Various interplays between relation and cylindric algebras

TL;DR

This paper investigates the non-elementarity of classes of cylindric and related algebras using model theoretic and rainbow construction techniques, extending previous results and exploring weakly vs. strongly representable atom structures.

Contribution

It generalizes non-elementarity results for classes of relation and cylindric algebras, introduces new blow-up and blur constructions, and discusses open problems in algebraic logic.

Findings

Classes of neat reducts are not elementary.

Certain classes of algebras are not closed under elementary equivalence.

Weakly representable atom structures can be formalized using blow-up and blur methods.

Abstract

Using model theoretic techniques that proved that the class of $n$ neat reducts of $m$ dimensional cylindric algebras, $\Nr_n\CA_m$, is not elementary, we prove the same result for $\Ra\CA_k$, $k\geq 5$, and we show that $\Ra\CA_k\subset S_c\Ra\CA_k$ for all $k\geq 5$. Conversely, using the rainbow construction for cylindric algebra, we show that several classes of algebras, related to the class $\Nr_n\CA_m$, $n$ finite and $m$ arbitrary, are not elementary. Our results apply to many cylindric-like algebras, including Pinter's substitution algebras and Halmos' polyadic algebras with and without equality. The techniques used are essentially those used by Hirsch and Hodkinson, and later by Hirsch in \cite{hh} and \cite{r}. In fact, the main result in \cite{hh} follows from our more general construction. Finally we blow up a little the {\it blow up and blur construction} of Andr\'eka nd N\'emeti, showing that various constructions of weakly representable atom structures that are not strongly representable, can be formalized in our blown up, blow up and blur construction, both for relation and cylindric algebras. Two open problems are discussed, in some detail, proposing ideas. One is whether class of subneat reducts are closed under completions, the other is whether there exists a weakly representable $\omega$ dimensional atom structure, that is not strongly representable. For the latter we propose a lifting argument, due to Monk, applied to what we call anti-Monk algebras (the algebras, constructed by Hirsch and Hodkinson, are atomic, and their atom structure is stongly representable.)