The elementary closure of the class Nr_nCA_m for m\geq n+1 is not finitely axiomatizable, futhermore for any finite k\geq 1, there is A\in Nr_{\omega}CA_{\omeg+k}that is not SNr_{\omega}CA_{\omega+k+1}

TL;DR

This paper demonstrates that the elementary closure of certain algebraic classes related to cylindric algebras is not finitely axiomatizable, providing constructions and theories for various finite and infinite dimensions.

Contribution

It introduces new pseudo-elementary theories for classes of cylindric algebras and proves their non-finite axiomatizability using Monk-like algebras and ultraproducts.

Findings

The class Nr_nCA_m is pseudo elementary for various n and m.

Finite Monk-like algebras exist that are in Nr_nCA_m but not in SNr_nCA_{m+1}.

There are algebras in infinite dimensions that are not in SNr_{α}CA_{α+k+1}.

Abstract

We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is pseudo elementary. When n and m are finite we use a two sorted theory, when n is finite and m infinite we use a three sorted one, and finally when both are infinite we use a four sorted defining theory. Our non finite axiomatizability result, follows from the fact that for 2<n<m, and any r\in \omega there exists a finite (Monk like) algebra C(m,n,r), such that C(m,n,r)\in Nr_nCA_m C(m,n,r)\notin SNr_nCA_{m+1}, and any non trivial ultraproduct on r of such algebras in in ElNr_nCA_m. Finally we use such algebras, to show that for infinite dimension there is an algebra A\in Nr_{\alpha}\CA_{\apha+k} that is not in SNr_{\alpha}CA_{\alpha+k+1} (\alpha an infinite ordinal).