A construction of cylindric and polyadic algebras from atomic relation algebras
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.
scientific article; zbMATH DE number 6132826
scientific article