Some construction methods of interval-valued implications on bounded lattices

Karacal, Funda; Karacair, Kübra; Ertuğrul, &Uuml;mit

doi:10.22111/ijfs.2024.48572.8569

Document Type : Research Paper

Authors

¹ Karadeniz Technical University

² Karadeniz Teknik Üniversitesi

³ Karadeniz Technical University Faculty of Scince Department of Mathematics TRABZON/TURKEY

https://doi.org/10.22111/ijfs.2024.48572.8569

Abstract

In recent years, implication operators have been studied intensively from many perspectives. These include the study of construction methods of implication operators. Additionally, it has been investigated whether some aggregation operators such as t-norm in interval-valued L-fuzzy set theory are representable. In this paper, we present some construction methods to obtain interval-valued implications, which are i-representable or not depending on the lattice and fuzzy logic operators, via some fuzzy logic operators, order preserving and/or order reversing functions. Moreover, many illustrative examples are included.

Keywords

Main Subjects

Fuzzy Math

References

[1] I. Aguil´o, V. K. Gupta, B. Jayaram, S. Massanet, J. V. Riera, N. R. Vemuri, Generating methods of some classes
of fuzzy implications obtained by unary functions and algebraic structures, Fuzzy Sets and Systems, 484 (2024),
108948. https://doi.org/10.1016/j.fss.2024.108948

[2] C. Alcaldea, A. Buruscob, R. Fuentes-Gonz´alez, A constructive method for the definition of interval-valued fuzzy
implication operators, Fuzzy Sets and Systems, 153(2) (2005), 211-227. https://doi.org/10.1016/j.fss.2005.
01.006

[3] M. Baczy´nski, B. Jayaram, Fuzzy implications, Studies in Fuzziness and Soft Computing, Heidelberg: Springer,
Berlin, 231 (2008). https://doi.org/10.1007/978-3-540-69082-5

[4] B. C. Bedregal, G. P. Dimuro, R. H. N. Santiago, R. H. S. Reiser, On interval fuzzy S-implications, Information
Sciences, 180(8) (2010), 1373-1389. https://doi.org/10.1016/j.ins.2009.11.035

[5] G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publishers, Providence, RI, 1967.

[6] H. Bustince, Interval-valued fuzzy sets in soft computing, International Journal of Computational Intelligence Systems,
3(2) (2010), 215-222. https://doi.org/10.1080/18756891.2010.9727692

[7] G. Deschrijver, A representation of t-norms in interval-valued L-fuzzy set theory, Fuzzy Sets and Systems, 159(13)
(2008), 1597-1618. https://doi.org/10.1016/j.fss.2007.09.017

[8] G. Deschrijver, Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory, Information
Sciences, 244 (2013), 48-59. https://doi.org/10.1016/j.ins.2013.04.033

[9] G. Deschrijver, C. Cornelis, Representability in interval-valued fuzzy set theory, International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems, 15(3) (2007), 345-361. https://doi.org/10.1142/S0218488507004716

[10] P. Dryga´s, Some class of uninorms in interval valued fuzzy set theory in intelligent systems’2014, Springer International Publishing, Cham, 322 (2015), 33-43. https://doi.org/10.1007/978-3-319-11313-5_4

[11] M. B. Gorza lczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets
and Systems, 21(1) (1987), 1-17. https://doi.org/10.1016/0165-0114(87)90148-5

[12] F. M. He, B. W. Fang, Characterizations for the alpha-cross-migrativity of continuous t-conorms over generated
implications, Iranian Journal of Fuzzy Systems, 21(1) (2024), 65-82. https://doi.org/10.22111/ijfs.2023.
44829.7899

[13] F. Kara¸cal, K. Karacair, Some implication operators on bounded lattices, International Journal of General Systems,
53(7-8) (2024), 898-927. https://doi.org/10.1080/03081079.2024.2335907

[14] F. Kara¸cal, M. N. Kesicio˘glu, ¨U. Ertu˘grul, The implications obtained by two given implications on bounded lattices,
International Journal of General Systems, 50(3) (2021), 281-299. https://doi.org/10.1080/03081079.
2021.1908279

[15] M. N. Kesicio˘glu, Construction methods for implications on bounded lattices, Kybernetika, 55(4) (2019), 641-667.
https://doi.org/10.14736/kyb-2019-4-0641

[16] M. N. Kesicio˘glu, ¨U. Ertu˘grul, F. Kara¸cal, Generalized convex combination of implications on bounded lattices,
Iranian Journal of Fuzzy Systems, 17(6) (2020), 75-91. https://doi.org/10.22111/ijfs.2020.5602

[17] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000. https:
//doi.org/10.1007/978-94-015-9540-7

[18] M. Mas, M. Monserrat, J. Torrens, The law of importation for discrete implications, Information Sciences, 179(24)
(2009), 4208-4218. https://doi.org/10.1016/j.ins.2009.08.028

[19] M. Mas, M. Monserrat, J. Torrens, E. Trillas, A survey on fuzzy implication functions, IEEE Transactions on Fuzzy
Systems, 15(6) (2007), 1107-1121. https://doi.org/10.1109/TFUZZ.2007.896304

[20] D. Maximov, V. I. Goncharenko, Y. S. Legovich, Multi-valued neural networks I a multi-valued associative memory,
Neural Computing and Applications, 33(16) (2021), 10189-10198. https://doi.org/10.48550/arXiv.2302.11909

[21] Z. Peng, X. Zhang, Study of the generalized hypothetical syllogism for some well known families of fuzzy implications
with respect to strict t-norm, Iranian Journal of Fuzzy Systems, 21(2) (2024), 181-200. https://doi.org/10.22111/
ijfs.2024.46361.8164

[22] J. V. Riera, J. Torrens, Residual implications on the set of discrete fuzzy numbers, Information Sciences, 247
(2013), 131-143. https://doi.org/10.1016/j.ins.2013.06.008

[23] A. Starkey, H. Hagras, S. Shakya, G. Owusu, A multi-objective genetic type-2 fuzzy logic based system for mobile
field workforce area optimization, Information Sciences, 329 (2016), 390-411. https://doi.org/10.1016/j.ins.
2015.09.014

[24] C. Y. Wang, L. Wan, Type-2 fuzzy implications and fuzzy-valued approximation reasoning, International Journal
of Approximate Reasoning, 102 (2018), 108-122. https://doi.org/10.1016/j.ijar.2018.08.004

[25] M. Wang, X. Zhang, H. Bustince, J. Fernandez, Construction methods of fuzzy implications on bounded posets,
International Journal of Approximate Reasoning, 164 (2024), 109064. https://doi.org/10.1016/j.ijar.2023.
109064

Some construction methods of interval-valued implications on bounded lattices

References

References

Volume 21, Issue 6November and December 2024Pages 33-54

Volume 21, Issue 6
November and December 2024
Pages 33-54