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On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

Tarek Sayed Ahmed

Mathematical Logic Quarterly 61 (6):418-477 (2015)

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Omitting types algebraically and more about amalgamation for modal cylindric algebras.Tarek Sayed Ahmed - 2021 - Mathematical Logic Quarterly 67 (3):295-312.details Let α be an arbitrary infinite ordinal, and. In [26] we studied—using algebraic logic—interpolation and amalgamation for an extension of first order logic, call it, with α many variables, using a modal operator of a unimodal logic that contributes to the semantics. Our algebraic apparatus was the class of modal cylindric algebras. Modal cylindric algebras, briefly, are cylindric algebras of dimension α, expanded with unary modalities inheriting their semantics from a unimodal logic such as, or. When modal cylindric algebras based (...) Download Export citation Bookmark
Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic.Tarek Sayed Ahmed - 2020 - Journal of Applied Non-Classical Logics 30 (3):223-271.details Fix 2 < n < ω and let C A n denote the class of cylindric algebras of dimension n. Roughly, C A n is the algebraic counterpart of the proof theory of first-order logic restricted to the first n var... Download Export citation Bookmark
Complexity of equational theory of relational algebras with standard projection elements.Szabolcs Mikulás, Ildikó Sain & András Simon - 2015 - Synthese 192 (7):2159-2182.details The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider the class $$\mathsf{TPA}^-$$ (...) Download Export citation Bookmark
Omitting Types in Fragments and Extensions of First Order Logic.Tarek Sayed Ahmed - 2021 - Bulletin of the Section of Logic 50 (3):249-287.details Fix \. Let \ denote first order logic restricted to the first n variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \ and for infinitary variants and extensions of \. Download Export citation Bookmark
On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results.Tarek Sayed Ahmed - 2021 - Bulletin of the Section of Logic 50 (4):465-511.details Fix a finite ordinal \ and let \ be an arbitrary ordinal. Let \ denote the class of cylindric algebras of dimension \ and \ denote the class of relation algebras. Let \\) stand for the class of polyadic algebras of dimension \. We reprove that the class \ of completely representable \s, and the class \ of completely representable \s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \ between polyadic algebras (...) Download Export citation Bookmark
Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L n.Tarek Sayed Ahmed - 2020 - Notre Dame Journal of Formal Logic 61 (4):503-519.details 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 (...) Download Export citation Bookmark
Complexity of equational theory of relational algebras with projection elements.Szabolcs Mikulás, Ildikó Sain & Andras Simon - 1992 - Bulletin of the Section of Logic 21 (3):103-111.details The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) Download Export citation Bookmark 5 citations

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