WAY-BELOW ESNEK KÜME BAĞINTISI

Öz

Sıralama, matematiksel objeler ve ilgili alanlarda sıklıkla kullanılmaktadır. Molodtsov [1] tarafında ortaya çıkartılan esnek kümelerde sıralama kullanıyoruz. İlk olarak Babitha and Sunil [2] esnek küme bağıntısı tanımını verdikten sonra Babitha and Sunil [3] kısmi sıralı esnek kümeyi tanımlamışlardır. “New structures on partially ordered soft sets and soft Scott topology” [4] isimli çalışmamızda, yönlendirilmiş esnek kümeleri tanıttık. Bundan başka, “Some new results on orderings on soft sets” [5] isimli çalışmamızda esnek latis, tam esnek latis tanımları verildi. Way-Below Esnek Küme bağıntısı hakkında ICRAPAM 2014’te Konferans Özet Kitabında sadece özetinin yayınlandığı bir sunum yaptık [6].  Bu makalede, bu çalışma genişletildi ve sunumumuzdaki tüm sonuçların ispatları verildi.  Ayrıca, sürekli esnek latis, L-esnek domain tanımlandı ve bazı önemli sonuçlar elde edilerek ispatlandı. 

Anahtar Kelimeler

• Esnek küme bağıntısı,Way-below esnek küme bağıntısı,Esnek domain,Yönlendirilmiş esnek küme

THE WAY-BELOW SOFT SET RELATION

Öz

Order is used frequently in mathematical objects and related fields. We use order in the soft set that is introduced by Molodtsov [1]. Babitha and Sunil [2] gave the definition of soft set relation, then Babitha and Sunil [3] defined partially ordered soft set. We introduced directed soft sets in [4]. Moreover, definition of soft lattice, complete soft lattice were given in [5]. We made a presentation, whose abstract was published only in the Abstract Book, in ICRAPAM 2014 [6] about the way-below soft set relation. In this paper, this study is  extended and the proofs of all results of our presentation is given. Moreover, continuous soft lattices, L-soft domain are defined, and some important results are obtained and proved in this study. 

Anahtar Kelimeler

• Soft set relation,Directed soft set,Soft domain,Way-below soft set relation

Kaynakça

1. [1] Molodtsov, D. (1999). “Soft set theory-first results.” Comput. Math. Appl. 37: 19–31
2. [2] Babitha, K. V. and Sunil, J. J. (2010). “Soft set relations and functions.” Comput. Math. Appl. 60: 1840–1849.
3. [3] Babitha, K.V. and Sunil, J.J. (2011). “Transitive closures and ordering on soft sets.” Comput. Math. Appl. 62: 2235–2239.
4. [4] Tanay, B. and Yaylalı, G. (2014). “New structures on partially ordered soft sets and soft Scott topology.” Ann. Fuzz. Math. Inform. 7(89-97).
5. [5] Tanay, B. and Yaylalı, G. (2015). “Some new results on orderings on soft sets.” Journal of Technology of Dumlupınar University 34.
6. [6] Tanay, B. and Yaylalı, G. (2014), “The Way-Below Soft Set Relation”, International Conferences on Recent Advances in Pure and Applied Mathematics, Antalya, Turkey (2014)
7. [7] Maji, P.K., Biswas, R. and Roy, A.R. (2003). “Soft set theory.” Comput. Math. Appl. 45: 555–562.
8. [8] Çağman N. and Enginoğlu S., “Soft matrix theory and its decision making”, Comput. Math. Appl. 59 (2010) 3308-3314.

9. [9] Aktaş H., Çağman N., “Soft Sets and Soft Groups”, Inform. Sciences. 177 2726-2735 (2007)
10. [10] Jun Y.B. and Park C.H., “Applications of Soft Sets in Ideal Theory of BCK/BCI Algebras”, Inform. Sciences, 178 2466-2475 (2008)
11. [11] Jun Y.B. and Park C.H., “Applications of Soft Sets in Hilbert Algebras”, Iran J. Fuzzy Syst. 6/2 75-88 (2009)
12. [12] Jun Y.B., Kim H.S. and Neggers J., “Pseudo d-algebras”, Inform. Sciences, 179 1751-1759 (2009)
13. [13] Jun, Y., B., Lee, K., J. and Khan, A., “Soft Ordered Semigroups”, Math. Log Quart. 56, No.1, 42-50 (2010)
14. [14] Roy, S. and Samanta, T.K. (2011). “An introduction of a soft topological spaces.” In Proceeding of UGC sponsored National seminar on Recent trends in Fuzzy set theory, Rough set theory and Soft set theory at Uluberia College on 23rd and 24th September, 2011.
15. [15] Onyeozili I.A. and Gwary T.M., “A study the Fundamentals of Soft Set Theory”, International of Sciences and Technology Research, 3-4 132-143 (2014)
16. [16] Maji P.K., Biswas R. and Roy A.R., “Fuzzy Soft sets”, Journal Fuzzy Mathematics, Vol.3, 589-602 (2001)
17. [17] Borah M.J., Neog T. J. and Sut D. K., “Relations on Fuzzy Soft Sets”, J. Math. Comput. Sci. 2 No: 3 515-534 (2012)
18. [18] Sut D. K., “An Application of Fuzzy Soft Relation in Decision Making Problems”, International Journal of Mathematics Trends and Technology Vol.3-2 (2012)
19. [19] Guan X., Li Y. and Feng F., “A new order relation on fuzzy soft sets and its applications”, Soft Compt. 17 63-70 (2013)
20. [20] Seselja B. and Tepavcecic A. “A note on a natural equivalence relation on fuzzy power set”, Fuzzy Sets and Systems, 148 201-210 (2004)
21. [21] Çağman N., Cıtak F. and Enginoğlu S., “Fuzzy parametrized fuzzy soft set theory and its applications”, Turk. J. Fuzzy Syst. 1/1 21-35 (2010)
22. [22] Ibrahim A.M, Dauda M.K. and Singh D., “Composition of Soft Set Relations and Construction of Transitive Closure”, Math Theory and Modeling, 2-7 (2012) 98-108.
23. [23] Fu Li, “Soft Lattices”, Global Journal of Science Frontier Research 56 Vol.10 Issue 4 (Ver 1,0) (2010)
24. [24] Ali, M., I., “Soft Ideals and Soft Filters of Soft Ordered Semigroups”, Comp. Math. Appl. 62 (2011) 3396-3403
25. [25] Yang, H. and Guo, Z., “Kernels and Closures of Soft Set Relations, and Soft Set Relation Mappings”, Comp. Math. Appl. 61(2011)651-662
26. [26] Zhang G. and Gao N., “Lattices and topological structures of soft sets over the power set of a universe”, Ann. Fuzzy Math. Inform., 8-2 319-337 (2014)
27. [27] Karaaslan F., Çağman N., and Enginoğlu S., “Soft Lattices”, Journal New Results in Science 1 5-17 (2012)
28. [28] Jobissh V. D., Babitha K. V. and Sunil J. J., “On soft Lattice Operation”, Journal Advanced Research in Pure Mathematics, 5-2 p.71 (2013)
29. [29] Park, J.H., Kim, O.H. and Kwun, Y.C., “Some properties of soft set relations”, Compt. Math. Appl. 63 (2012) 1079-1088
30. [30] Sayed, A. F. (2014). “Continuity of partially ordered soft sets via soft scott topology and soft sobrification.” Bulletin of Mathematical Sciences and Applications 3: 98–113.
31. [31] Karel Hrbacek, Thomas Jech, “Introduction to Set Theory”, Marcel Dekker Inc., 1984.
32. [32] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (1993). “Continuous Lattices and Domains. Cambrigde University Press.
33. [33] Yaylalı G., Tanay B., “Results on soft continuous functions in the soft topological spaces equipped with soft Scott topology”, J. Nonlinear Sci. Appl., 10, 1183-1194, doi:10.22436/jnsa.010.03.26 (2017)