Relation algebras from cylindric algebras. II
If \(K\) and \(L\) are elementary classes with \(K\subseteq L\), then \(K\) is ``not finitely axiomatizable relative to'' \(L\) iff there is no set \(\Sigma\) of first-order sentences with the property that an algebra is in \(K\) iff it is in \(L\) and is also a model of \(\Sigma\). By definition, if \(3\leq m<n<\omega\), then \(\mathbf{S}{\mathsf RaCA}_n\) is the class of subalgebras of relation-algebraic reducts of cylindric algebras of dimension \(n\), \(\mathbf{S}{\mathsf Nr}_m{\mathsf CA}_n\) is the class of subalgebras of neat \(m\)-reducts of cylindric algebras of dimension \(n\), and \({\mathsf RA}_n\) is the class of relation algebras whose perfect extensions have an \(n\)-dimensional relational basis. In this paper it is proved that if \(3\leq m<n<\omega\), then \(\mathbf{S}{\mathsf Nr}_m{\mathsf CA}_{n+1}\) is not finitely axiomatizable relative to \(\mathbf{S}{\mathsf Nr}_m{\mathsf CA}_n\) and \(\mathbf{S}{\mathsf RaCA}_{n+1}\) is not finitely axiomatizable relative to \(\mathbf{S}{\mathsf RaCA}_n\). The construction also shows that \(\mathbf{S}{\mathsf RaCA}_{n+1}\) is not finitely axiomatizable relative to \({\mathsf RA}_{n+1}\) and \(\mathbf{S}{\mathsf RaCA}_{n+1}\) is not finitely axiomatizable relative to \({\mathsf RA}_{n+1}\cap\mathbf{S}{\mathsf RaCA}_n\). The key to these results is the construction, for \(4\leq n<\omega\) and \(0<r<\omega\), of a complete atomic symmetric integral relation algebra \(\mathfrak{A}(n,r)\) whose atoms are \(1\text{{}'}\) and \(a^k(i,j)\) for \(i<n-1\), \(j<r\), and \(k<(nr)^{nr}\), and whose forbidden cycles are \((1\text{{}'},x,y)\) where \(x\neq y\), and also \((a^k(i,j),a^{k'}(i,j),a^{k''}(i,j'))\) where \(j\leq j'\). The algebra \(\mathfrak{A}(n,r)\) is in \({\mathsf RA}_{n+1}\cap\mathbf{S}{\mathsf RaCA}_n\) and not in \(\mathbf{S}{\mathsf RaCA}_{n+1}\), but an ultraproduct of these algebras does lie in \(\mathbf{S}{\mathsf RaCA}_{n+1}\). These are significant results that considerably expand the hierarchies of non-representable relation algebras and cylindric algebras. NEWLINENEWLINENEWLINEPart I is reviewed above.
scientific article; zbMATH DE number 1713720
scientific article