Relation algebras from cylindric algebras. I (Q5956889)

For every \(n\geq 3\), \(\mathbf{S}\mathcal RaCA_n\) is, by definition, the class of subalgebras of relation-algebraic reducts of cylindric algebras of dimension \(n\). It was known that \(\mathbf{S}\mathcal RaCA_4\) is the class of relation algebras, and that, for every \(n\geq 4\), \(\mathbf{S}\mathcal RaCA_n\) is a canonical variety, that is, it is an equational class closed under the formation of perfect extensions. This paper includes a method for constructing a recursive equational axiomatization of \(\mathbf{S}\mathcal RaCA_n\), as well as characterizations of \(\mathbf{S}\mathcal RaCA_n\) in terms of hyperbasis and relativized representations. In particular, for \(n\geq 5\), an algebra \(\mathfrak{A}\) is in \(\mathbf{S}\mathcal RaCA_n\) iff the perfect extension of \(\mathfrak{A}\) has an \(n\)-dimensional hyperbasis (a set of labelled \(n\)-node networks with special properties) iff \(\mathfrak{A}\) has an \(n\)-flat relativized representation iff \(\mathfrak{A}\) has an \(n\)-smooth relativized representation. NEWLINENEWLINENEWLINEPart II is reviewed below.