Kybernetika 57 no. 3, 404-425, 2021

Compositions of ternary relations

Norelhouda Bakri, Lemnaouar Zedam and Bernard De BaetsDOI: 10.14736/kyb-2021-3-0404

Abstract:

In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introduced by Zedam et al. We study the properties of these compositions, in particular the link with the usual composition of binary relations through the use of the operations of projection and cylindrical extension.

Keywords:

binary relation, ternary relation, relational compositions

Classification:

03E20, 97E60

paper.pdf

References:

  1. S. Alvarez-Garcia, G. de Bernardo, N. R. Brisaboa and G. Navarro: A succinct data structure for self-indexing ternary relations. J. Discrete Algorithms 43 (2017), 38-53.   DOI:10.1016/j.jda.2016.10.002
  2. W. Bandler and L. J. Kohout: Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems. In: Theory and Application to Policy Analysis and Information Systems (P. Wang and S. Chang, eds.), Plenum Press, New York 1980, pp3 41-367.   DOI:10.1007/978-1-4684-3848-2\_1
  3. W. Bandler and L. J. Kohout: Semantics of implication operators and fuzzy relational product. Int. J. Man-Machine Studies 12 (1980), 89-116.   DOI:10.1016/S0020-7373(80)80055-1
  4. J. Beall, R. Brady, J. M. Dunn, A. P. Hazen, E. Mares, R. K. Meyer, G. Priest, G. Restall, D. Ripley, J. Slaney and R. Sylvan: On the ternary relation and conditionality. J. Philosoph. Logic 41 (2012), 595-612.   DOI:10.1007/s10992-011-9191-5
  5. L. Běhounek, U. Bodenhofer and P. Cintula: Relations in fuzzy class theory: initial steps. Fuzzy Sets Systems 159 (2008), 1729-1772.   DOI:10.1016/j.fss.2007.10.012
  6. L. Běhounek and M. Daňková: Relational compositions in fuzzy class theory. Fuzzy Sets Systems 160 (2008), 1005-1036.   DOI:10.1016/j.fss.2008.06.013
  7. R. Bělohlávek: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers/Plenum Publishers, New York 2002.   CrossRef
  8. R. Bělohlávek and P. Osicka: Triadic fuzzy Galois connections as ordinary connections. Fuzzy Sets Systems 249 (2014), 83-99.   DOI:10.1016/j.fss.2014.02.003
  9. I. Cristea: Several aspects on the hypergroups associated with $n$-ary relations. Analele Stiintifice ale Universitatii Ovidius Constanta 17 (2009), 99-110.   CrossRef
  10. B. De Baets and E. E. Kerre: Fuzzy relational compositions. Fuzzy Sets Systems 60 (1993, 109-120.   DOI:0.1016/0165-0114(93)90296-T
  11. B. De Baets: Analytical solution methods for fuzzy relational equations. In: Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series (D. Dubois and H. Prade, eds.), Vol. 1, Kluwer Academic Publishers, Dordrecht 2000, pp. 291-340.   DOI:10.1007/978-1-4615-4429-6\_7
  12. A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht 1989.   CrossRef
  13. J. P. Doignon, B. Monjardet, M. Roubens and Ph. Vincke: Biorder families, valued relations and preference modelling. J. Math. Psychol. 30 (1986), 435-480.   DOI:10.1016/0022-2496(86)90020-9
  14. J. M. Dunn: Ternary relational semantics and beyond. Logical Studies 7 (2001), 1-20.   CrossRef
  15. I. Düntsch: Relation algebras and their application in temporal and spatial reasoning. Artif. Intell. Rev. 23 (2005), 315-357.   DOI:10.1007/s10462-004-5899-8
  16. J. Fodor: Traces of fuzzy binary relations. Fuzzy Sets Systems 50 (1992), 331-341.   DOI:10.1016/0165-0114(92)90229-W
  17. B. Ganter and R. Wille: Formal Concept Analysis. Springer-Verlag, Berlin, Heidelberg 1999.   CrossRef
  18. J. A. Goguen: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174.   DOI:10.1016/0022-247X(67)90189-8
  19. D. I. Ignatov, D. V. Gnatyshak, S. O. Kuznetsov and B. G. Mirkin: Triadic formal concept analysis and triclustering: searching for optimal patterns. Machine Learning 101 (2015), 271-302.   DOI:10.1007/s10994-015-5487-y
  20. A. Isli and A. Cohn: A new approch to cyclic ordering of 2D orientations using ternary relation algebra. Artif. Intell. 122 (2000), 137-187.   CrossRef
  21. J. Kim, A. Amir, J. C. Na, K. Park and J. S. Sim: On representations of ternary order relations in numeric strings. Math. Computer Sci. 11 (2017), 127-136.   DOI:10.1007/s40096-017-0217-1
  22. J. Konecny and P. Osicka: Triadic concept lattices in the framework of aggregation structures. Inform. Sci. 279 (2014), 512-527.   DOI:10.1016/j.ins.2014.04.006
  23. V. Novák and M. Novotný: On representation of cyclically ordered sets. Czechoslovak Math. J. 39 (1989), 127-132.   DOI:10.1016/j.ins.2014.04.006
  24. V. Novák and M. Novotný: Pseudodimension of relational structures. CzechoslovakMath. J. 49 (1999), 547-560.   DOI:10.1023/A:1022471319218
  25. C. S. Peirce: On the algebra of logic. Amer. J. Math. 3 (1880), 15-58.   CrossRef
  26. R. Pérez-Fernández and B. De Baets: On the role of monometrics in penalty-based data aggregation. IEEE Trans. Fuzzy Systems 27 (2018), 7, 1456-1468.   DOI:10.1109/TFUZZ.2018.2880716
  27. R. Pérez-Fernández, M. Rademaker and B. De Baets: Monometrics and their role in the rationalisation of ranking rules. Inform. Fusion 34 (2017), 16-27.   DOI:10.1016/j.inffus.2016.06.001
  28. O. Pivert and P. Bosc: Fuzzy Preference Queries to Relational Databases. Imperial College Press, London 2012.   DOI:10.1142/p840
  29. A. Pourabdollah: Theory and Practice of The Ternary Relations Model of Information Management. PhD Thesis, University of Nottingham, 2009.   CrossRef
  30. S. Powers: Practical RDF. O'Reilly, Beijing 2003.   CrossRef
  31. B. S. Schröder: Ordered Sets. Birkhauser, Boston 2002.   CrossRef
  32. M. Steel: Phylogeny: Discrete and Random Processes in Evolution. SIAM, Philadelphia 2016.   DOI:10.1137/1.9781611974485
  33. M. Štěpnička and B. Jayaram: On the suitability of the Bandler-Kohout subproduct as an inference mechanism. IEEE Trans. Fuzzy Systems 18 (2010), 285-298.   DOI:10.1109/TFUZZ.2010.2041007
  34. M. Štěpnička and B. De Baets: Implication-based models of monotone fuzzy rule bases. Fuzzy Sets Systems 232 (2013), 134-155.   DOI:10.1016/j.fss.2013.07.019
  35. M. Štěpnička and M. Holčapek: Fuzzy relational compositions based on generalized quantifiers. In: Information Processing and Management of Uncertainty in Knowledge-Based Systems PT II (IPMU'14), Communications in Computer and Information Science, Vol. 443, Springer, Berlin 2014, pp. 224-233.   DOI:10.1007/978-3-319-08855-6\_23
  36. X. Wang and Y. Xue: Traces and property indicators of fuzzy relations. Fuzzy Sets Systems 246 (2014), 78-90.   DOI:10.1016/j.fss.2014.01.008
  37. L. A. Zadeh: Fuzzy sets. Inform. Control 8 (1965), 338-353.   DOI:10.1016/S0019-9958(65)90241-X
  38. L. Zedam, O. Barkat and B. De Baets: Traces of ternary relations. Int. J. General Systems 47 (2018), 350-373.   DOI:10.1080/03081079.2018.1446433