×

Various notions of represetability for cylindric and polyadic algebras. (English) Zbl 1438.03060

Summary: For \(\beta\) an ordinal, let PEA\(_{\beta}\) (SetPEA\(_{\beta}\)) denote the class of polyadic equality (set) algebras of dimension \(\beta\). We show that for any infinite ordinal \(\alpha\), if \(\mathfrak{a}\in\) PEA\(_{\alpha}\) is atomic, then for any \(n < \omega\), the \(n\)-neat reduct of \(\mathfrak{A}\), in symbols \(\mathfrak{Nr}_n\mathfrak{A}\), is a completely representable PEA\(_n\) (regardless of the representability of \(\mathfrak{A}\)). That is to say, for all non-zero \(a\in 2 \mathfrak{Nr}_n\mathfrak{A}\) there is a \(\mathfrak{B}_a\in \) SetPEA\(_n\) and a homomorphism \(f_a: \mathfrak{Nr}_n\mathfrak{A}\to \mathfrak{B}\) such that \(f_a(a)\ne 0\) and \(f_a(\sum X)=\cup_{x\in X}f_a(x)\) for any \(X \subseteq \mathfrak{A}\) for which \(\sum X\) exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under \(\equiv_{\infty,\omega}\). Various notions of representability (such as satisfying the Lyndon conditions, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CA\(_n\)s satisfying the Lyndon conditions coincides with the class of atomic algebras in ElS\(_c\)Nr\(_n\)CA\(_{\omega}\), where El denotes elementary closure and S\(_c\) is the operation of forming complete subalgebras.

MSC:

03G15 Cylindric and polyadic algebras; relation algebras
PDFBibTeX XMLCite
Full Text: DOI