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 $A⊆Ω$ 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.

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 $A⊂Ω$ une relation d'équivalence $≡A$ sur U par $u≡u′$ si $a(u)=a(u′)$ pour toute fonction $a∈A$. Au moyen de cette famille de relations, nous introduisons la relation d'indistinguabilité ≈ sur l'ensemble puissance $P(Ω)$ de la façon suivante : pour $A,A′∈P(Ω)$, nous posons $A≈A′$ 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.

Accepted:
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
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.010/

[1] G.E. Andrews Euler's “De Partitio numerorum”, Bull. Amer. Math. Soc., Volume 44 (2007) no. 4, pp. 561-573

[2] N. Apollonio; M. Caramia Recognizing Helly edge path tree graphs and their clique graphs, Discrete Appl. Math., Volume 159 (2011), pp. 1166-1175

[3] N. Apollonio; B. Simeone Improved approximation of maximum vertex coverage problem on bipartite graphs, SIAM J. Discrete Math., Volume 28 (2014) no. 3, pp. 1137-1151

[4] N. Apollonio; B. Simeone The maximum vertex coverage problem on bipartite graphs, Discrete Appl. Math., Volume 165 (2014), pp. 37-48

[5] N. Apollonio; M. Caramia; P.G. Franciosa On the Galois lattice of bipartite distance hereditary graphs, Discrete Appl. Math., Volume 190 (2015), pp. 13-23

[6] R.A. Bayley Orthogonal partitions in designed experiments, Des. Codes Cryptogr., Volume 8 (1996) no. 3, pp. 45-77

[7] R.A. Bayley Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, Cambridge, UK, 2004 (387 p)

[8] C. Berge Hypergraphs: Combinatorics of Finite Sets, Elsevier, Amsterdam, 1984

[9] L. Biacino Generated envelopes, J. Math. Anal. Appl., Volume 172 (1993), pp. 179-190

[10] L. Biacino; G. Gerla An extension principle for closure operators, J. Math. Anal. Appl., Volume 198 (1996), pp. 1-24

[11] G. Birkhoff Lattice Theory, American Mathematical Society, Providence, Rhode Island, 1967

[12] C. Bisi; G. Chiaselotti A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture, Adv. Geom., Volume 13 (2013) no. 1, pp. 1-27

[13] C. Bisi; G. Chiaselotti; G. Marino; P.A. Oliverio A natural extension of the Young partition lattice, Adv. Geom., Volume 15 (2015) no. 3, pp. 263-280

[14] C. Bisi; G. Chiaselotti; D. Ciucci; T. Gentile; F. Infusino Micro and macro models of granular computing induced by the indiscernibility relation, Inf. Sci., Volume 388–389 (2017), pp. 247-273

[15] C. Bisi; G. Chiaselotti; T. Gentile; P.A. Oliverio Dominance order on signed partitions, Adv. Geom., Volume 17 (2017) no. 1, pp. 5-29

[16] T. Boykett Orderly algorithm to enumerate central groupoids and their graphs, Acta Math. Sin. Engl. Ser., Volume 23 (2007) no. 2, pp. 249-264

[17] T. Boykett Rectangular groupoids and related structures, Discrete Math., Volume 313 (2013), pp. 1409-1418

[18] G. Cattaneo; G. Chiaselotti; D. Ciucci; T. Gentile On the connection of hypergraph theory with formal concept analysis and rough set theory, Inf. Sci., Volume 330 (2016), pp. 342-357

[19] G. Cattaneo; G. Chiaselotti; P.A. Oliverio; F. Stumbo A new discrete dynamical system of signed integer partitions, Eur. J. Comb., Volume 55 (2016), pp. 119-143

[20] B. Chen; M. Yan Eulerian stratification of polyhedra, Adv. Appl. Math., Volume 21 (1998), pp. 22-57

[21] B. Chen; M. Yan The geometric cone relations for simplicial and cubical complexes, Discrete Math., Volume 183 (1998), pp. 39-46

[22] B. Chen; S.T. Yau; Y.N. Yeh Graph homotopy and Graham homotopy, Discrete Math., Volume 241 (2001), pp. 153-170

[23] G. Chiaselotti; T. Gentile Intersection properties of maximal directed cuts in digraphs, Discrete Math., Volume 340 (2017), pp. 3171-3175

[24] G. Chiaselotti; G. Infante; G. Marino New results related to a conjecture of Manickam and Singhi, Eur. J. Comb., Volume 29 (2008) no. 2, pp. 361-368

[25] G. Chiaselotti; T. Gentile; P.A. Oliverio Parallel and sequential dynamics of two discrete models of signed integer partitions, Appl. Math. Comput., Volume 232 (2014), pp. 1249-1261

[26] G. Chiaselotti; D. Ciucci; T. Gentile Simple graphs in granular computing, Inf. Sci., Volume 340–341 (2016), pp. 279-304

[27] G. Chiaselotti; T. Gentile; F. Infusino Dependency structures for decision tables, Int. J. Approx. Reason., Volume 88 (2017), pp. 333-370

[28] G. Chiaselotti; T. Gentile; F. Infusino Knowledge pairing systems in granular computing, Knowl.-Based Syst., Volume 124 (2017), pp. 144-163

[29] G. Chiaselotti; T. Gentile; F. Infusino; P.A. Oliverio The adjacency matrix of a graph as a data table. A geometric perspective, Ann. Mat. Pura Appl., Volume 196 (2017) no. 3, pp. 1073-1112

[30] G. Chiaselotti, T. Gentile, F. Infusino, P. Oliverio, Local dissymmetry on graphs and related algebraic structures, preprint.

[31] R. Diestel Graph Theory, Grad. Texts Math., Springer, 2010

[32] T. Eiter; G. Gottlob Identifying the minimal transversals of a hypergraph and related problems, SIAM J. Comput., Volume 24 (1995), pp. 1278-1304

[33] K. Elbassioni On the complexity of monotone dualization and generating minimal hypergraph transversals, Discrete Appl. Math., Volume 32 (2008) no. 2, pp. 171-187

[34] P. Erdös; A. Rényi Asymmetric graphs, Acta Math. Hung., Volume 14 (1963) no. 3–4, pp. 295-315

[35] B. Ganter; R. Wille Formal Concept Analysis. Mathematical Foundations, Springer-Verlag, 1999

[36] L. Guo; F. Huang; Q. Lia; G. Zhang Power contexts and their concept lattices, Discrete Math., Volume 311 (2011), pp. 2049-2063

[37] A. Gyárfás; J. Lehel Hypergraph families with bounded edge cover or transversal number, Combinatorica, Volume 3 (1983) no. 3–4, pp. 351-358

[38] M. Hagen Lower bounds for three algorithms for transversal hypergraph generation, Discrete Appl. Math., Volume 157 (2009), pp. 1460-1469

[39] Graph Symmetry. Algebraic Methods and Applications (G. Hahn; G. Sabidussi, eds.), NATO ASI Ser., vol. 497, Springer, 1997

[40] P.W. Harley Metric and symmetric spaces, Proc. Amer. Math. Soc., Volume 43 (1974) no. 2, pp. 428-430

[41] A. Huang; H. Zhao; W. Zhu Nullity-based matroid of rough sets and its application to attribute reduction, Inf. Sci., Volume 263 (2014), pp. 153-165

[42] W.J. Keith A bijective toolkit for signed partitions, Ann. Comb., Volume 15 (2011), pp. 95-117

[43] A. Kelarev; C.E. Praeger On transitive Cayley graphs of groups and semigroups, Eur. J. Comb., Volume 24 (2003) no. 1, pp. 59-72

[44] A. Kelarev; S.J. Quinn Directed graphs and combinatorial properties of semigroups, J. Algebra, Volume 251 (2002) no. 1, pp. 16-26

[45] A. Kelarev; J. Ryan; J. Yearwood Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math., Volume 309 (2009), pp. 5360-5369

[46] H.W. Martin Metrization of symmetric spaces and regular maps, Proc. Amer. Math. Soc., Volume 35 (1972), pp. 269-274

[47] L. Molnár Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn's version of Wigner's theorem, J. Funct. Anal., Volume 194 (2002), pp. 248-262

[48] P. Pagliani; M.K. Chakraborty A Geometry of Approximation. Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns, Springer, 2008

[49] B. Poonen Union-closed families, J. Comb. Theory, Ser. A, Volume 59 (1992), pp. 253-268

[50] C.M. Reidys Sequential dynamical systems over words, Ann. Comb., Volume 10 (2006) no. 4, pp. 481-498

[51] C.M. Reidys Combinatorics of sequential dynamical systems, Discrete Math., Volume 308 (2008) no. 4, pp. 514-528

[52] J. Tanga; K. Shea; F. Min; W. Zhu A matroidal approach to rough set theory, Theor. Comput. Sci., Volume 471 (2013), pp. 1-11

[53] P.M. Van den Broek Symmetry transformations in indefinite metric spaces: a generalization of Wigner's theorem, Physica A, Volume 127 (1984) no. 3, pp. 599-612

[54] D.J.A. Welsh Matroid Theory, Academic Press, 1976

[55] W. Zhu; S. Wang Rough matroids based on relations, Inf. Sci., Volume 232 (2013), pp. 241-252

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