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

Summary/Abstract: Fix a finite ordinal n≥3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, 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 CAns. 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∈PEAα, then for any n<ω, NrnA is a completely representable PEAn. We show that for any α≥ω, the class of completely representable algebras in certain reducts of PAαs, 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 PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second.

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