On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Sayed Ahmed, Tarek  

Bulletin of the Section of Logic, Tome 50 (2021) no. 4, pp. 465-511

Résumé

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_ns, and the class CRRA of completely representable 𝖱𝖠s 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_ns. 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 𝔄∈𝖯𝖤𝖠_α, then for any n, 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_αs, 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_αs for α an infinite ordinal, proving negative results for the first and positive ones for the second.

Détail

Keywords: Algebraic logic, relation algebras, cylindric algebras, polyadic algebras, complete representations

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