Martin's axiom, omitting types, and complete representations in algebraic logic
type: Item
instance of: scholarly article
title: Martin's axiom, omitting types, and complete representations in algebraic logic
zbMATH Open document ID: 1019.03047
DOI: 10.1023/A:1021368713305
author: Tarek Sayed Ahmed
published in: Studia Logica
publication date: 2003-03-12T00:00:00Z
review text: An \(n\)-dimensional (\(n\)-finite) cylindric algebra \(A\) is said to be completely representable if, for every non-zero element \(a \in A\), there is a completely additive homomorphism \(f\) from \(A\) to an \(n\)-dimensional cylindric set algebra with \(f(a) \neq 0\). One of author's results gives a new characterisation of countable completely representable finite-dimensional cylindric (and relational) algebras via special neat embeddings. It is also shown that a certain statement concerning the existence of representations for cylindric algebras is equivalent to a certain form of Martin's axiom (restricted to countable Boolean algebras) and turns out to be connected also with omitting types for \(L_{\omega\omega}\). As an application, Henkin and Orey's omitting types theorem is shown to be false for first-order logics with a finite number of variables. Some open problems are posed, and recent results by H. Andréka and by R. Hirsch related to two of them are discussed.
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 03C25
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 1880460
zbMATH Keywords: algebraic logic
zbMATH Keywords: cylindric algebra
zbMATH Keywords: neat reducts
zbMATH Keywords: neat embeddings
zbMATH Keywords: complete representations
zbMATH Keywords: Martin's axiom
zbMATH Keywords: omitting types
MaRDI profile type: Publication
full work available at URL: a:1021368713305
OpenAlex ID: W1544936696
description: scientific article; zbMATH DE number 1880460