A Note on Proper Relational Structures

TL;DR

This paper presents an algorithm to convert relational structures into proper ones, preserving key properties, facilitating completeness proofs in modal logic through applications in Simplicial Semantics.

Contribution

The paper introduces a novel translation algorithm that produces proper relational structures while maintaining important classical properties.

Findings

Preserves transitivity and Euclidean properties in the translation.

Enables construction of proper canonical structures for modal logic.

Facilitates completeness proofs in Simplicial Semantics.

Abstract

In this note we provide an algorithm for translating relational structures into "proper" relational structures, i.e., those such that there is no pair of worlds w and u such that w is accessible from u for every agent. In particular, our method of translation preserves many classical properties of relational structures, such as transitivity and the Euclidean property. As a result, this method of translation has many applications in the literature on Simplicial Semantics for modal logic, where the creation of proper canonical relational structures is a common step in proofs of completeness.