# zbMATH — the first resource for mathematics

Relational compositions in fuzzy class theory. (English) Zbl 1185.03079
The paper gives a unified, yet elementary, treatment of fuzzy relation theory. The formal background is the fuzzy class theory as conceived by L. Běhounek and P. Cintula [Fuzzy Sets Syst. 154, No. 1, 34–55 (2005; Zbl 1086.03043)], and developed within the framework of the mathematical fuzzy logic $$\mathrm{MTL}_{\triangle}$$.
This paper treats a series of notions related to relation composition, like different composition operations, images and preimages. That this elementary treatment is possible shows that the chosen framework is well suited to do, in a serious way, fuzzy mathematics within this framework.

##### MSC:
 3e+72 Theory of fuzzy sets, etc. 3e+70 Nonclassical and second-order set theories
Full Text:
##### References:
 [1] W. Bandler, L.J. Kohout, Mathematical relations, their products and generalized morphisms, Technical Report EES-MMS-REL 77-3, Man-Machine Systems Laboratory, Department of Electrical Engineering, University of Essex, Essex, Colchester, 1977. [2] W. Bandler, L.J. Kohout, Fuzzy relational products and fuzzy implication operators, in: Internat. Workshop of Fuzzy Reasoning Theory and Applications, Queen Mary College, University of London, London, 1978. [3] Bandler, W.; Kohout, L.J., Fuzzy power sets and fuzzy implication operators, Fuzzy sets and systems, 4, 183-190, (1980) [4] Bandler, W.; Kohout, L.J., Fast fuzzy relational algorithms, (), 123-131 · Zbl 0853.68163 [5] Bandler, W.; Kohout, L.J., On the general theory of relational morphisms, Internat. J. gen. systems, 13, 47-66, (1986) · Zbl 0875.04002 [6] Bandler, W.; Kohout, L.J., Special properties, closures and interiors of crisp and fuzzy relations, Fuzzy sets and systems, 26, 317-331, (1988) · Zbl 0664.04001 [7] Bandler, W.; Kohout, L.J., On the universality of the triangle superproduct and the square product of relations, Internat. J. gen. systems, 25, 399-403, (1997) · Zbl 0874.04001 [8] L. Běhounek, Extensionality in graded properties of fuzzy relations, in: Proc. 11th IPMU Conf., Paris, Edition EDK, 2006, pp. 1604-1611. [9] L. Běhounek, Relative interpretations over first-order fuzzy logics, in: F. Hakl (Ed.), Doktorandský den ’06, Prague, 2006, pp. 1-6. ICS AS CR/Matfyzpress. Available at $$\langle$$http://www.cs.cas.cz/hakl/doktorandsky-den/history.html⟩. [10] Běhounek, L., On the difference between traditional and deductive fuzzy logic, Fuzzy sets and systems, 159, 1153-1164, (2008) · Zbl 1175.03012 [11] Běhounek, L.; Bodenhofer, U.; Cintula, P., Relations in fuzzy class theory: initial steps, Fuzzy sets and systems, 159, 10, 1729-1772, (2008) · Zbl 1185.03078 [12] Běhounek, L.; Cintula, P., Fuzzy class theory, Fuzzy sets and systems, 154, 1, 34-55, (2005) · Zbl 1086.03043 [13] L. Běhounek, P. Cintula, Fuzzy class theory as foundations for fuzzy mathematics, in: Fuzzy Logic, Soft Computing and Computational Intelligence: 11th IFSA World Congress, Vol. 2, Tsinghua University Press, Springer, Beijing, Berlin, 2005, pp. 1233-1238. [14] L. Běhounek, P. Cintula, General logical formalism for fuzzy mathematics: methodology and apparatus, in: Fuzzy Logic, Soft Computing and Computational Intelligence: 11th IFSA World Congress, Vol. 2, Tsinghua University Press, Springer, Beijing, Berlin, 2005, pp. 1227-1232. [15] Běhounek, L.; Cintula, P., From fuzzy logic to fuzzy mathematics: a methodological manifesto, Fuzzy sets and systems, 157, 5, 642-646, (2006) · Zbl 1108.03027 [16] L. Běhounek, P. Cintula, Fuzzy class theory: a primer v1.0, Technical Report V-939, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, 2006. Available at $$\langle$$www.cs.cas.cz/research/library/reports_900.shtml⟩. [17] Běhounek, L.; Cintula, P., Features of mathematical theories in formal fuzzy logic, (), 523-532 · Zbl 1202.03034 [18] R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, IFSR International Series on Systems Science and Engineering, Vol. 20, Kluwer Academic, Plenum Press, Dordrecht, New York, 2002. · Zbl 1067.03059 [19] Bodenhofer, U.; De Cock, M.; Kerre, E.E., Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems, Internat. J. gen. systems, 32, 4, 343-360, (2003) · Zbl 1110.03048 [20] Daňková, M., On approximate reasoning with graded rules, Fuzzy sets and systems, 158, 652-673, (2007) · Zbl 1111.68133 [21] De Baets, B.; Kerre, E., Fuzzy relational compositions, Fuzzy sets and systems, 60, 1, 109-120, (1993) · Zbl 0794.04004 [22] De Cock, M.; Kerre, E.E., Fuzzy modifiers based on fuzzy relations, Inform. sci., 160, 173-199, (2004) · Zbl 1076.68083 [23] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 3, 271-288, (2001) · Zbl 0994.03017 [24] Fodor, J., Traces of fuzzy binary relations, Fuzzy sets and systems, 50, 3, 331-341, (1992) · Zbl 0794.04003 [25] Fodor, J., Contrapositive symmetry of fuzzy implications, Fuzzy sets and systems, 69, 141-156, (1995) · Zbl 0845.03007 [26] Frascella, A.; Guido, C., Transporting many-valued sets along many-valued relations, Fuzzy sets and systems, 159, 1-22, (2008) · Zbl 1176.03029 [27] S. Gottwald, A Treatise on Many-Valued Logics, Studies in Logic and Computation, Vol. 9, Research Studies Press, Baldock, 2001. · Zbl 1048.03002 [28] S. Gottwald, Some general considerations on the evaluation of fuzzy rule systems, in: Proc. of Joint EUSFLAT-LFA Conf., Barcelona, 2005, pp. 639-644. [29] Gottwald, S., Universes of fuzzy sets and axiomatizations of fuzzy set theory, part I: model-based and axiomatic approaches, Studia logica, 82, 2, 211-244, (2006) · Zbl 1111.03047 [30] P. Hájek, Metamathematics of Fuzzy Logic, Trends in Logic, Vol. 4, Kluwer, Dordrecht, 1998. [31] Hájek, P., Making fuzzy description logic more general, Fuzzy sets and systems, 154, 1, 1-15, (2005) · Zbl 1094.03014 [32] P. Hájek, P. Cintula, Triangular norm predicate fuzzy logics, in: Proc. of Linz Seminar 2005, 2005, to appear. · Zbl 1200.03020 [33] L.P. Holmblad, J.J. Ostergaard, Control of a cement kiln by fuzzy logic, in: M.M. Gupta, E. Sanchez (Eds.), Fuzzy Information and Decision Processes, Amsterdam, 1982, pp. 389-399. [34] Klawonn, F.; Castro, J.L., Similarity in fuzzy reasoning, Mathware soft comput., 3, 2, 197-228, (1995) · Zbl 0859.04006 [35] Kohout, L.J.; Bandler, W., Fuzzy relational products in knowledge engineering, (), 51-66 · Zbl 0800.68936 [36] Kohout, L.J.; Kim, E., The role of BK-products of relations in soft computing, Soft comput., 6, 92-115, (2002) · Zbl 1001.68047 [37] Magrez, P.; Smets, Ph., Fuzzy modus ponens: a new model suitable for applications in knowledge-based systems, Internat. J. intell. systems, 4, 181-200, (1989) · Zbl 0672.03010 [38] Mamdani, E.H.; Assilian, S., An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. man—machine stud., 7, 1-13, (1975) · Zbl 0301.68076 [39] Morsi, N.N.; Fahmy, A.A., On generalized modus ponens with multiple rules and a residuated implication, Fuzzy sets and systems, 129, 267-274, (2002) · Zbl 1001.03024 [40] Morsi, N.N.; Lotfallah, W.; El-Zekey, M.S., The logic of tied implications, part 2: syntax, Fuzzy sets and systems, 157, 2030-2057, (2006) · Zbl 1111.03033 [41] Takeuti, G.; Titani, S., Fuzzy logic and fuzzy set theory, Arch. math. logic, 32, 1-32, (1992) · Zbl 0786.03039 [42] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.