Simplicial complexes and closure systems induced by indistinguishability relations

Abstract

In this paper, we develop in a more general mathematical context the notion of indistinguishability, which in graph theory has recently been investigated as a symmetry relation with respect to a fixed vertex subset. The starting point of our analysis is to consider a set Ω of functions defined on a universe set U and to define an equivalence relation ≡A on U for any subset A⊆Ω in the following way: u≡Au′ if a(u)=a(u′) for any function a∈A. By means of this family of relations, we introduce the indistinguishability relation ≈ on the power set P(Ω) as follows: for A,A′∈P(Ω), we set A≈A′ if the relations ≡A and ≡A′ coincide. We use then the indistinguishability relation ≈ to introduce several set families on Ω that have interesting order, matroidal and combinatorial properties. We call the above set families the indistinguishability structures of the function system(U,Ω). Furthermore, we obtain a closure system and an abstract simplicial complex interacting each other by means of three hypergraphs having relevance in both theoretical computer science and graph theory. The first part of this paper is devoted to investigate the basic mathematical properties of the indistinguishability structures for arbitrary function systems. The second part deals with some specific cases of study derived from simple undirected graphs and the usual Euclidean real line.