Algebra universalis

, Volume 68, Issue 3, pp 257-285

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

A construction of cylindric and polyadic algebras from atomic relation algebras

  • Ian HodkinsonAffiliated withDepartment of Computing, Imperial College London Email author 


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