Atom structures of cylindric algebras and relation algebras
type: Item
instance of: scholarly article
title: Atom structures of cylindric algebras and relation algebras
zbMATH Open document ID: 0898.03025
DOI: 10.1016/S0168-0072(97)00015-8
published in: Annals of Pure and Applied Logic
publication date: 1998-11-01T00:00:00Z
review text: In 1970, \textit{J. D. Monk} [Math. Nachr. 46, 47-55 (1970; Zbl 0182.32301)] proved that relation algebras and cylindric algebras have completions, and asked whether completions of representable algebras are representable. This paper shows the answer is ``no''. There is a countable infinite relational structure \({\mathfrak S}\) whose (uncountable) complex algebra is a relation algebra that is not representable, and yet \({\mathfrak S}\) is the atom structure of a countable representable relation algebra. It follows that there is a countable atomic representable relation algebra whose completion is not representable. Similar results hold for cylindric algebras and solve a problem of \textit{L. Henkin, J. D. Monk}, and \textit{A. Tarski} [Cylindric algebras. Part I (1971; Zbl 0214.01302), Remarks 2.7.25 and Problem 2.16] by showing that the classes of representable relation algebras and representable cylindric algebras of finite dimension \(n\) \((n\geq 3)\) are not axiomatizable by equations which are ``positive in the wider sense''.
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 03C05
zbMATH DE Number: 1109906
zbMATH Keywords: representability
zbMATH Keywords: axiomatizability
zbMATH Keywords: relation algebras
zbMATH Keywords: cylindric algebras
zbMATH Keywords: completion
author: Ian Hodkinson
reviewed by: Roger Maddux
MaRDI profile type: Publication
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full work available at URL: s0168-0072(97)00015-8
OpenAlex ID: W2120627125
description: scientific article; zbMATH DE number 1109906