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New perspectives of granular computing in relation geometry induced by pairings

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Abstract

Given an arbitrary set \(\Omega \), we call a triple \({\mathfrak {P}}=(U,F, \Lambda )\), where U and \(\Lambda \) are two non-empty sets and F is a map from \(U\times \Omega \) into \(\Lambda \), a pairing on \(\Omega \). A pairing is an abstract mathematical generalization of the notion of information table, classically used in several scopes of granular computing and rough set theory. In this paper we undertake the study of pairings in relation to specific types of set operators, set systems and binary relations appearing in several branches of pure mathematics and information sciences. For example, an intersection-closed system \(MAXP({\mathfrak {P}})\) on \(\Omega \) can be canonically associated with any pairing \({\mathfrak {P}}\) on \(\Omega \) and we showed that for any intersection-closed system \(\mathfrak {S}\) on an arbitrary (even infinite) set \(\Omega \) there exists a pairing \({\mathfrak {P}}\) on \(\Omega \) such that \(MAXP({\mathfrak {P}})=\mathfrak {S}\). Next, we introduce some classes of pairings whose properties have a close analogy with corresponding notions derived from topology and matroid theory. We describe such classifications by means of a binary relation \(\leftarrow _{{\mathfrak {P}}}\) on the power set \(\mathcal {P}(\Omega )\) canonically associated with any pairing \({\mathfrak {P}}\). Using such a relation, we analyze new properties of intersection-closed systems and related operators, both within concrete models induced by metric spaces and also in connection with basic notions of common interest in several scopes of pure and applied mathematics and information sciences.

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References

  1. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D. M., Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, Vol. 3. Oxford University Press, Oxford (1994)

  2. Aledo, J.A., Martínez, S., Valverde, J.C.: Parallel dynamical systems over directed dependency graphs. Appl. Math. Comput. 129(3), 1114–1119 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Aledo, J.A., Diaz, L.G., Martínez, S., Valverde, J.C.: On periods and equilibria of computational sequential systems. Inf. Sci. 409, 27–34 (2017)

    Article  MathSciNet  Google Scholar 

  4. Apollonio, N., Caramia, M., Franciosa, P.G.: On the Galois lattice of bipartite distance hereditary graphs. Discrete Appl. Math. 190, 13–23 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Armstrong, W.W.: Dependency Structures of Database Relationships, Information Processing, pp. 580–583. North-Holland, Amsterdam (1974)

    Google Scholar 

  6. de Bakker, J.W., de Vink, E.P.: Control Flow Semantics. The MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  7. Bayley, R.A.: Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8(3), 45–77 (1996)

    MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Rhode Island (1967)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Bisi, C.: On commuting polynomial automorphisms of \({\mathbb{C}}^k\), \(k \ge 3\). Math. Z. 258(4), 875–891 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Cattaneo, G., Chiaselotti, G., Oliverio, P.A., Stumbo, F.: A new discrete dynamical system of signed integer partitions. Eur. J. Comb. 55, 119–143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G., Zhong, N., Yao, Y.: A Hypergraph Model of Granular Computing. In: Proceedings IEEE International Conference on Granular Computing, pp 130–135 (2008)

  15. Chen, J., Li, J.: An application of rough sets to graph theory. Inf. Sci. 201, 114–127 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chiaselotti, G., Ciucci, D., Gentile, T., Infusino, F.: Preclusivity and Simple Graphs. In: Proceedings RSFDGrC 2015, Lecture Notes in Computer Science, Vol. 9437, 127–137, Springer (2015)

  17. Chiaselotti, G., Gentile, T., Infusino, F.: Knowledge pairing systems in granular computing. Knowl. Based Syst. 124, 144–163 (2017)

    Article  MATH  Google Scholar 

  18. Chiaselotti, G., Gentile, T., Infusino, F.: Dependency structures for decision tables. Int. J. Approx. Reason. 88, 333–370 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Chiaselotti, G., Gentile, T., Infusino, F.: Pairings and related symmetry notions. Ann. dell’Univ. Ferrara 64(2), 285–322 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chiaselotti, G., Gentile, T., Infusino, F.: Granular computing on information tables: families of subsets and operators. Inf. Sci. 442–443, 72–102 (2018)

    Article  MathSciNet  Google Scholar 

  24. Chiaselotti, G., Gentile, T., Infusino, F.: Decision systems in rough set theory. A set operatorial perspective. J. Algebra Appl. 18(01), 1950004 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chiaselotti, G., Infusino, F.: Notions from rough set theory in a generalized dependency relation context. Int. J. Approx. Reason. 98, 25–61 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chiaselotti, G., Gentile, T., Infusino, F.: Symmetry geometry by pairings. J. Aust. Math. Soc., 1–19. https://doi.org/10.1017/S1446788718000137

  27. Doust, I., Sánchez, S., Weston, A.: Asymptotic negative type properties of finite ultrametric spaces. J. Math. Anal. Appl. 446, 1776–1793 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  29. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  30. Hahn, G., Sabidussi, G. (Eds.): Graph Symmetry. Algebraic Methods and Applications, NATO ASI Series, Vol. 497, Springer, Berlin (1997)

  31. Huang, A., Zhao, H., Zhu, W.: Nullity-based matroid of rough sets and its application to attribute reduction. Inf. Sci. 263, 153–165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kelarev, A., Quinn, S.J.: Directed graphs and combinatorial properties of semigroups. J. Algebra 251(1), 16–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kelarev, A., Praeger, C.E.: On transitive Cayley graphs of groups and semigroups. Eur. J. Combin. 24(1), 59–72 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: the influence of asymmetries. Discrete Math. 309, 5360–5369 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Larsen, K.G., Winskell, G.: Using information systems to solve recursive domain equations. Inf. Comput. 91, 232–258 (1991)

    Article  MathSciNet  Google Scholar 

  36. Li, X., Liu, S.: Matroidal approaches to rough sets via closure operators. Int. J. Approx. Reason. 53, 513–527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, X., Yi, H., Liu, S.: Rough sets and matroids from a Lattice-theoretic viewpoint. Inf. Sci. 342, 37–52 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, X., Yi, H., Wang, Z.: Approximation via a double-matroid structure. Soft Comput. (2019). https://doi.org/10.1007/s00500-018-03749-8

    Article  MATH  Google Scholar 

  39. Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems, p. xii+248. Springer, New York (2008)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  41. Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inf. Sci. 177, 3–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pedrycz, W.: Granular Computing: An Emerging Paradigm. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  43. Pedrycz, W.: Granular Computing : Analysis and Design of Intelligent Systems. CRC Press, Boca Raton (2013)

    Book  Google Scholar 

  44. Pedrycz, W.: Granular computing for data analytics: a manifesto of human-centric computing. EEE/CAA J. Autom. Sin. 5(6), 1025–1034 (2018)

    Article  MathSciNet  Google Scholar 

  45. Polkowski, L.: Rough Sets: Mathematical Foundations. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  46. Polkowski, L.: On fractal dimension in information systems. Toward exact sets in infinite information systems. Fundam. Inf. 50(3–4), 305–314 (2002)

    MathSciNet  MATH  Google Scholar 

  47. Polkowski, L., Artiemjew, P.: Granular Computing in Decision Approximation. An Application of Rough Mereology. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  48. Rodríguez-López, J., Schellekens, M.P., Valero, O.: An extension of the dual complexity space and an application to Computer Science. Topol. Appl. 156, 3052–3061 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Romaguera, S., Sapena, A., Tirado, P.: The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words. Topol. Appl. 154, 2196–2203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Romaguera, S., Schellekens, M., Valero, O.: Complexity spaces as quantitative domains of computation. Topol. Appl. 158, 853–860 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  52. Sapena, A.: A contribution to the study of fuzzy metric spaces. Appl. Gen. Topol. 2, 63–76 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  53. Schellekens, M.: The Smyth completion: A common foundation for denotational semantics and complexity analysis. Electron. Notes Theor. Comput. Sci. 1, 211–232 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  54. Schellekens, M.: A characterization of partial metrizability. Domains are quantifiable. Theor. Comput. Sci. 305, 409–432 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  55. Scott, D.S.: Data types as lattices. SIAM J. Comput. 5(3), 522–587 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  56. Scott, D. S.: Domains for Denotational Semantics, in Automata, Languages and Programming, Lecture Notes in Computer Science, Vol.140, 577–613 (1982)

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

    Book  MATH  Google Scholar 

  58. Skowron, A., Rauszer, C.: The Discernibility Matrices and Functions in Information Systems, Intelligent Decision Support, Theory and Decision Library series, vol. 11, Springer, Netherlands, pp. 331–362 (1992)

  59. Ślezak, D.: Approximate entropy reducts. Fundam. Inf. 53, 365–390 (2002)

    MathSciNet  MATH  Google Scholar 

  60. Ślezak, D.: On Generalized Decision Functions: Reducts, Networks and Ensembles, RSFDGrC, 13–23 (2015)

  61. Stawicki, S., Ślezak, D., Janusz, A., Widz, S.: Decision bireducts and decision reducts—a comparison. Int. J. Approx. Reason. 84, 75–109 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tanga, J., Shea, K., Min, F., Zhu, W.: A matroidal approach to rough set theory. Theor. Comput. Sci. 471(3), 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  63. Wang, J., Zhu, W.: Applications of bipartite graphs and their adjacency matrices to covering-based rough sets. Fundam. Inf. 156, 237–254 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  64. Weston, A.: On the generalized roundness of finite metric spaces. J. Math. Anal. Appl. 192, 323–334 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  65. Yao, Y.Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16(1), 87–104 (2001)

    Article  MATH  Google Scholar 

  66. Yao, Y. Y., Zhong, N.: Granular Computing using Information Tables, in Data Mining, Rough Sets and Granular Computing, Physica-Verlag, pp. 102–124 (2002)

  67. Yao, Y.: A Partition Model of Granular Computing. In: Transactions on Rough Sets I, Lecture Notes in Computer Science, vol. 3100, Springer-Verlag, 232–253 (2004)

  68. Yao, Y., Zhao, Y.: Discernibility matrix simplification for constructing attribute reducts. Inf. Sci. 179, 867–882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Xu, L.: Continuity of posets via scott topology and sobrification. Topol. Appl. 153, 1886–1894 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  71. Zhu, W., Wang, S.: Rough matroids based on relations. Inf. Sci. 232, 241–252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  72. Ziarko, W.: Variable precision rough set model. J. Comput. Syst. Sci. 46, 39–59 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  73. Ziarko, W.: Probabilistic approach to rough sets. Int. J. Approx. Reason. 49, 272–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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  1. Department of Mathematics and Informatics, University of Calabria, Via Pietro Bucci, Cubo 30B, 87036, Arcavacata di Rende, CS, Italy

    G. Chiaselotti, T. Gentile & F. Infusino

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  1. G. Chiaselotti
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Chiaselotti, G., Gentile, T. & Infusino, F. New perspectives of granular computing in relation geometry induced by pairings. Ann Univ Ferrara 65, 57–94 (2019). https://doi.org/10.1007/s11565-019-00314-5

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