Monotonic Distributive Semilattices

TL;DR

This paper introduces and studies a new class of distributive meet-semilattices with a monotonic modal operator, providing representation theory and a topological duality, extending to subclasses.

Contribution

It develops the theory of distributive meet-semilattices with a monotonic modal operator, including representation and duality results, advancing algebraic semantics for non-classical logics.

Findings

Established a representation theory for these semilattices.

Developed a topological duality extending to subclasses.

Connected the algebraic structures to logical semantics.

Abstract

In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the $\{\rightarrow,\wedge,\top\}$-fragment of intuitionistic logic is the variety of implicative meet-semilattices \cite{CelaniImplicative} \cite{ChajdaHalasKuhr}. In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator $m$. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we show how our new duality extends to some particular subclasses.