Algebra universalis

, Volume 68, Issue 3–4, pp 257–285

A construction of cylindric and polyadic algebras from atomic relation algebras

Open Access
Article

Abstract

Given a simple atomic relation algebra \({\mathcal{A}}\) and a finite n ≥ 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra \({\mathcal{P}}\) such that for any subsignature L of the signature of \({\mathcal{P}}\) that contains the boolean operations and cylindrifications, 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 (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.

2010 Mathematics Subject Classification

Primary: 03G15 

Key words and phrases

algebraic logic algebras of relations cylindric algebra diagonal free cylindric algebra polyadic algebra polyadic equality algebra complete representation undecidable non-elementary 

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© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Department of ComputingImperial College LondonLondonUK

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