Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
ordinal sum; implication; bounded lattice
ordinal sum; implication; bounded lattice
Summary:
In this paper, the ordinal sum construction methods of implications on bounded lattices are studied. Necessary and sufficient conditions of an ordinal sum for obtaining an implication are presented. New ordinal sum construction methods on bounded lattices which generate implications are discussed. Some basic properties of ordinal sum implications are studied.
References:
[1] Baczyński, M., Drygaś, P., Król, A., Mesiar, R.: New types of ordinal sum of fuzzy implications. In: Fuzzy systems (FUZZ-IEEE), 2017 IEEE International Conference, 2017. DOI 10.1109/fuzz-ieee.2017.8015700
[2] Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing 231, Springer, Berlin, Heidelberg, 2008. MR 2428086 | Zbl 1293.03012
[3] Birkhoff, G.: Lattice Theory. Third edition. Providence, 1967. DOI 10.1090/coll/025 | MR 0227053
[4] Çaylı, G. D.: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets and Systems 132 (2018), 129-143. DOI 10.1016/j.fss.2017.07.015 | MR 3732255
[5] Drygaś, P., Król, A.: Various kinds of fuzzy implications. In: Novel Developments in Uncertainty Represent, and Processing (K. T. Atanassov et al., eds.), Advences in Intelligent Systems and Computing 401, Springer Internat. Publ. AG, 2016, pp. 37-49. DOI 10.1007/978-3-319-26211-6\_4
[6] Drygaś, P., Król, A.: Generating fuzzy implications by ordinal sums. Tatra Mt. Math. Publ. 66 (2016), 39-50. DOI 10.1515/tmmp-2016-0018 | MR 3591073
[7] Dubois, D., Prade, H.: A review of fuzzy set aggregation connectives. Inform. Sci. 36 (1985), 85-121. DOI 10.1016/0020-0255(85)90027-1 | MR 0813766 | Zbl 0582.03040
[8] Dubois, D., Prade, H.: Fuzzy sets in approximate reasoning, Part 1: inference with possibility distributions. Fuzzy Sets and Systems 40 (1991), 143-202. DOI 10.1016/0165-0114(91)90050-z | MR 1103660
[9] Ertuğrul, Ü., Kesicioğlu, M. N., Karaçal, F.: Ordering based on uninorms. Inform. Sci. 330 (2016), 315-327. DOI 10.1016/j.ins.2015.10.019
[10] Ertuğrul, Ü., Karaçal, F., Mesiar, R.: Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. Int. J. Intell. Systems 30 (2015), 807-817. DOI 10.1002/int.21713
[11] Fodor, J., Rudas, I. J.: Migrative t-norms with respect to continuous ordinal sums. Inform. Sci. 181 (2011), 4860-4866. DOI 10.1016/j.ins.2011.05.014 | MR 2825865
[12] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, 2009. DOI 10.1109/sisy.2008.4664901 | MR 2538324 | Zbl 1206.68299
[13] Kesicioğlu, M. N., Mesiar, R.: Ordering based on implications. Inform. Sci. 276 (2014), 377-386. DOI 10.1016/j.ins.2013.12.047 | MR 3206505
[14] Klement, E. P., (eds.), R. Mesiar: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier, Amsterdam 2005. MR 2165231
[15] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. MR 1790096 | Zbl 1087.20041
[16] Klement, E. P., Mesiar, R., Pap, E.: Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford. Semigroup Forum 65 (2002), 71-82. DOI 10.1007/s002330010127 | MR 1903555
[17] Ma, Z., Wu, W. M.: Logical operators on complete lattices. Inform. Sci. 55 (1991), 77-97. DOI 10.1016/0020-0255(91)90007-h | MR 1080449 | Zbl 0741.03010
[18] Mas, M., Monserrat, M., Torrens, J.: The law of importation for discrete implications. Inform. Sci. 179 (2009), 4208-4218. DOI 10.1016/j.ins.2009.08.028 | MR 2722378
[19] Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15 (2007), 1107-1121. DOI 10.1109/tfuzz.2007.896304
[20] Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets and Systems 202 (2012), 75-88. DOI 10.1016/j.fss.2012.03.002 | MR 2934787
[21] Mesiar, R., Mesiarová, A.: Residual implications and left-continuous t-norms which are ordinal sum of semigroups. Fuzzy Sets and Systems 143 (2004), 47-57. DOI 10.1016/j.fss.2003.06.008 | MR 2060272
[22] Mesiarová-Zemánková, A.: Ordinal sum construction for uninorms and generalized uninorms. Int. J. Approx. Reason. 76 (2016), 1-17. DOI 10.1016/j.ijar.2016.04.007 | MR 3521044
[23] Mesiarová-Zemánková, A.: Ordinal sums of representable uninorms. Fuzzy Sets and Systems 308 (2017), 42-53. DOI 10.1016/j.fss.2016.07.006 | MR 3579153
[24] Riera, J. V., Torrens, J.: Residual implications on the set of discrete fuzzy numbers. Inform. Sci. 247 (2013), 131-143. DOI 10.1016/j.ins.2013.06.008 | MR 3085948
[25] Saminger, S.: On ordinal sum of triangular norms on bounded lattices. Fuzzy Sets and Systems 157 (2006), 1403-1416. DOI 10.1016/j.fss.2005.12.021 | MR 2226983
[26] Su, Y., Xie, A., Liu, H.: On ordinal sum implications. Inform. Sci. 293 (2015), 251-262. DOI 10.1016/j.ins.2014.09.021 | MR 3273566
[27] Xie, A., Liu, H., Zhang, F., Li, C.: On the distributivity of fuzzy implications over continuous Archimedean t-conorms and continuous t-conorms given as ordinal sums. Fuzzy Sets and Systems 205 (2012), 76-100. DOI 10.1016/j.fss.2012.01.009 | MR 2960108
[28] Yager, R. R.: Aggregation operators and fuzzy systems modelling. Fuzzy Sets and Systems 67 (1994), 129-145. DOI 10.1016/0165-0114(94)90082-5 | MR 1302575
[29] Yager, R. R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120. DOI 10.1016/0165-0114(95)00133-6 | MR 1389951 | Zbl 0871.04007
Search
Partner of
