Probability over Plonka sums of Boolean algebras: states, metrics and topology

TL;DR

This paper explores the relationship between states and probability measures on involutive bisemilattices, represented as Plonka sums of Boolean algebras, and investigates their topological and metric properties.

Contribution

It introduces the concept of states for involutive bisemilattices and analyzes their connections with probability measures and topological structures.

Findings

States correspond to probability measures on component Boolean algebras.

Involutive bisemilattices can be completed as pseudometric spaces.

The induced topology relates to the algebraic and probabilistic structure.

Abstract

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