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November 2020 Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L n
Tarek Sayed Ahmed
Notre Dame J. Formal Logic 61(4): 503-519 (November 2020). DOI: 10.1215/00294527-2020-0022

Abstract

Fix 2 < n < ω . Let CA n denote the class of cylindric algebras of dimension n , and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n , an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n , namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n . Unlike Boolean algebras having a Stone representation theorem, RCA n CA n . Using combinatorial game theory, we show that the existence of certain finite relation algebras RA , which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n . Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 0 , and F = Γ i : i < λ is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T .

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Tarek Sayed Ahmed. "Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L n ." Notre Dame J. Formal Logic 61 (4) 503 - 519, November 2020. https://doi.org/10.1215/00294527-2020-0022

Information

Received: 12 November 2019; Accepted: 10 September 2020; Published: November 2020
First available in Project Euclid: 17 December 2020

Digital Object Identifier: 10.1215/00294527-2020-0022

Subjects:
Primary: 03G15
Secondary: 03B45

Rights: Copyright © 2020 University of Notre Dame

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Vol.61 • No. 4 • November 2020
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