Representations of polyadic-like equality algebras

Ferenczi, Miklós

doi:10.1007/s00012-015-0360-1

Cited by 3 publications

(1 citation statement)

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“…Our results further emphasizes the dichotomy existing between the cylindric paradigm and the polyadic one, a phenomena recurrent in the literature of Tarski's cylindric algebras and Halmos' polyadic algebras, with algebras 'in between' such as Ferenzci's cylindric-polyadic algebras with and without equality, aspiring to share only nice desirable properties of both. Such properties, some of which are thoroughly investigated below, include (not exclusively) finite axiomatizablity of the variety of representable algebras, the canonicity and atom-canonicity of such varieties, decidability of its equational/ or and universal theory, and the first order definability of the notion of complete representability [4,5,6].…”

Section: Polyadic Paradigmmentioning

confidence: 99%

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

Ahmed

2021

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Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), 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 \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.

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Section: Polyadic Paradigmmentioning

confidence: 99%