Description

• Title:
  A Category of Ordered Algebras Equivalent to the Category of Multialgebras
• Creator:
  Coniglio, Marcelo Esteban;
  de Toledo, Guilherme Vicentin
• Contributor:
  Coniglio, Marcelo Esteban - University of Campinas (Unicamp), Institute of Philosophy and the Humanities (IFCH) and Centre for Logic, Epistemology and the History of Science (CLE)(UNICAMP);
  de Toledo, Guilherme Vicentin - Bar Ilan University, Department of Computer Science, The Spiegel Mathematics & Computer Center;
  Coniglio, Marcelo Esteban - coniglio@unicamp.br;
  de Toledo, Guilherme Vicentin - guivtoledo@gmail.com
• Description:
  It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s.We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\).The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nc-nd/4.0
• Date:
  2024.01.04;
  2023.08.16
• Type:
  Other
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Wydawnictwo Uniwersytetu Łódzkiego
• Relation:
  Bulletin of the Section of Logic;4;
  52
• Subject:
  multialgebras;
  ordered algebras;
  non-deterministic semantics
• Identifier:
  0138-0680;
  http://hdl.handle.net/11089/49161 ;
  2449-836X;
  10.18778/0138-0680.2023.23
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  check license
• Type:
  other