Dependency relations


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.

This is a preview of subscription content, log in to check access.


  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)

    Google Scholar 

  3. 3.

    Baldwin, J.T.: First-order theories of abstract dependence relations. Ann. Pure Appl. Logic 26, 215–243 (1984)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Biacino, L.: Generated envelopes. J. Math. Anal. Appl. 172, 179–190 (1993)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Biacino, L., Gerla, G.: An extension principle for closure operators. J. Math. Anal. Appl. 198, 1–24 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bielawski, R.: Simplicial convexity and its applications. J. Math. Anal. Appl. 127, 155–171 (1987)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Birkhoff, G., Frink, O.: Representations of lattices by sets. Trans. Am. Math. Soc. 64, 299–316 (1948)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bisi, C.: On commuting polynomial automorphisms of \(\mathbb{C}^2\). Publ. Mat. 48(1), 227–239 (2004)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bisi, C.: On commuting polynomial automorphisms of \(\mathbb{C}^k\), \(k \ge 3\). Math. Z. 258(4), 875–891 (2007)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bisi, C.: On closed invariant sets in local dynamics. J. Math. Anal. Appl. 350(1), 327–332 (2009)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Boudabbous, Y., Delhommé, C.: \((\le k)\)-reconstructible binary relations. Eur. J. Combin. 37, 43–67 (2014)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Chiaselotti, G., Gentile, T., Infusino, F.: Pairings and related symmetry notions. Ann. Univ. Ferrara 64(2), 285–322 (2018)

    MathSciNet  Article  Google 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).

    Article  MATH  Google Scholar 

  18. 18.

    Chiaselotti, G., Gentile, T., Infusino, F.: Symmetry Geometry by Pairings, Journal of the Australian Mathematical Society, 1–19,

    MathSciNet  Article  Google Scholar 

  19. 19.

    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)

    Google 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)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Jelíneka, V., Klazar, M.: Embedding dualities for set partitions and for relational structures. Eur. J. Combin. 32, 1084–1096 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Metakides, G., Nerode, A.: Recursion theory on fields and abstract dependence. J. Algebra 65, 36–59 (1980)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Pawlak, Z.: Rough Sets—Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  25. 25.

    Rado, R.: Abstract linear dependence. Colloquium Math. 14(1), 257–264 (1966)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Sanahuja, S.M.: New rough approximations for \(n\)-cycles and \(n\)-paths. Appl. Math. Comput. 276, 96–108 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Schmidt, J.: Mehrstufige Austauschstrukturen. Z. Math. Logik Grundlagen Math. 2, 233–249 (1956)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Simovici, D.A., Djeraba, C.: Mathematical Tools for Data Mining. Springer, London (2014)

    Google Scholar 

  29. 29.

    Tholen, W.: Ordered topological structures. Topol. Appl. 156, 2148–2157 (2009)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Tholen, W.: Closure operators and their middle-interchange law. Topol. Appl. 158, 2437–2441 (2011)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)

    Google Scholar 

  32. 32.

    Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57, 509–533 (1935)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Xu, L., Mao, X.: Strongly continuous posets and the local scott topology. J. Math. Anal. Appl. 345, 816–824 (2008)

    MathSciNet  Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to G. Chiaselotti.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chiaselotti, G., Infusino, F. & Oliverio, P.A. Dependency relations. Boll Unione Mat Ital 12, 525–548 (2019).

Download citation


  • Abstract dependency
  • Closure systems
  • Abstract simplicial complexes
  • Graphs

Mathematics Subject Classification

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