The finitely axiomatizable complete theories of non-associative arrow frames
type: Item
instance of: scholarly article
title: The finitely axiomatizable complete theories of non-associative arrow frames
zbMATH Open document ID: 1457.03043
author: Mohamed Khaled
published in: Advances in Mathematics
publication date: 2019-03-13T00:00:00Z
review text: The paper studies the non-associative arrow logic NAL. One of the important notions used for studying NAL is a notion of normal forms. Let \(F_k(X)\) denote a set of normal forms of degree \(k\) in variable from \(X\). Then the following holds: There is an algorithm that, given a formula \(\varphi\), a set of propositional variables \(X \supseteq \mathrm{var}(\phi)\) and a finite ordinal \(k\) bigger than or equal to the maximum depth of modal operators nesting in \(\phi\), generates a finite set \(\Sigma \subseteq F_k(X)\) such that \(\vDash \varphi \leftrightarrow \bigvee\Sigma\). A game theory semantic of NAL has been intensively studied. Using it, an alternative proof [\textit{I. Németi}, Proc. Am. Math. Soc. 100, 340--344 (1987; Zbl 0638.03061)] of the following result is presented: NAL has the finite model property and it is decidable. In addition, the descriptions of all finitely axiomatizable, all complete and all consistent theories of NAL are given. A negative answer to a problem concerning the atomicity of the free non-associative relation algebras is given.
reviewed by: Alex Citkin
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 06E25
zbMATH DE Number: 7035898
zbMATH Keywords: arrow logic
zbMATH Keywords: relation algebras
zbMATH Keywords: finitely axiomatizable theories
zbMATH Keywords: finite model property
MaRDI profile type: Publication
full work available at URL: j.aim.2019.01.014
OpenAlex ID: W2914527465
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Wikidata QID: Q128424799
DOI: 10.1016/J.AIM.2019.01.014
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description: scientific article; zbMATH DE number 7035898