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(\(S,N\))-implications on bounded lattices. (English) Zbl 1307.03015

Baczyński, Michał (ed.) et al., Advances in fuzzy implication functions. Berlin: Springer (ISBN 978-3-642-35676-6/hbk; 978-3-642-35677-3/ebook). Studies in Fuzziness and Soft Computing 300, 101-124 (2013).
This paper intends to explore usual fuzzy logics from the viewpoint of arbitrary bounded lattices. In the framework of bounded lattice-valued fuzzy logic, the authors propose the class of (\(S,N\))-implications on the unit interval, and a detailed characterization of the (\(S,N\))-implications in a simpler way. Moreover, the authors show that some properties adapted for the bounded lattice framework are preserved, while some properties are not completely equivalent to their fuzzy counterparts. This work can bring a new theory foundation of fuzzy logics, such as intuitionistic fuzzy sets, fuzzy multisets and fuzzy rough sets.
For the entire collection see [Zbl 1257.03013].

MSC:

03B52 Fuzzy logic; logic of vagueness
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[1] Abbot, J. C., Semi-Boolean algebras, Mathematichki Vesnik, 4, 177-198 (1967) · Zbl 0153.02704
[2] Alcalde, C.; Burusco, A.; Fuentes-González, R., A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets and Systems, 153, 2, 211-227 (2005) · Zbl 1086.03017
[3] Alsina, C.; Trillas, E., When (S,N)-implications are (T,T_1)-conditional functions?, Fuzzy Sets and Systems, 134, 2, 305-310 (2003) · Zbl 1014.03026
[4] Alsina, C.; Trillas, E.; Valverde, L., On non-distributive logical connectives for fuzzy set theory, Busefal, 3, 18-29 (1980)
[5] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986) · Zbl 0631.03040
[6] Atanassov, K. T.; Gargov, G., Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31, 343-349 (1989) · Zbl 0674.03017
[7] Baczyński, M., Residual implications revised. Notes on the Smets-Magrez theorem, Fuzzy Sets and Systems, 145, 2, 267-277 (2004) · Zbl 1064.03019
[8] Baczyński, M.; Jayaram, B., On the characterizations of (S,N)-implications, Fuzzy Sets and Systems, 158, 1713-1727 (2007) · Zbl 1122.03021
[9] Baczyński, M.; Jayaram, B., Fuzzy Implications (2008), Heidelberg: Springer, Heidelberg · Zbl 1147.03012
[10] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy Sets and Systems, 104, 1, 61-75 (1999) · Zbl 0935.03060
[11] Bedregal, B.; Beliakov, G.; Bustince, H.; Calvo, T.; Mesiar, R.; Paternain, D., A class of fuzzy multisets with a fixed number of memberships, Information Sciences, 189, 1, 1-17 (2012) · Zbl 1247.03113
[12] Bedregal, B. C., On interval fuzzy negations, Fuzzy Sets and Systems, 161, 2290-2313 (2010) · Zbl 1204.03030
[13] Bedregal, B. C.; Dimuro, G. P.; Santiago, R. H.N.; Reiser, R. H.S., On interval fuzzy S-implications, Information Sciences, 180, 8, 1373-1389 (2010) · Zbl 1189.03027
[14] Bedregal, B. C.; Reiser, R. H.S.; Dimuro, G. P., Xor-implications and E-implications: Classes of fuzzy implications based on fuzzy xor, Elec. Notes in Theoretical Computer Science, 247, 5-18 (2009) · Zbl 1311.03052
[15] Bedregal, B.C., Santos, H.S., Callejas-Bedregal, R.: T-norms on bounded lattices: t-norm morphisms and operators. In: Proceedings of 2006 IEEE International Conference on Fuzzy Systems, Vancouver, Canada, pp. 22-28 (2006)
[16] Birkhoff, G., Lattice Theory (1973), Providence: American Mathematical Society, Providence
[17] Bustince, H.; Barrenechea, E.; Mohedano, V., Intuitionistic fuzzy implication operators – an expression and main properties, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, 13, 387-406 (2004) · Zbl 1054.03022
[18] Bustince, H.; Burillo, P.; Soria, F., Automorphisms, negations and implication operators, Fuzzy Sets and Systems, 134, 2, 209-229 (2003) · Zbl 1010.03017
[19] Chang, C. C., Algebraic analisys of many valued logics, Transactions of the American Mathematical Society, 88, 467-490 (1958) · Zbl 0084.00704
[20] De Cooman, G.; Kerre, E. E., Order norms on bounded partially ordered sets, Fuzzy Mathematics, 2, 281-310 (1994) · Zbl 0814.04005
[21] Da Costa, C. G.; Bedregal, B. C.; Doria Neto, A. D., Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms, International Journal of Approximate Reasoning, 52, 4, 473-487 (2010) · Zbl 1228.03038
[22] Davey, B. A.; Priestley, H. A., Introduction to Lattices and Order (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1002.06001
[23] Deschrijver, G., A representation of t-norms in interval-valued L-fuzzy set theory, Fuzzy Sets and Systems, 159, 1597-1618 (2008) · Zbl 1176.03027
[24] Deschrijver, G.; Cornelis, C.; Kerre, E. E., On the representation of intuitionistic fuzzy t-norms and t-conorms, IEEE Transactions on Fuzzy Systems, 12, 1, 45-61 (2004)
[25] Dienes, Z. P., On an implication function in many-valued systems of logic, Journal of Symbolic Logics, 14, 95-97 (1949) · Zbl 0037.00301
[26] Dujet, C..; Vincent, N., Force implication: A new approach to human reasoning, Fuzzy Sets and Systems, 69, 1, 53-63 (1995)
[27] Esteva, F.; Godo, L., Monoidal t-norm based logic: Toward a logic for left-continuous t-norms, Fuzzy Sets and Systems, 123, 3, 271-288 (2001) · Zbl 0994.03017
[28] Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Dordrecht: Kluwer Academic Publisher, Dordrecht · Zbl 0827.90002
[29] Font, J. M.; Rodriguez, A. J.; Torrens, A., Wajsberg algebra, Stochastica, 8, 1, 5-31 (1984) · Zbl 0557.03040
[30] Gödel, K., Zum intuitionistischen Aussagenkalkül, Anzeiger Akademie der Wissenschaften Wien, 69, 65-66 (1932) · JFM 58.1001.03
[31] Goguen, J., L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18, 1, 145-174 (1967) · Zbl 0145.24404
[32] Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices. Dover Publications (2009) · Zbl 0232.06001
[33] Grätzer, G.: Lattice Theory: Foundation. Birkhäuser (2011) · Zbl 1233.06001
[34] Hájek, P., Metamathematics of Fuzzy Logic (2001), Heildelberg: Springer, Heildelberg · Zbl 0937.03030
[35] Hellmann, M.: Fuzzy logic introduction. Epsilon Nought - Radar Remote Sensing Tutorial (2001)
[36] Hungerford, T. W., Algebra (1974), New York: Springer, New York · Zbl 0293.12001
[37] Grattan-Guiness, I., Fuzzy membership mapped onto interval and many-valued quantities, Z. Math. Logik. Grundladen Math., 22, 149-160 (1975) · Zbl 0334.02011
[38] Karaçal, F., On the directed decomposability of strong negations and S-implications operators on product lattices, Information Sciences, 176, 20, 3011-3025 (2006) · Zbl 1104.03016
[39] Kitainik, L., Fuzzy Decision Procedures with Binary Relations (1993), Dordrecht: Kluwer Academic Publisher, Dordrecht · Zbl 0821.90001
[40] Kleene, S. C., On a notation for ordinal numbers, Journal of Symbolic Logics, 3, 150-155 (1938) · Zbl 0020.33803
[41] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Dordrecht: Kluwer, Dordrecht · Zbl 0972.03002
[42] Li, D. C.; Li, Y. M.; Xie, Y. J., Robustness of interval-valued fuzzy inference, Information Sciences, 181, 20, 4754-4764 (2011) · Zbl 1242.68334
[43] Lin, L.; Xia, Z. Q., Intuitionistic fuzzy implication operators expressions and properties, Journal of Applied Mathematics & Computing, 22, 3, 325-338 (2006) · Zbl 1120.03040
[44] Liu, H.-W.; Xue, P.-J.; Cao, B.-Y.; Wang, G.-J.; Chen, S.-L.; Guo, S.-Z., T-Seminorms and Implications on a Complete Lattice, Quantitative Logic and Soft Computing 2010, 215-225 (2010), Heidelberg: Springer, Heidelberg · Zbl 1253.03048
[45] Łukasiewicz, J.: O logice trójwartościowej. Ruch Filozoficzny 5, 170-171 (1920); English translation: On three-valued logic, In: Borkowski, L. (ed.), Jan Łukasiewicz Selected Works, pp. 87-88. North Holland (1990)
[46] Ma, Z.; Wu, W., Logical operators on complete lattices, Information Sciences, 55, 1, 77-97 (1991) · Zbl 0741.03010
[47] Mas, M.; Monserrat, M.; Torrens, J., A characterization of (U,N), RU, QL and D-implications derived from uninorms satisfying the law of importation, Fuzzy Sets and Systems, 161, 10, 1369-1387 (2010) · Zbl 1195.03031
[48] Mas, M.; Monserrat, M.; Torrens, J.; Trillas, E., A survey on fuzzy implication functions, IEEE Transactions on Fuzzy Systems, 15, 6, 1107-1121 (2007)
[49] Massanet, S.; Torrens, J., On a new class of fuzzy implications: h-implications and generalizations, Information Sciences, 181, 11, 2111-2127 (2011) · Zbl 1252.03060
[50] Menger, K., Statistical metrics, Proceedings of the National Academy of Sciences, 28, 12, 535-537 (1942) · Zbl 0063.03886
[51] Mesiar, R.; Komorníková, M.; Cornelis, C.; Deschrijver, G.; Nachtegael, M.; Schockaert, S.; Shi, Y., Aggregation Functions on Bounded Posets, 35 Years of Fuzzy Set Theory, 3-17 (2010), Heidelberg: Springer, Heidelberg · Zbl 1231.03048
[52] Nguyen, H. T.; Walker, E. A., A First Course in Fuzzy Logics (2000), Boca Raton: CRC Press, Boca Raton · Zbl 0927.03001
[53] Palmeira, E. S.; Bedregal, B., Extension of fuzzy logic operators defined on bounded lattices via retractions, Computers and Mathematics with Applications, 63, 6, 1026-1038 (2011) · Zbl 1247.03037
[54] Post, E. L., Introduction to a general theory of elementary propositions, American Journal of Mathematics, 43, 163-185 (1921) · JFM 48.1122.01
[55] Reiser, R. H.S.; Dimuro, G. P.; Bedregal, B.; Santos, H. S.; Callejas-Bedregal, R., S-implications on complete lattice and the interval constructor, Tendências em Matemática Aplicada e Computacional - TEMA, 9, 1, 143-154 (2008) · Zbl 1208.03029
[56] Ruan, D.; Kerre, E. E., Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy Sets and Systems, 54, 1, 23-37 (1993) · Zbl 0784.68078
[57] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Dedrecen, 10, 69-81 (1963) · Zbl 0119.14001
[58] Shang, Y.; Yuan, X.; Lee, E. S., The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets, Computers & Mathematics with Applications, 60, 442-463 (2010) · Zbl 1201.03048
[59] Shi, Y.; Van Gasse, B.; Ruan, D.; Cornelis, C.; Deschrijver, G.; Nachtegael, M.; Schockaert, S.; Shi, Y., Implications in Fuzzy Logic: Properties and a New Class, 35 Years of Fuzzy Set Theory, 83-103 (2010), Heidelberg: Springer, Heidelberg · Zbl 1256.03029
[60] Shi, Y.; Van Gasse, B.; Ruan, D.; Kerre, E. E., On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets and Systems, 161, 10, 1388-1405 (2010) · Zbl 1195.03032
[61] Shi, Y.; Ruan, D.; Kerre, E. E., On the characterizations of fuzzy implications satisfying I(x,y) = I(x,I(x,y)), Information Sciences, 177, 14, 2954-2978 (2007) · Zbl 1122.03023
[62] Sussner, P., Nachtegael, M., Mélange, T., Deschrijver, G., Esmi, E., Kerre, E.E.: Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of \(\mathbb{L} \)-fuzzy mathematical morphology. Mathematical Imaging and Vision (2011), doi:10.1007/s10851-011-0283-1 · Zbl 1255.68267
[63] Titani, S., A lattice-valued set theory, Archive for Mathematical Logic, 38, 395-421 (1999) · Zbl 0936.03048
[64] Trillas, E., Sobre funciones de negación en la teoria de los conjuntos difusos, Stochastica, 3, 1, 47-59 (1979) · Zbl 0419.03035
[65] Trillas, E., Valverde, L.: On some functionally expressable implications for fuzzy set theory. In: Klement, E. (ed.) Proc. Third International Seminar on Fuzzy Set Theory, Linz, Austria, pp. 173-190 (1981) · Zbl 0498.03015
[66] Trillas, E.; Valverde, L., On Implication and Indistinguishability in the Setting of Fuzzy Logic, Management Decision Support Systems Using Fuzzy Sets and Possibility Theory (1985), Rheinland: Verlag TÜV, Rheinland
[67] Wang, Z.; Yu, Y., Pseudo-t-norms and implications operators on a complete Browerian lattice, Fuzzy Sets and Systems, 132, 113-124 (2002) · Zbl 1013.03020
[68] Xu, Y., Ruan, D., Qim, K., Liu, J.: Lattice-Valued Logic. Springer (2010)
[69] Yager, R. R., On the implication operator in fuzzy logic, Information Sciences, 31, 2, 141-164 (1983) · Zbl 0557.03015
[70] Yager, R. R., On some new class of implication operators and their role in approximate reasoning, Information Sciences, 167, 193-216 (2004) · Zbl 1095.68119
[71] Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
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