An algebraic study of S5-modal G\"odel logic

TL;DR

This paper explores the algebraic structures underlying S5-modal G"odel logic, introducing subvarieties, duality, and properties of monadic G"odel algebras, advancing the understanding of their logical and algebraic features.

Contribution

It develops a topological duality for monadic G"odel algebras and characterizes subvarieties and free algebras, extending the algebraic understanding of S5-modal G"odel logic.

Findings

Identified three locally finite subvarieties of monadic G"odel algebras.

Established a topological duality for these algebras.

Characterized free algebras and proved a Glivenko-type theorem.

Abstract

In this paper we continue the study of the variety $\mathbb{MG}$ of monadic G\"odel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of G\"odel logic, which is equivalent to the one-variable monadic fragment of first-order G\"odel logic. We show three families of locally finite subvarieties of $\mathbb{MG}$ and give their equational bases. We also introduce a topological duality for monadic G\"odel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic G\"odel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over $n$ generators.