Research Article
BibTex RIS Cite

Sınırlı Kafesler Üzerinde Kapanış ve İç Operatörlere Dayanan Uninormların İnşaası

Year 2024, Volume: 24 Issue: 6, 1323 - 1332, 02.12.2024

Gül Deniz Çaylı

https://doi.org/10.35414/akufemubid.1439061

Abstract

Sınırlı kafesler üzerinde üçgensel normları ve üçgensel konormları genelleştiren uninormlar son zamanlarda oldukça ilgi çekmiştir. Bu makalede bir sınırlı kafes üzerinde bir birim elemanlı uninormları üreten iki yeni yaklaşım önerilmektedir. Bu yaklaşımlar, bir sınırlı kafes üzerinde bir üçgensel normun (üçgensel konormun) ve bir kapanış operatörün (iç operatörün) varlıklarından yararlanmaktadır. Bu esnada, sınırlı kafesler üzerinde idempotent uninormların iki yapısı elde edilmektedir. Ayrıca, önerilen yaklaşımlar ve mevcut inşaalar arasındaki ilişki araştırılmaktadır.

Keywords

Sınırlı kafes uninorm kapanış operatörü iç operatör

References

  • Beliakov, G., Pradera, A. and Calvo, T., 2007. Aggregation Functions: A guide for Practitioners, Springer, Berlin. Benítez, J.M., Castro, J.L. and Requena, I., 1997. Are artificial neural networks black boxes? IEEE Transactions on Neural Networks and Learning Systems, 8, 1156–1163. https://doi.org/10.1109/72.623216
  • Birkhoff, G., 1967. Lattice Theory. American Mathematical Society Colloquium Publishers, Providence, RI. Bodjanova, S. and Kalina, M., 2018. Uninorms on bounded lattices-recent development. In: Kacprzyk, J., et al. (Eds.) EUSFLAT 2017, AISC, vol 641, Springer, Cham, 224–234.
  • Bodjanova, S. and Kalina, M., 2019. Uninorms on bounded lattices with given underlying operations. In: Halaś, R., et al. (Eds.), AGOP 2019, AISC, vol. 981, Springer, Cham, 183-194.
  • Çaylı, G.D., Karaçal, F. and Mesiar, R., 2019. On internal and locally internal uninorms on bounded lattices. International Journal of General Systems, 48, 235-259. https://doi.org/10.1080/03081079.2018.1559162
  • Çaylı, G.D., 2019. Alternative approaches for generating uninorms on bounded lattices. Information Sciences, 488, 111-139. https://doi.org/10.1016/j.ins.2019.03.007
  • Çaylı, G.D., 2020. Uninorms on bounded lattices with the underlying t-norms and t-conorms. Fuzzy Sets and Systems, 395, 107-129. https://doi.org/10.1016/j.fss.2019.06.005
  • Çaylı, G.D., 2021. New construction approaches of uninorms on bounded lattices, International Journal of General Systems, 50, 139-158. https://doi.org/10.1080/03081079.2020.1863397
  • Dan, Y. and Hu, B.Q., 2020. A new structure for uninorms on bounded lattices. Fuzzy Sets and Systems, 386, 77-94. https://doi.org/10.1016/j.fss.2019.02.001
  • Dan, Y., Hu, B.Q. and Qiao, J., 2019. New constructions of uninorms on bounded lattices. International Journal of Approximate Reasoning, 110, 185-209. https://doi.org/10.1016/j.ijar.2019.04.009
  • De Baets, B., 1999. Idempotent uninorms. European Journal of Operational Research, 118, 631-642. https://doi.org/10.1016/S0377-2217(98)00325-7
  • De Baets, B., Fodor, J., Ruiz-Aguilera, D. and Torrens, J., 2009. Idempotent uninorms on finite ordinal scales. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 17, 1-14. https://doi.org/10.1142/S021848850900570X
  • Drewniak, J. and Drygaś, P., 2002. On a class of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 10, 5-10. https://doi.org/10.1142/S021848850200179X
  • Drossos, C. A., 1999. Generalized t-norm structures. Fuzzy Sets and Systems, 104, 53–59. https://doi.org/10.1016/S0165-0114(98)00258-9
  • Drossos, C. A. and Navara, M. 1996. Generalized t-conorms and closure operators. Proceedings of EUFIT’96, Aachen, Germany, 22–26.
  • Dubios, D. and Prade, H., 1995. A review of fuzzy set aggregation connectives. Information Sciences, 3, 85–121. https://doi.org/10.1016/0020-0255(85)90027-1
  • Dubios, D. and Prade, H., 2000. Fundamentals of Fuzzy Sets Kluwer Academic Publisher, Boston. Everett, C.J., 1944. Closure operators and Galois theory in lattices. Transactions of the American Mathematical Society, 55, 514–525. https://doi.org/10.1090/S0002-9947-1944-0010556-9
  • Fodor, J., Yager, R.R. and Rybalov, A., 1997. Structure of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 5, 411-427. https://doi.org/10.1142/S0218488597000312
  • González-Hidalgo, M., Massanet, S., Mir, A. and Ruiz-Aguilera, D., 2015. On the choice of the pair conjunction–implication into the fuzzy morphological edge detector. IEEE Transactions on Fuzzy Systems, 23, 872–884. https://doi.org/10.1109/TFUZZ.2014.2333060
  • He, P. and Wang, X.P., 2021. Constructing uninorms on bounded lattices by using additive generators. International Journal of Approximate Reasoning, 136, 1-13. https://doi.org/10.1016/j.ijar.2021.05.006
  • Hua, X.J. and Ji, W., 2022. Uninorms on bounded lattices constructed by t-norms and t-subconorms. Fuzzy Sets and Systems, 427, 109-131. https://doi.org/10.1016/j.fss.2020.11.005
  • Karaçal, F. and Mesiar, R., 2015. Uninorms on bounded lattices. Fuzzy Sets and Systems, 261, 33–43. https://doi.org/10.1016/j.fss.2014.05.001
  • Klement, E.P., Mesiar, R. and Pap, E., 2000. Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000. Klement, E.P., Mesiar, R. and Pap, E., 2004a. Triangular norms. Position article I: basic analytical and algebraic properties. Fuzzy Sets and Systems, 143, 5–26. https://doi.org/10.1016/j.fss.2003.06.007
  • Klement, E.P., Mesiar, R. and Pap, E., 2004b. Triangular norms. Position article II: general construc- tions and parametrized families. Fuzzy Sets and Systems, 145, 411–438. https://doi.org/10.1016/S0165-0114(03)00327-0
  • Menger, K., 1942. Statistical metrics. Proceedings of the National Academy of Sciences, 8, 535–537. https://doi.org/10.1073/pnas.28.12.535
  • Ouyang, Y. and Zhang, H.P., 2020. Constructing uninorms via closure operators on a bounded lattice. Fuzzy Sets and Systems, 395, 93–106. https://doi.org/10.1016/j.fss.2019.05.006
  • Schweizer, B. and Sklar, A., 1963. Associative functions and abstract semigroups. Publicationes Mathematicae Debrecen, 10, 69–81. https://doi.org/10.5486/PMD.1963.10.1-4.09
  • Schweizer, B. and Sklar, A., 1983. Probabilistic Metric Spaces. Elsevier North-Holland, New York. Sun, X.R. and Liu, H.W., 2022. Further characterization of uninorms on bounded lattices. Fuzzy Sets and Systems, 427, 96-108. https://doi.org/10.1016/j.fss.2021.01.006
  • Takács, M., 2008. Uninorm-based models for FLC systems. Journal of Intelligent & Fuzzy Systems, 19, 65-73. Yager, R.R., 1994. Aggregation operators and fuzzy systems modelling. Fuzzy Sets and Systems, 67, 129–145. https://doi.org/10.1016/0165-0114(94)90082-5
  • Yager, R.R. and Rybalov, A., 1996. Uninorm aggregation operators. Fuzzy Sets and Systems, 80, 111-120. https://doi.org/10.1016/0165-0114(95)00133-6
  • Yager, R.R., 2001. Uninorms in fuzzy systems modelling. Fuzzy Sets and Systems, 122, 167–175. https://doi.org/10.1016/S0165-0114(00)00027-0
  • Yager, R.R., 2003. Defending against strategic manipulation in uninorm-based multi-agent decision making. Fuzzy Sets and Systems, 140, 331-339. https://doi.org/10.1016/S0377-2217(01)00267-3
  • Zhao, B. and Wu, T., 2021. Some further results about uninorms on bounded lattices. International Journal of Approximate Reasoning, 130, 22-49. https://doi.org/10.1016/j.ijar.2020.12.008

Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices

Year 2024, Volume: 24 Issue: 6, 1323 - 1332, 02.12.2024

Gül Deniz Çaylı

https://doi.org/10.35414/akufemubid.1439061

Abstract

Uninorms generalizing triangular norms and triangular conorms on bounded lattices have attracted considerable attention recently. In this article, two new approaches are suggested to generate uninorms with an identity element on a bounded lattice. These approaches exploit the existences of a triangular norm (triangular conorm) and a closure operator (interior operator) on a bounded lattice. Meanwhile, two structures of idempotent uninorms on bounded lattices are obtained. In addition, the relationship between the proposed approaches and the existing constructions is investigated.

Keywords

Bounded lattice Uninorm Closure operator Interior operator

References

  • Beliakov, G., Pradera, A. and Calvo, T., 2007. Aggregation Functions: A guide for Practitioners, Springer, Berlin. Benítez, J.M., Castro, J.L. and Requena, I., 1997. Are artificial neural networks black boxes? IEEE Transactions on Neural Networks and Learning Systems, 8, 1156–1163. https://doi.org/10.1109/72.623216
  • Birkhoff, G., 1967. Lattice Theory. American Mathematical Society Colloquium Publishers, Providence, RI. Bodjanova, S. and Kalina, M., 2018. Uninorms on bounded lattices-recent development. In: Kacprzyk, J., et al. (Eds.) EUSFLAT 2017, AISC, vol 641, Springer, Cham, 224–234.
  • Bodjanova, S. and Kalina, M., 2019. Uninorms on bounded lattices with given underlying operations. In: Halaś, R., et al. (Eds.), AGOP 2019, AISC, vol. 981, Springer, Cham, 183-194.
  • Çaylı, G.D., Karaçal, F. and Mesiar, R., 2019. On internal and locally internal uninorms on bounded lattices. International Journal of General Systems, 48, 235-259. https://doi.org/10.1080/03081079.2018.1559162
  • Çaylı, G.D., 2019. Alternative approaches for generating uninorms on bounded lattices. Information Sciences, 488, 111-139. https://doi.org/10.1016/j.ins.2019.03.007
  • Çaylı, G.D., 2020. Uninorms on bounded lattices with the underlying t-norms and t-conorms. Fuzzy Sets and Systems, 395, 107-129. https://doi.org/10.1016/j.fss.2019.06.005
  • Çaylı, G.D., 2021. New construction approaches of uninorms on bounded lattices, International Journal of General Systems, 50, 139-158. https://doi.org/10.1080/03081079.2020.1863397
  • Dan, Y. and Hu, B.Q., 2020. A new structure for uninorms on bounded lattices. Fuzzy Sets and Systems, 386, 77-94. https://doi.org/10.1016/j.fss.2019.02.001
  • Dan, Y., Hu, B.Q. and Qiao, J., 2019. New constructions of uninorms on bounded lattices. International Journal of Approximate Reasoning, 110, 185-209. https://doi.org/10.1016/j.ijar.2019.04.009
  • De Baets, B., 1999. Idempotent uninorms. European Journal of Operational Research, 118, 631-642. https://doi.org/10.1016/S0377-2217(98)00325-7
  • De Baets, B., Fodor, J., Ruiz-Aguilera, D. and Torrens, J., 2009. Idempotent uninorms on finite ordinal scales. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 17, 1-14. https://doi.org/10.1142/S021848850900570X
  • Drewniak, J. and Drygaś, P., 2002. On a class of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 10, 5-10. https://doi.org/10.1142/S021848850200179X
  • Drossos, C. A., 1999. Generalized t-norm structures. Fuzzy Sets and Systems, 104, 53–59. https://doi.org/10.1016/S0165-0114(98)00258-9
  • Drossos, C. A. and Navara, M. 1996. Generalized t-conorms and closure operators. Proceedings of EUFIT’96, Aachen, Germany, 22–26.
  • Dubios, D. and Prade, H., 1995. A review of fuzzy set aggregation connectives. Information Sciences, 3, 85–121. https://doi.org/10.1016/0020-0255(85)90027-1
  • Dubios, D. and Prade, H., 2000. Fundamentals of Fuzzy Sets Kluwer Academic Publisher, Boston. Everett, C.J., 1944. Closure operators and Galois theory in lattices. Transactions of the American Mathematical Society, 55, 514–525. https://doi.org/10.1090/S0002-9947-1944-0010556-9
  • Fodor, J., Yager, R.R. and Rybalov, A., 1997. Structure of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systsems, 5, 411-427. https://doi.org/10.1142/S0218488597000312
  • González-Hidalgo, M., Massanet, S., Mir, A. and Ruiz-Aguilera, D., 2015. On the choice of the pair conjunction–implication into the fuzzy morphological edge detector. IEEE Transactions on Fuzzy Systems, 23, 872–884. https://doi.org/10.1109/TFUZZ.2014.2333060
  • He, P. and Wang, X.P., 2021. Constructing uninorms on bounded lattices by using additive generators. International Journal of Approximate Reasoning, 136, 1-13. https://doi.org/10.1016/j.ijar.2021.05.006
  • Hua, X.J. and Ji, W., 2022. Uninorms on bounded lattices constructed by t-norms and t-subconorms. Fuzzy Sets and Systems, 427, 109-131. https://doi.org/10.1016/j.fss.2020.11.005
  • Karaçal, F. and Mesiar, R., 2015. Uninorms on bounded lattices. Fuzzy Sets and Systems, 261, 33–43. https://doi.org/10.1016/j.fss.2014.05.001
  • Klement, E.P., Mesiar, R. and Pap, E., 2000. Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000. Klement, E.P., Mesiar, R. and Pap, E., 2004a. Triangular norms. Position article I: basic analytical and algebraic properties. Fuzzy Sets and Systems, 143, 5–26. https://doi.org/10.1016/j.fss.2003.06.007
  • Klement, E.P., Mesiar, R. and Pap, E., 2004b. Triangular norms. Position article II: general construc- tions and parametrized families. Fuzzy Sets and Systems, 145, 411–438. https://doi.org/10.1016/S0165-0114(03)00327-0
  • Menger, K., 1942. Statistical metrics. Proceedings of the National Academy of Sciences, 8, 535–537. https://doi.org/10.1073/pnas.28.12.535
  • Ouyang, Y. and Zhang, H.P., 2020. Constructing uninorms via closure operators on a bounded lattice. Fuzzy Sets and Systems, 395, 93–106. https://doi.org/10.1016/j.fss.2019.05.006
  • Schweizer, B. and Sklar, A., 1963. Associative functions and abstract semigroups. Publicationes Mathematicae Debrecen, 10, 69–81. https://doi.org/10.5486/PMD.1963.10.1-4.09
  • Schweizer, B. and Sklar, A., 1983. Probabilistic Metric Spaces. Elsevier North-Holland, New York. Sun, X.R. and Liu, H.W., 2022. Further characterization of uninorms on bounded lattices. Fuzzy Sets and Systems, 427, 96-108. https://doi.org/10.1016/j.fss.2021.01.006
  • Takács, M., 2008. Uninorm-based models for FLC systems. Journal of Intelligent & Fuzzy Systems, 19, 65-73. Yager, R.R., 1994. Aggregation operators and fuzzy systems modelling. Fuzzy Sets and Systems, 67, 129–145. https://doi.org/10.1016/0165-0114(94)90082-5
  • Yager, R.R. and Rybalov, A., 1996. Uninorm aggregation operators. Fuzzy Sets and Systems, 80, 111-120. https://doi.org/10.1016/0165-0114(95)00133-6
  • Yager, R.R., 2001. Uninorms in fuzzy systems modelling. Fuzzy Sets and Systems, 122, 167–175. https://doi.org/10.1016/S0165-0114(00)00027-0
  • Yager, R.R., 2003. Defending against strategic manipulation in uninorm-based multi-agent decision making. Fuzzy Sets and Systems, 140, 331-339. https://doi.org/10.1016/S0377-2217(01)00267-3
  • Zhao, B. and Wu, T., 2021. Some further results about uninorms on bounded lattices. International Journal of Approximate Reasoning, 130, 22-49. https://doi.org/10.1016/j.ijar.2020.12.008
There are 32 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory, Pure Mathematics (Other)
Journal Section Articles
Authors

Gül Deniz Çaylı 0000-0002-7918-9752

Early Pub Date November 11, 2024
Publication Date December 2, 2024
Submission Date February 18, 2024
Acceptance Date August 6, 2024
Published in Issue Year 2024 Volume: 24 Issue: 6

Cite

APA Çaylı, G. D. (2024). Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 24(6), 1323-1332. https://doi.org/10.35414/akufemubid.1439061
AMA Çaylı GD. Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. December 2024;24(6):1323-1332. doi:10.35414/akufemubid.1439061
Chicago Çaylı, Gül Deniz. “Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 24, no. 6 (December 2024): 1323-32. https://doi.org/10.35414/akufemubid.1439061.
EndNote Çaylı GD (December 1, 2024) Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 24 6 1323–1332.
IEEE G. D. Çaylı, “Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 24, no. 6, pp. 1323–1332, 2024, doi: 10.35414/akufemubid.1439061.
ISNAD Çaylı, Gül Deniz. “Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 24/6 (December 2024), 1323-1332. https://doi.org/10.35414/akufemubid.1439061.
JAMA Çaylı GD. Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2024;24:1323–1332.
MLA Çaylı, Gül Deniz. “Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 24, no. 6, 2024, pp. 1323-32, doi:10.35414/akufemubid.1439061.
Vancouver Çaylı GD. Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2024;24(6):1323-32.
 Download Cover Image


This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.