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Algebraic Characterization of the Local Craig Interpolation Property

100%

Gyenis Z.

Bulletin of the Section of Logic

2018

tom 47

nr 1

The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain.

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

88%

Sayed Ahmed T.

Bulletin of the Section of Logic

2021

tom 50

nr 4

465-511

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.

Ograniczanie wyników

2 Bulletin of the Section of Logic

1 Gyenis Z.

1 Sayed Ahmed T.

1 2021

1 2018

Wyniki wyszukiwania

Algebraic Characterization of the Local Craig Interpolation Property

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