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

Abstract

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 Jonsson-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. The author is grateful to Professor Susumu Hayashi for his encouragement, to Shohei Izawa for his comments and discussions, and to Kentaro Sato for his suggesting a similar result to Theorem 2.5 for the category of algebras of Lukasiewicz n-valued logic.