On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results - Sayed Ahmed, Tarek (Bulletin of the Section of Logic, 2021)
Fix a finite ordinal \(n≥ 3\) and let \(α\) be an arbitrary ordinal. Let \(CA_n\) denote the class of cylindric algebras of dimension \(n\) and \(RA\) denote the class of relation algebras. Let \(PA_α(PEA_α)\) stand for the class of polyadic (equality) algebras of dimension \(α\). We reprove that the class \(CRCA_n\) of completely representable \(CA_n\)s, and the class \(CRRA\) of completely representable \(RA\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(V\) between polyadic algebras of dimension \(n\) and diagonal free \(CA_n\)s. We show that that the class of completely and strongly representable algebras in \(V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(CRRA\) is not closed under \(_∞,ω\). In contrast, we show that given \(α≥ ω\), and an atomic \(APEA_α\), then for any \(n<ω\), \(Nr_nA\) is a completely representable \(PEA_n\). We show that for any \(α≥ ω\), the class of completely representable algebras in certain reducts of \(PA_α\)s, that happen to be varieties, is elementary. We show that for \(α≥ ω\), the the class of polyadic-cylindric algebras dimension \(α\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(PA_α\)s for \(α\) an infinite ordinal, proving negative results for the first and positive ones for the second.