From Multirelations to Meet-Relations: A Relational Duality for Semilattices with Adjunctions

TL;DR

This paper establishes a duality framework connecting semilattices with adjunctions to relational structures, enabling a better understanding of their semantics and categorical equivalences.

Contribution

It introduces a duality for semilattices with adjunctions using meet-relations and relates it to multirelational semantics, unifying different frameworks.

Findings

Dual equivalence between modal semilattices and MoS-spaces.

Categories RelSP and SLata are shown to be isomorphic.

Multirelational structures can be recovered from meet-relations under normality conditions.

Abstract

We develop a relational duality for semilattices with adjunctions (SLatas) based on binary meet-relations. First, we introduce the category of MoS-spaces and establish a dual equivalence with modal semilattices. Then, by means of A-relations, we define the category RelSP and prove a dual equivalence between SLata and RelSP. To compare this framework with the multirelational semantics previously developed for SLatas, we introduce the notion of normal mS-space and show that, under this condition, the multirelational structure can be canonically recovered from a meet-relation, and conversely. As a consequence, we prove that the categories RelSP and SLataSp are isomorphic.