Probability over Płonka sums of Boolean algebras: States, metrics and topology

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.