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Keywords:
binary relation; ternary relation; relational compositions
binary relation; ternary relation; relational compositions
Summary:
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.
References:
[1] Alvarez-Garcia, S., Bernardo, G. de, Brisaboa, N. R., Navarro, G.: A succinct data structure for self-indexing ternary relations. J. Discrete Algorithms 43 (2017), 38-53. DOI
[2] Bandler, W., Kohout, L. J.: 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
[3] Bandler, W., Kohout, L. J.: Semantics of implication operators and fuzzy relational product. Int. J. Man-Machine Studies 12 (1980), 89-116. DOI
[4] Beall, J., Brady, R., Dunn, J. M., Hazen, A. P., Mares, E., Meyer, R. K., Priest, G., Restall, G., Ripley, D., Slaney, J., Sylvan, R.: On the ternary relation and conditionality. J. Philosoph. Logic 41 (2012), 595-612. DOI
[5] Běhounek, L., Bodenhofer, U., Cintula, P.: Relations in fuzzy class theory: initial steps. Fuzzy Sets Systems 159 (2008), 1729-1772. DOI
[6] Běhounek, L., Daňková, M.: Relational compositions in fuzzy class theory. Fuzzy Sets Systems 160 (2008), 1005-1036. DOI
[7] Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers/Plenum Publishers, New York 2002.
[8] Bělohlávek, R., Osicka, P.: Triadic fuzzy Galois connections as ordinary connections. Fuzzy Sets Systems 249 (2014), 83-99. DOI
[9] Cristea, I.: Several aspects on the hypergroups associated with $n$-ary relations. Analele Stiintifice ale Universitatii Ovidius Constanta 17 (2009), 99-110.
[10] Baets, B. De, Kerre, E. E.: Fuzzy relational compositions. Fuzzy Sets Systems 60 (1993, 109-120. DOI
[11] Baets, B. De: 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 | Zbl 0970.03044
[12] Nola, A. Di, Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publishers, Dordrecht 1989. Zbl 0694.94025
[13] Doignon, J. P., Monjardet, B., Roubens, M., Vincke, Ph.: Biorder families, valued relations and preference modelling. J. Math. Psychol. 30 (1986), 435-480. DOI
[14] Dunn, J. M.: Ternary relational semantics and beyond. Logical Studies 7 (2001), 1-20.
[15] Düntsch, I.: Relation algebras and their application in temporal and spatial reasoning. Artif. Intell. Rev. 23 (2005), 315-357. DOI
[16] Fodor, J.: Traces of fuzzy binary relations. Fuzzy Sets Systems 50 (1992), 331-341. DOI
[17] Ganter, B., Wille, R.: Formal Concept Analysis. Springer-Verlag, Berlin, Heidelberg 1999.
[18] Goguen, J. A.: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 145-174. DOI | MR 0224391
[19] Ignatov, D. I., Gnatyshak, D. V., Kuznetsov, S. O., Mirkin, B. G.: Triadic formal concept analysis and triclustering: searching for optimal patterns. Machine Learning 101 (2015), 271-302. DOI
[20] Isli, A., Cohn, A.: A new approch to cyclic ordering of 2D orientations using ternary relation algebra. Artif. Intell. 122 (2000), 137-187. DOI 10.1016/S0004-3702(00)00044-8
[21] Kim, J., Amir, A., Na, J. C., Park, K., Sim, J. S.: On representations of ternary order relations in numeric strings. Math. Computer Sci. 11 (2017), 127-136. DOI
[22] Konecny, J., Osicka, P.: Triadic concept lattices in the framework of aggregation structures. Inform. Sci. 279 (2014), 512-527. DOI
[23] Novák, V., Novotný, M.: On representation of cyclically ordered sets. Czechoslovak Math. J. 39 (1989), 127-132. DOI
[24] Novák, V., Novotný, M.: Pseudodimension of relational structures. CzechoslovakMath. J. 49 (1999), 547-560. DOI
[25] Peirce, C. S.: On the algebra of logic. Amer. J. Math. 3 (1880), 15-58. DOI 10.2307/2369442
[26] Pérez-Fernández, R., Baets, B. De: On the role of monometrics in penalty-based data aggregation. IEEE Trans. Fuzzy Systems 27 (2018), 7, 1456-1468. DOI
[27] Pérez-Fernández, R., Rademaker, M., Baets, B. De: Monometrics and their role in the rationalisation of ranking rules. Inform. Fusion 34 (2017), 16-27. DOI
[28] Pivert, O., Bosc, P.: Fuzzy Preference Queries to Relational Databases. Imperial College Press, London 2012. DOI | Zbl 1246.68011
[29] Pourabdollah, A.: Theory and Practice of The Ternary Relations Model of Information Management. PhD Thesis, University of Nottingham, 2009.
[30] Powers, S.: Practical RDF. O'Reilly, Beijing 2003.
[31] Schröder, B. S.: Ordered Sets. Birkhauser, Boston 2002.
[32] Steel, M.: Phylogeny: Discrete and Random Processes in Evolution. SIAM, Philadelphia 2016. DOI
[33] Štěpnička, M., Jayaram, B.: On the suitability of the Bandler-Kohout subproduct as an inference mechanism. IEEE Trans. Fuzzy Systems 18 (2010), 285-298. DOI
[34] Štěpnička, M., Baets, B. De: Implication-based models of monotone fuzzy rule bases. Fuzzy Sets Systems 232 (2013), 134-155. DOI
[35] Štěpnička, M., Holčapek, M.: 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
[36] Wang, X., Xue, Y.: Traces and property indicators of fuzzy relations. Fuzzy Sets Systems 246 (2014), 78-90. DOI
[37] Zadeh, L. A.: Fuzzy sets. Inform. Control 8 (1965), 338-353. DOI | Zbl 0942.00007
[38] Zedam, L., Barkat, O., Baets, B. De: Traces of ternary relations. Int. J. General Systems 47 (2018), 350-373. DOI
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