A construction of cylindric and polyadic algebras from atomic relation algebras
type: Item
instance of: scholarly article
title: A construction of cylindric and polyadic algebras from atomic relation algebras
zbMATH Open document ID: 1263.03058
published in: Algebra Universalis
publication date: 2013-01-30T00:00:00Z
review text: The purpose of this paper is to present a construction that does preserve and reflect representability in higher dimensions than 3. Given a simple atomic relation algebra \({\mathcal A}\) and a finite \(n\geq 3\), the author constructs effectively an atomic \(n\)-dimensional polyadic equality-type algebra \({\mathcal P}\) that for any signature \(L\) of the signature of \({\mathcal P}\) contains the Boolean operations and cylindrifications, and the \(L\)-reduct of \({\mathcal P}\) is completely representable if and only if \({\mathcal A}\) is completely representable. If \({\mathcal A}\) is finite then so is \({\mathcal P}\). It follows that there is no algorithm to determine whether a finite \(n\)-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable. The author also obtains a new proof that the classes of completely representable \(n\)-dimensional algebras of these types are non-elementary.
reviewed by: Lavinia Ciungu
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 6132826
zbMATH Keywords: algebraic logic
zbMATH Keywords: algebras of relations
zbMATH Keywords: cylindric algebra
zbMATH Keywords: diagonal-free algebra
zbMATH Keywords: polyadic algebra
zbMATH Keywords: polyadic equality algebra
zbMATH Keywords: complete representation
author: Ian Hodkinson
MaRDI profile type: Publication
full work available at URL: s00012-012-0202-3
OpenAlex ID: W2039637520
Wikidata QID: Q59291449
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DOI: 10.1007/S00012-012-0202-3
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description: scientific article; zbMATH DE number 6132826