Dependency relations

  • G. ChiaselottiEmail author
  • F. Infusino
  • P. A. Oliverio


In this paper, we introduce a notion of dependency between subsets of an arbitrary fixed non-empty set \(\Omega \). To be more detailed, we introduce a preorder \(\leftarrow \) on the power set \(\mathcal {P}(\Omega )\) having the further property that \(B \leftarrow A\) if and only if \(\{b\} \leftarrow A\) for any \(b \in B\). We shall argue that this relation generalizes well-studied notions of dependence occurring in such fields as linear algebra, topology, and combinatorics. Furthermore, we show that this relation is characterized by two set operators whose fixed points have interesting geometric and order-theoretic properties. After giving some some elementary results about such a dependency relation, we provide some specific examples taken from graph theory. An interesting property we will provide consists of the possibility to characterize partial orders on a finite lattice in terms of a suitable dependency relation. Finally, we introduce and analyze some specific classes of dependency relations, namely attractive and anti-attractive dependency relations.


Abstract dependency Closure systems Abstract simplicial complexes Graphs 

Mathematics Subject Classification

Primary 08A02 08A05 06A06 Secondary 05C50 05C75 



We are extremely thankful to the unknown reviewers whose thorough objections and suggestions have been very useful in order to improve the quality of our paper.


  1. 1.
    Armstrong, W.W.: Dependency Structures of Data Base Relationships. Information Processing, vol. 74, pp. 580–583. North-Holland, Amsterdam (1974)Google Scholar
  2. 2.
    Baldwin, J.T.: Recursion theory and abstract dependence. In: Metakides, G. (ed.) Patras Logic Symposyum, pp. 67–76. North-Holland, Amsterdam (1982)CrossRefGoogle Scholar
  3. 3.
    Baldwin, J.T.: First-order theories of abstract dependence relations. Ann. Pure Appl. Logic 26, 215–243 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biacino, L.: Generated envelopes. J. Math. Anal. Appl. 172, 179–190 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biacino, L., Gerla, G.: An extension principle for closure operators. J. Math. Anal. Appl. 198, 1–24 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bielawski, R.: Simplicial convexity and its applications. J. Math. Anal. Appl. 127, 155–171 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Birkhoff, G., Frink, O.: Representations of lattices by sets. Trans. Am. Math. Soc. 64, 299–316 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bisi, C.: On commuting polynomial automorphisms of \(\mathbb{C}^2\). Publ. Mat. 48(1), 227–239 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bisi, C.: On commuting polynomial automorphisms of \(\mathbb{C}^k\), \(k \ge 3\). Math. Z. 258(4), 875–891 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bisi, C.: On closed invariant sets in local dynamics. J. Math. Anal. Appl. 350(1), 327–332 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boudabbous, Y., Delhommé, C.: \((\le k)\)-reconstructible binary relations. Eur. J. Combin. 37, 43–67 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cattaneo, G., Chiaselotti, G., Oliverio, P.A., Stumbo, F.: A new discrete dynamical system of signed integer partitions. Eur. J. Combin. 55, 119–143 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chiaselotti, G., Gentile, T., Infusino, F., Oliverio, P.A.: The adjacency matrix of a graph as a data table. A geometric perspective. Ann. Mat. Pura Appl. 196(3), 1073–1112 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chiaselotti, G., Gentile, T., Infusino, F.: Simplicial complexes and closure systems induced by indistinguishability relations. C. R. Acad. Sci. Paris Ser. I 355, 991–1021 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chiaselotti, G., Gentile, T., Infusino, F., Oliverio, P.A.: Dependency and accuracy measures for directed graphs. Appl. Math. Comput. 320, 781–794 (2018)MathSciNetGoogle Scholar
  16. 16.
    Chiaselotti, G., Gentile, T., Infusino, F.: Pairings and related symmetry notions. Ann. Univ. Ferrara 64(2), 285–322 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chiaselotti, G., Gentile, T., Infusino, F.: Decision systems in rough set theory. A set operatorial perspective. J. Algebra Appl. 1950004, 48 (2018). Google Scholar
  18. 18.
    Chiaselotti, G., Gentile, T., Infusino, F.: Symmetry Geometry by Pairings, Journal of the Australian Mathematical Society, 1–19,
  19. 19.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)CrossRefzbMATHGoogle Scholar
  20. 20.
    Diestel, R.: Graph Theory, Graduate Text in Mathematics, 4th edn. Springer, Berlin (2010)Google Scholar
  21. 21.
    Finkbeiner, D.: Dependence relation for lattices. Proc. Am. Math. Soc. 2(5), 756–759 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jelíneka, V., Klazar, M.: Embedding dualities for set partitions and for relational structures. Eur. J. Combin. 32, 1084–1096 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Metakides, G., Nerode, A.: Recursion theory on fields and abstract dependence. J. Algebra 65, 36–59 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pawlak, Z.: Rough Sets—Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  25. 25.
    Rado, R.: Abstract linear dependence. Colloquium Math. 14(1), 257–264 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sanahuja, S.M.: New rough approximations for \(n\)-cycles and \(n\)-paths. Appl. Math. Comput. 276, 96–108 (2016)MathSciNetGoogle Scholar
  27. 27.
    Schmidt, J.: Mehrstufige Austauschstrukturen. Z. Math. Logik Grundlagen Math. 2, 233–249 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Simovici, D.A., Djeraba, C.: Mathematical Tools for Data Mining. Springer, London (2014)CrossRefzbMATHGoogle Scholar
  29. 29.
    Tholen, W.: Ordered topological structures. Topol. Appl. 156, 2148–2157 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tholen, W.: Closure operators and their middle-interchange law. Topol. Appl. 158, 2437–2441 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)zbMATHGoogle Scholar
  32. 32.
    Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57, 509–533 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Xu, L., Mao, X.: Strongly continuous posets and the local scott topology. J. Math. Anal. Appl. 345, 816–824 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CalabriaArcavacata di RendeItaly

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