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Register a new userA brief history of algebraic logic from neat embeddings to rainbow constructions
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Authors
Tarek Sayed Ahmed
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We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow constructions invented by Hirsch and Hodkinson. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. The paper has a survey character but it contains new approaches to old ones. We aspire to make our survey fairly comprehensive, at least in so far as Tarskian algebraic logic, specifically, the theory of cylindric algebras, is concerned. Other topics, such as abstract algebraic logic, modal logic and the so--called (central) finitizability problem in algebraic logic will be dealt with; the last in some detail. Rainbow constructions are used to solve problems adressing classes of cylindric--like algebras consisting of algebras having a neat embedding property. The hitherto obtained results generalize seminal results of Hirsch and Hodkinson on non--atom canonicity, non--first order definabiity and non--finite axiomatizability, proved for classes of representable cylindric algebras of finite dimension$>2$. We show that such results remain valid for cylindric algebras possesing relativized {it clique guarded} representations that are {it only locally} well behaved. The paper is written in a way that makes it accessible to non--specialists curious about the state of the art in Tarskian algebraic logic. Reaching the boundaries of current research, the paper also aspires to be informative to the practitioner, and even more, stimulates her/him to carry on further research in main stream algebraic logic.
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Fix 2<n<omega. Let L_n denote first order logic restricted to the first n variables. CA_n denotes the class of cylindric algebras of dimension n and for m>n, Nr_nCA_m(subseteq CA_n) denotes the class of n-neat reducts of CA_ms. The existence of certain finite relation algebras and finite CA_ns lacking relativized complete representations is shown to imply that the omitting types theorem (OTT) fails for L_n with respect to clique guarded semantics (which is an equivalent formalism of its packed fragments), and for the multi-dimensional modal logic S5^n. Several such relation and cylindric algebras are explicitly exhibited using rainbow constructions and Monk-like algebras. Certain CA_n constructed to show non-atom canonicity of the variety SNr_nCA_{n+3} are used to show that Vaughts theorem (VT) for L_{omega, omega}, looked upon as a special case of OTT for L_{omega, omega}, fails almost everywhere (a notion to be defined below) when restricted to L_n. That VT fails everywhere for L_n, which is stronger than failing almost everywhere as the name suggests, is reduced to the existence, for each n<m<omega, of a finite relation algebra R_m having a so-called m-1 strong blur, but R_m has no m-dimensional relational basis. VT for other modal fragments and expansions of L_n, like its guarded fragments, n-products of uni-modal logics like K^n, and first order definable expansions, is approached. It is shown that any multi-modal canonical logic L, such that $K^nsubseteq Lsubseteq S5^n$, L cannot be axiomatized by canonical equations. In particular, L is not Sahlqvist. Elementary generation and di-completeness for L_n and its clique guarded fragments are proved. Positive omitting types theorems are proved for L_n with respect to standard semantics.
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