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Relation algebras from cylindric algebras, II

Robin Hirsch & Ian Hodkinson

Annals of Pure and Applied Logic 112 (2-3):267-297 (2001)

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Algebraic Logic, Where Does It Stand Today?Tarek Sayed Ahmed - 2005 - Bulletin of Symbolic Logic 11 (3):465-516.details This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...) Download Export citation Bookmark 22 citations
First-Order Axiomatisations of Representable Relation Algebras Need Formulas of Unbounded Quantifier Depth.Rob Egrot & Robin Hirsch - 2022 - Journal of Symbolic Logic 87 (3):1283-1300.details Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables. Download Export citation Bookmark
Amalgamation Theorems in Algebraic Logic, an overview.Tarek Sayed-Ahmed - 2005 - Logic Journal of the IGPL 13 (3):277-286.details We review, and in the process unify two techniques , for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results of this (...) Download Export citation Bookmark 2 citations
Relation algebras from cylindric algebras, I.Robin Hirsch & Ian Hodkinson - 2001 - Annals of Pure and Applied Logic 112 (2-3):225-266.details We characterise the class S Ra CA n of subalgebras of relation algebra reducts of n -dimensional cylindric algebras by the notion of a ‘hyperbasis’, analogous to the cylindric basis of Maddux, and by representations. We outline a game–theoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of S Ra CA n. Download Export citation Bookmark 6 citations
Omitting types for finite variable fragments and complete representations of algebras.Hajnal Andréka, István Németi & Tarek Sayed Ahmed - 2008 - Journal of Symbolic Logic 73 (1):65-89.details We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order (...) Download Export citation Bookmark 13 citations
Positive fragments of relevance logic and algebras of binary relations.Robin Hirsch & Szabolcs Mikulás - 2011 - Review of Symbolic Logic 4 (1):81-105.details We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations. Download Export citation Bookmark 1 citation
Neat embedding is not sufficient for complete representability.T. Sayed Ahmed - 2007 - Bulletin of the Section of Logic 36 (1/2):29-35.details Download Export citation Bookmark 1 citation
Neat Embeddings, Omitting Types, and Interpolation: An Overview.Tarek Sayed Ahmed - 2003 - Notre Dame Journal of Formal Logic 44 (3):157-173.details We survey various results on the relationship among neat embeddings (a notion special to cylindric algebras), complete representations, omitting types, and amalgamation. A hitherto unpublished application of algebraic logic to omitting types of first-order logic is given. Download Export citation Bookmark 4 citations
Neat reducts and amalgamation in retrospect, a survey of results and some methods Part I: Results on neat reducts.Judit Madarász & Tarek Ahmed - 2009 - Logic Journal of the IGPL 17 (4):429-483.details Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new (...) Download Export citation Bookmark

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