Atom-canonicity and complete representations for cylindric-like algebras, and omitting types for the clque guarded fragment of first order logic

TL;DR

This paper demonstrates limitations of atom-canonicity and elementary properties in cylindric algebras, using rainbow constructions, and explores implications for omitting types in various fragments of first-order logic.

Contribution

It establishes new non-atom-canonicity and non-elementarity results for classes of cylindric algebras, extending these findings to transfinite cases and connecting them to logical omitting types theorems.

Findings

SNr_nCA_{n+k} is not atom-canonical for k≥3

S_cNr_nCA_{n+k} is not elementary for k≥3

Omitting types theorems are affected in various logical fragments

Abstract

Fix a finite ordinal n>2. We show that there exists an atomic, simple and countable representable CA_n, such that its minimal completion is outside SNr_nCA_{n+3}. Hence, for any finite k\geq 3, the variety SNr_nCA_{n+k} is not atom-canonical, so that the variety of CA_n's having n+k-flat representations is not atom-canonical, too. We show, for finite k\geq 3, that S_cNr_nCA_{n+k} is not elementary, hence the class of CA_n's having complete n+3-smooth representations is not elementary. We obtain analogous results by replacing flat and smooth, respectively, by (the weaker notion of) square; this give a stronger result in both cases and here we can allow k to be infinite. Our results are proved using rainbow constructions for CA's. We lift the negative result on atom-canonicity to the transfinite. We also show that for any ordinal \alpha\geq \omega, for any finite k\geq 1, and for any r\in \omega, there exists an atomic algebra A_r\in SNr_\alphaCA_{\alpha+k}\sim SNr_nCA_{\alpha+k+1}, such that \Pi_{r/U} A_r\in RCA_{\alpha} where U is any non--principal ultrafilter on \omega. Reaping the harvest of our algebraic results we investigate a plethora of omitting types theorems for variants of first logic including its finite variable fragments and its packed fragment.