On Bochvar Algebras and Regular Double Stone Algebras

Bochvar algebras and regular double Stone algebras are the equivalent algebraic semantics of logics widely used for reasoning about partial information in computer science and AI. Viewed as categories, they are both equivalent to certain categories based on Boolean algebras. Despite this similarity, the precise relationship between Bochvar algebras and regular double Stone algebras has not been clearly established. In this paper, we obtain the following main results: i) we provide a different proof of the categorical equivalence result for Bochvar algebras, using a modified twist product construction in place of a Płonka sum construction; ii) we show that the functor witnessing this equivalence can be decomposed into a standard twist product and a translation map yielding Bochvar algebras as term reducts of regular double Stone algebras; iii) we prove that the algebraic category of regular double Stone algebras is equivalent to a full and reflective subcategory of the category of Bochvar algebras, and (with certain provisos) that the embedding functor is dense – namely, every Bochvar algebra arises as a colimit of its regular double Stone subalgebras.