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ON THE STRUCTURE OF BOCHVAR ALGEBRAS

Published online by Cambridge University Press:  09 May 2024

STEFANO BONZIO*
Affiliation:
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF CAGLIARI VIA OSPEDALE 72 09124 CAGLIARI ITALY
MICHELE PRA BALDI
Affiliation:
FISPPA DEPARTMENT UNIVERSITY OF PADUA PIAZZA CAPITANIATO 3 35139 PADOVA ITALY E-mail: michele.prabaldi@unipd.it

Abstract

Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP).

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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