Unibelirleyici

Abstract

Bu çalışmada, hem t-belirleyici (t-definer) hem de t-eşbelirleyici (t-codefiner) yapılarını kapsayan birleşik bir çerçeve sağlayan ve [0,∞]aralığında tanımlanan unibelirleyici (unidefiner) kavramını incelemekteyiz. [0,1]aralığında tanımlanan uninormlara benzer bir yaklaşımla, unibelirleyici, [0,∞] üzerinde tanımlı, değişmeli (komütatif), birleşmeli (assosiatif) ve monoton olan bir ikili işlemdir ve e∈[0,∞] olmak üzere birim (nötr) elemana sahiptir. Buna bağlı olarak, e=0 durumunda bir t-belirleyici, e=∞ durumunda ise bir t-eşbelirleyici elde edilir. Bu çalışma, unibelirleyicilerin teorik özelliklerini ele almakta ve t-belirleyici ile t-eşbelirleyici yapılarla olan ilişkilerini incelemektedir. Sonuç olarak, unibelirleyicilerin, [0,∞] aralığında uygun bir birim değeri (e ≠ 0,e ≠ ∞) seçildiğinde, uninormlara benzer şekilde "genelleştirilmiş bir bağlayıcı" olarak işlev görebileceği vurgulanmaktadır.

Keywords

• Unibelirleyici
• t-belirleyici
• t-eşbelirleyici
• birim eleman
• monoid

Unidefiners

Abstract

In this study, we examine the concept of a unidefiner, defined on the interval [0,∞], which provides a unified framework for both t-definer and t-codefiner structures. Similar to the approach of uninorms on [0,1], a unidefiner is a binary operation on [0,∞], with a neutral element e∈[0,∞],, which is associative, commutative, and monotonic. Consequently, when e = 0, it yields a t-definer, and when e =∞, it yields a t-codefiner. This paper discusses the theoretical properties of unidefiners and explores their relationship with t-definer and t-codefiner examples. In conclusion, it emphasizes that unidefiners can serve as a “generalized connective” analogous to uninorms for a “proper” identity value (i.e., e ≠ 0, e≠∞) in the [0,∞] range.

Keywords

• Unidefiner
• t-definer
• t-codefiner
• identity element
• monoid

References

1. [1] Aşıcı E., Mesiar R. 2021. On the direct product of uninorms on bounded lattices. Kybernetika 57(6), 989–1004
2. [2] Aşıcı E., Mesiar R. 2024. Some investigations on the U-partial order induced by uninorms. Aequa-tiones mathematicae 1-14.
3. [3] Cao M., Du WS. 2023. On residual implications derived from 2-uninorms. International Journal of Approximate Reasoning 159, 108926.
4. [4] Çaylı GD. 2023. An alternative construction of uninorms on bounded lattices. International Journal of General Systems 52(5), 574–596.
5. [5] Çaylı GD. 2024. Constructing uninorms on bounded lattices through closure and interior op-erators. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 32(1), 109–129.
6. [6] Çaylı GD., Ertuğrul U., Karaçal F. 2023. Some fur-ther construction methods for uninorms on bounded lattices. International Journal of General Systems 52(4), 414–442.
7. [7] Csiszar O, Pusztahazi LS., Denes-Fazakas L., Gashler MS., Kreinovich V, Csisz´ar G. 2023. Un-inorm-like parametric activation functions for human-understandable neural models. Knowledge-Based Systems 260, 110095.
8. [8] Dan Y. 2023. A unified way to studies of t-seminorms, t-semiconorms and semi-uninorms on a complete lattice in terms of behaviour oper-ations. International Journal of Approximate Rea-soning 156, 61–76.

9. [9] Dan Y., Hu BQ., De Baets B. 2022. Nullnorms on bounded lattices constructed by means of closure and interior operators. Fuzzy Sets and Systems 439, 142–156.
10. [10] De Campos Souza PV., Lughofer E. 2022. An ad-vanced interpretable fuzzy neural network model based on uni-nullneuron constructed from n-uninorms. Fuzzy Sets and Systems 426, 1–26.
11. [11] Dvorak A., Holcapek M., Paseka J. 2022. On ordi-nal sums of partially ordered monoids: A unified approach to ordinal sum constructions of t-norms, t-conorms and uninorms. Fuzzy Sets and Systems 446, 4–25.
12. [12] Fodor J., De Baets B. 2007. Uninorm basics. In Fuzzy Logic: A Spectrum of Theoretical & Practical Issues 49–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 15s.
13. [13] Fodor JC., Yager RR., Rybalov A. 1997. Structure of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5(4), 411–427.
14. [14] Gürdal U., Oğur O., Çetin S. Unima on [0,1]. Manu-script submitted for publication.
15. [15] He SY., Xie LH., Yan PF. 2022. On ⋆-metric spaces. Filomat 36(18), 6173–6185.
16. [16] Hlinena D., Kalina M. 2011. Characterization of uninorms on bounded lattices and pre-order they induce. International Journal of Computational In-telligence Systems 14(1), 148–158.
17. [17] İnce MA., Karaçal F. 2023. Determination of the smallest-greatest uni-nullnorms and null-uninorms on an arbitrary bounded lattice L. Inter-national Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 31(1), 103–119.
18. [18] Jiang DX., Liu HW. 2024. Migrativity of uninorms not internal on the boundary over continuous t-(co)norms. Iranian Journal of Fuzzy Systems 21(3), 103–121.
19. [19] ] Jocic D., Stajner-Papuga I. 2023. Distributivity of a uni-nullnorm with continuous and Archimedean underlying T-norms and T-conorms over an arbi-trary uninorm. Mathematica Slovaca 73(6), 1527–1544.
20. [20] Karaçal F., Ertuğrul U., Arpacı S., Kesicioğlu MN. 2023. Congruence relations and direct decomposi-tion of uninorms on bounded lattices. Interna-tional Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 31(6), 1033–1059.
21. [21] Karaçal F., Ertuğrul U., Kesicioğlu M. 2021. Gener-ating methods for principal topologies on bound-ed lattices. Kybernetika 57(4), 714-736.
22. [22] Karaçal F., Köroğlu T. 2022. A principal topology obtained from uninorms. Kybernetika 58(6), 863–882.
23. [23] Khatami SMA., Mirzavaziri M. 2020. Yet another generalization of the notion of a metric space. arXiv preprint. arXiv:2009.00943
24. [24] Mesiarova-Zemankova A., Mesiar R., Su Y., Wang ZD. 2024. Idempotent uninorms on bounded lat-tices with at most single point incomparable with the neutral element: Part I. International Journal of General Systems 1-34.
25. [25] Ouyang Y., Zhang HP., De Baets B. 2024. Decom-position and construction of uninorms on the unit interval. Fuzzy Sets and Systems 493–494, 109083.
26. [26] Wang SM. 2019. The logic of pseudo-uninorms and their residua, Symmetry 11(3), 368-380.
27. [27] Wen H., Wu X., Çaylı GD. 2023. Characterizing some types of uninorms on bounded lattices. In-ternational Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 31(4), 533–549.
28. [28] Xie A., Zhang JQ. 2024. On modularity property for uninorms with continuous underlying functions. Iranian Journal of Fuzzy Systems 21(2), 105–116.
29. [29] Yang E. 2021. Micanorm aggregation operators: basic logico-algebraic properties. Soft Computing 25, 13167–13180.
30. [30] Yang B., Lı W., Liu YH., Xu J. 2023. The distributiv-ity of extended semi-t-operators over extended S-uninorms on fuzzy truth values. Soft Computing 28(4), 2823–2841.
31. [31] Zhang HP., Ouyang Y., De Baets B. 2021. Construc-tions of uni-nullnorms and null-uninorms on a bounded lattice. Fuzzy Sets and Systems 403, 78–87.
32. [32] Zong WW., Su Y., Liu HW. 2024. Conditionally distributive uninorms locally internal on the boundary. Semigroup Forum 1–9.