On the representation of Bochvar algebras and their modal expansions

This thesis is a study of weak Kleene logics. In particular, I consider their external versions, which are obtained introducing classical recapture operators to the standard propositional connectives. The increased expressive power obtained greatly strengthens the otherwise weak algebraic properties of weak Kleene logics, which in their external versions are algebraizable in the sense of Blok and Pigozzi, and it makes possible to employ techniques from abstract algebraic logic. Therefore the focus will be on the equivalent algebraic semantics of these logics, namely the quasivariety of Bochvar algebras BCA. Starting from recent results by Bonzio and Pra Baldi about the structure theory of BCA in terms of Płonka sums, I provide a new representation in the form of twist product between a Boolean subalgebra of a BCA and a meet-subsemilattice of the former. In the second part of the thesis I consider the further addition of modalities to the language, obtaining modal external weak Kleene logics. The interaction between external and modal operators will be investigated first by providing these logics with a three-valed Kripke-style semantics, then focusing on their global versions and studying their equivalent algebraic semantics.