Author:
Tarek Sayed Ahmed Department of Mathematics, Faculty of Science, Cairo University

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Abstract

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever A ∈ V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that NrnCAω ∩ CRCAn ¯ K ¯ ScNrnCAn+3, where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω ¯ K ¯ ScRaCA5 is not elementary, where Sd denotes the operation of forming dense subalgebra.

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    Sayed Ahmed, T., The class of 2-dimensional polyadic algebras is not elementary, Fundamenta Mathematica, 172 (2002), 6181.

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    Sayed Ahmed, T., RaCA n is not elementary for n ≥ 5, Bulletin Section of Logic, 37(2) (2008), 123136.

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    Sayed Ahmed, T., Neat reducts and neat embeddings in cylindric algebras, in: Andréka, H., Ferenczi, M. and Németi, I., (Editors), Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22, 105134, 2013.

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    Sayed Ahmed, T., On notions of representability for cylindric–polyadic algebras and a solution to the finitizability problem for first order logic with equality, Mathematical Logic Quarterly, 61(6) (2015), 418447.

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    Sayed Ahmed, T., Notions of Representability for cylindric and polyadic algebras, Studia Mathematicea Hungarica, 56(3) (2019), 335363.

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    Sayed Ahmed, T. and Samir, B., Omitting types for first order logic with infinitary predicates, Mathematical Logic Quaterly, 53(6) (2007) 564576.

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    Venema, Y., Atom structures and Sahlqvist equations, Algebra Universalis, 38 (1997), 185199.

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Studia Scientiarum Mathematicarum Hungarica
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