Connectivity structure of multiple relations

TL;DR

This paper introduces the concept of connectivity structures for multiple relations across indexed sets, demonstrating that any such structure can be represented by a compatible relation, with a proof akin to Brunn's theorem.

Contribution

It formalizes the notion of connectivity structures for multiple relations and proves a fundamental theorem linking these structures to compatible relations.

Findings

Every connectivity structure corresponds to a compatible multiple relation.

The paper establishes a Brunn-type theorem for these structures.

Provides a mathematical foundation for analyzing compatibility in complex systems.

Abstract

The prime purpose of this paper is to define the connectivity structure , on a set E, of any multiple relation defined on a family of sets indexed by E, such a relation expressing compatibility between the states of different systems (thus a full compatibility indicates absence of any connection). We then demonstrate a "Brunn's theorem" for those multiple relations, that is the fact that every connectivity structure is the connectivity structure of such a relation.