Comptes Rendus
Topology/Computer science
Simplicial complexes and closure systems induced by indistinguishability relations
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 991-1021.

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 AΩ in the following way: uAu if a(u)=a(u) for any function aA. By means of this family of relations, we introduce the indistinguishability relation ≈ on the power set P(Ω) as follows: for A,AP(Ω), we set AA 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.

Nous développons dans ce texte la notion d'indistinguabilité dans un contexte mathématique plus général. Cette notion a en effet été récemment étudiée en théorie des graphes, comme une relation de symétrie relativement aux sommets fixés. Le point de départ de notre analyse est de considérer un ensemble Ω de fonctions définies sur un ensemble univers U et de définir pour tout sous-ensemble AΩ une relation d'équivalence A sur U par uu si a(u)=a(u) pour toute fonction aA. Au moyen de cette famille de relations, nous introduisons la relation d'indistinguabilité ≈ sur l'ensemble puissance P(Ω) de la façon suivante : pour A,AP(Ω), nous posons AA si les relations A et A coïncident. Nous utilisons cette relation d'indistinguabilité ≈ pour définir plusieurs familles d'ensembles sur Ω ayant d'intéressantes propriétés d'ordre, de matroïde et combinatoires. Nous appelons les familles d'ensembles ci-dessus les structures indistinguables du système de fonctions (U,Ω). De plus, nous obtenons un système de clôture et un complexe simplicial abstrait interagissant l'un l'autre au travers de trois hypergraphes, qui sont significatifs aussi bien en théorie des graphes qu'en informatique théorique. La première partie du texte est dédiée à l'étude les propriétés mathématiques élémentaires des structures d'indistinguabilité pour les systèmes de fonctions arbitraires. La seconde partie traite de quelques cas particuliers dérivés des graphes non orientés simples et de la droite euclidienne réelle usuelle.

Published online:
DOI: 10.1016/j.crma.2017.09.010

Giampiero Chiaselotti 1; Tommaso Gentile 1; Federico Infusino 1

1 Department of Mathematics and Informatics, University of Calabria, Via Pietro Bucci, Cubo 30B, 87036 Arcavacata di Rende (CS), Italy
     author = {Giampiero Chiaselotti and Tommaso Gentile and Federico Infusino},
     title = {Simplicial complexes and closure systems induced by indistinguishability relations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {991--1021},
     publisher = {Elsevier},
     volume = {355},
     number = {9},
     year = {2017},
     doi = {10.1016/j.crma.2017.09.010},
     language = {en},
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TI  - Simplicial complexes and closure systems induced by indistinguishability relations
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PB  - Elsevier
DO  - 10.1016/j.crma.2017.09.010
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%T Simplicial complexes and closure systems induced by indistinguishability relations
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Giampiero Chiaselotti; Tommaso Gentile; Federico Infusino. Simplicial complexes and closure systems induced by indistinguishability relations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 991-1021. doi : 10.1016/j.crma.2017.09.010.

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