algebraic topology, where the main constructions and theorems are algebraic in nature, but the main intuitions...

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