Bochvar algebras: A categorical equivalence and the generated variety

TL;DR

This paper establishes a categorical equivalence for Bochvar algebras, provides an axiomatisation of the generated variety, and explores the subvariety lattice involving Boolean algebras and semilattices.

Contribution

It introduces a categorical equivalence for Bochvar algebras and axiomatizes the variety generated by them, advancing the algebraic understanding of Bochvar's external logic.

Findings

Categorical equivalence between Bochvar algebras and pairs of Boolean algebras and meet-subsemilattices.

Axiomatisation of the variety generated by Bochvar algebras.

Characterization of the join of Boolean algebras and semilattices within the subvariety lattice.

Abstract

The proper quasivariety BCA of Bochvar algebras, which serves as the equivalent algebraic semantics of Bochvar's external logic, was introduced by Finn and Grigolia in and extensively studied in a recent work by two of these authors. In this paper, we show that the algebraic category of Bochvar algebras is equivalent to a category whose objects are pairs consisting of a Boolean algebra and a meet-subsemilattice (with unit) of the same. Furthermore, we provide an axiomatisation of the variety $V(BCA) generated by Bochvar algebras. Finally, we axiomatise the join of Boolean algebras and semilattices within the lattice of subvarieties of V(BCA).