Description

• Title:
  On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results
• Creator:
  Sayed Ahmed, Tarek
• Contributor:
  Cairo University, Department of Mathematics, Faculty of Science;
  rutahmed@gmail.com
• Description:
  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/p>
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nc-nd/4.0
• Date:
  2022.03.10;
  2021.07.21
• Type:
  Other
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Wydawnictwo Uniwersytetu Łódzkiego
• Relation:
  Bulletin of the Section of Logic;4;
  50
• Subject:
  Algebraic logic;
  relation algebras;
  cylindric algebras;
  polyadic algebras;
  complete representations
• Identifier:
  0138-0680;
  http://hdl.handle.net/11089/41067 ;
  2449-836X;
  10.18778/0138-0680.2021.17
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  check license
• Type:
  other