, Volume 68, Issue 3-4, pp 257-285,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 17 Oct 2012

A construction of cylindric and polyadic algebras from atomic relation algebras


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.

Presented by J. Raftery.
The research was partially supported by UK EPSRC grant GR/S19905/01.