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

dc.contributor.authorSayed Ahmed, Tarek
dc.date.accessioned2022-03-10T17:28:35Z
dc.date.available2022-03-10T17:28:35Z
dc.date.issued2021-07-21
dc.identifier.issn0138-0680
dc.identifier.urihttp://hdl.handle.net/11089/41067
dc.description.abstractFix 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/p>en
dc.language.isoen
dc.publisherWydawnictwo Uniwersytetu Łódzkiegopl
dc.relation.ispartofseriesBulletin of the Section of Logic;4en
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0
dc.subjectAlgebraic logicen
dc.subjectrelation algebrasen
dc.subjectcylindric algebrasen
dc.subjectpolyadic algebrasen
dc.subjectcomplete representationsen
dc.titleOn Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Resultsen
dc.typeOther
dc.page.number465-511
dc.contributor.authorAffiliationCairo University, Department of Mathematics, Faculty of Scienceen
dc.identifier.eissn2449-836X
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dc.contributor.authorEmailrutahmed@gmail.com
dc.identifier.doi10.18778/0138-0680.2021.17
dc.relation.volume50


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