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On the Compactness Property of Mereological Spaces

Polkowski Lech

Fundamenta Informaticae

2020

Vol. 172, nr 1

73--95

Continuing our work on mass-based rough mereologies, we make use of the Stone representation theorem for complete Boolean algebras and we exhibit the existence of a finite base in each mereological space. Those bases in turn allow for the introduction of distributed mereologies; regarding each element of the base as a mereological space, we propose a mechanism for fusing those mereological spaces into a global distributed mereological space. We define distributed mass-assignments and rough inclusions pointing to possible applications.

Dualities for Algebras of Fitting's Many-Valued Modal Logics

Maruyama Y.

Fundamenta Informaticae

2011

Vol. 106, nr 2-4

273-294

Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting’s L-valued logic and L-valued modal logic for a finite distributive lattice L. Building upon the theory of natural dualities, which is a universal algebraic theory of categorical dualities, we establish a Jonsson-Tarski-style duality for algebras of L-valued modal logic, which encompasses J´onsson-Tarski duality for modal algebras as the case L = 2. We also discuss how the dualities change when the algebras are enriched by truth constants. Topological perspectives following from the dualities provide compactness theorems for the logics and the effective classification of categories of algebras involved, which tells us that Stone-type duality makes it possible to use topology for logic and algebra in significant ways.