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.

中文翻译：

---

布尔代数 Płonka 和的概率：状态、度量和拓扑

本文介绍了对合双半格的状态概念，该变体扮演弱 Kleene 逻辑的代数对应物的角色，其元素表示为布尔代数的 Płonka 和。我们研究了 Płonka 和表示中的（布尔）代数的对合二分格上的状态和概率测度之间的关系，以及这些代数的直接限制。此外，我们研究了对合双半格的度量完成，作为伪度量空间，以及由伪度量引起的拓扑。