Two New Versions of the Pasting Lemma via Soft Mixed Structure

In this paper, we present two new generalizations of the pasting lemma using soft mixed structure. To do this, we introduce the notions of a $(\tau _{1},\tau _{2})$-$g$-closed soft set and a $(\tau _{1},\tau _{2})$-$gpr$% -closed soft set. We establish the notions of mixed $g$-soft continuity and mixed $gpr$-soft continuity between two soft topological spaces $(X,\tau _{1},\Delta _{1})$, $(X,\tau _{2},\Delta _{1})$ and a soft topological space $(X,\tau ,\Delta _{2})$. Finally we prove two new versions of the pasting lemma using the mixed $g$-soft continuous mapping and the mixed $gpr$-soft continuous mapping.

Keywords:

Pasting lemma, $(\tau _{1}\tau _{2})$-$g$-closed soft $ (\tau _{1}\tau _{2})$-$gpr$-closed soft,

PDF

___

[1] D. Molodtsov, Soft set theory - first results, Comput. Math. Appl., 37 (1999), 19-31.
[2] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547-1553.
[3] I˙. Demir, O. B. Özbakır, İ. Yıldız, A contribution to the study of soft proximity spaces, Filomat, 31 (2017), 2023-2034.
[4] A. Ç . Güler, G. Kale, Regularity and normality on soft ideal topological spaces, Ann. Fuzzy Math. Inform., 9 (2015), 373-383.
[5] Ç. Gündüz, T. Y. Öztürk, S. Bayramov, Separation axioms on neutrosophic soft topological spaces, Turk. J. Math., 43 (2019), 498-510.
[6] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. ABD El-Latif, g - operation and decompositions of some forms of soft continuity in soft topological spaces, Ann. Fuzzy Math. Inform., 7 (2014), 181-196.
[7] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555-562.
[8] S. K. Nazmul, S. K. Samanta, Neighbourhood properties of soft topological spaces, Ann. Fuzzy Math. Inform., 6 (2013), 1-15.
[9] A. Özkan, On near soft sets, Turk. J. Math., 43 (2019), 1005-1017.
[10] M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786-1799.
[11] J. Thomas, S. J. John, On soft generalized topological spaces, J. New Results Sci., 4 (2014), 1-15.
[12] E. D. Yıldırım, A. Ç. Güler, O. B. Özbakir, On soft Ie-Baire spaces. Ann. Fuzzy Math. Inform., 10 (2015), 109-121.
[13] E. D. Yıldırım, O. B. Özbakir, On soft Ie-scattered spaces, An. Univ. Oradea, Fasc. Mat., 23 (2016), 137-146.
[14] İ. Zorlutuna, M. Akdağ, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., 3 (2012), 171-185.
[15] İ Zorlutuna, H. Ç akır, On continuity of soft mappings, Appl. Math. Inf. Sci., 9 (2015), 403-409.
[16] J. C. R. Alcantud, S. C. Rambaud, M. J. M. Torrecillas, Valuation fuzzy soft sets: a flexible fuzzy soft set based decision making procedure for the valuation of assets, Symmetry, 9 (2017), 253.
[17] ˙I. Demir, N-soft mappings with application in medical diagnosis, Math. Methods Appl. Sci., 44(8) (2021), 7343-7358.
[18] R. I˙rkin, N. Y. Özgür, N. Taş, Optimization of lactic acid bacteria viability using fuzzy soft set modelling, Int. J. Optim. Control, Theor. Appl. (IJOCTA), 8 (2018), 266-275.
[19] Dr. A. Kalaichelvi, P. H. Malini, Application of fuzzy soft sets to investment decision making problem, Int. J. Math. Sci. Appl., 1 (2011), 1583-1586.
[20] F. Karaca, N. Taş, Decision making problem for life and non-life insurances, J. BAUN Inst. Sci. Technol., 20 (2018), 572-588.
[21] N. Y. Özgür, N. Taş, A note on “application of fuzzy soft sets to investment decision making problem”, J. New Theory, 7 (2015), 1-10.
[22] N. Taş, N. Y. Özgür, P. Demir, An application of soft set and fuzzy soft set theories to stock management, Süleyman Demirel University J. Nat. Appl. Sci., 21 (2017), 791-196.
[23] M. Ali, H. Khan, L. H. Son, F. Smarandache, W. B. Vasantha Kandasamy, New soft set based class of linear algebraic codes, Symmetry, 10 (2018), 510.
[24] P. Mani, K. Muthusamy, S. Jafari, F. Smarandache, U. Ramalingam, Decision-making via neutrosophic support soft topological spaces, Symmetry, 10 (2018), 217.
[25] G. Ali, M. Akram, A. N. A. Koam, J. C. R. Alcantud, Parameter reductions of bipolar fuzzy soft sets with their decision-making algorithms, Symmetry, 11 (2019), 949.
[26] T. H. Şimşekler, Fuzzy soft topological spaces and the related category FST, Turk. J. Math., 43 (2019), 871-878.
[27] M. Riaz, K. Naeem, M. Aslam, D. Afzal, F. A. Ahmed Almahdi, S. Shaukat Jamal, Multi-criteria group decision making with Pythagorean fuzzy soft topology, J. Intell. Fuzzy Syst. (Preprint), (2020), 1-18.
[28] M. Riaz, S. T. Tehrim, On bipolar fuzzy soft topology with decision-making, Soft Comput., 24(24) (2020), 18259-18272.
[29] M. Riaz, K. Naeem, Measurable soft mappings, J. Math., Punjab Univ., 48(2) (2020).
[30] M. Riaz, N. Ç ağman, I. Zareef, M. Aslam, N-soft topology and its applications to multi-criteria group decision making, J. Intell. Fuzzy Syst., 36(6) (2019), 6521-6536.
[31] A. Açıkgöz, N. Taş, Some new mixed soft sets, MSAEN, 2 (2014), 105-118.
[32] A. Açıkgöz, N. A. Taş, T. Noiri, A decomposition of some types of mixed soft continuity in soft topological spaces, Filomat, 30 2016, 379-385.
[33] A. Csaszar, Mixed constructions for generalized topologies, Acta Math. Hung., 122 (2009), 153-159.
[34] W. K. Min, Mixed q-continuity on generalized topological spaces, Math. Comput. Modelling, 54 (2011), 2597-2601.
[35] W. K. Min, Mixed weak continuity on generalized topological spaces, Acta Math. Hung., 132 (2011), 339-347.
[36] N. A. Taş, A. Açıkgöz, Some mixed soft operations and extremally soft disconnectedness via two soft topologies, Appl. Math. 5 (2014), 490-500.
[37] N. A. Taş, O. B. Özbakır, On some mixed types of continuity on generalized neighborhood systems, J. Adv. Stud. Topol., 5 (2014), 32-43.
[38] M. Anitha, R. Selvi, P. Thangavelu, Pasting lemmas for g-continuous functions, Missouri J. Math. Sci., 21 (2009), 28-33.
[39] K. Balachandran, P. Sundaram, H. Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci., Kochi Univ., Ser. A 12 (1991), 5-13.
[40] Y. Gnanambal, K. Balachandran, On gpr-continuous functions in topological spaces, Indian J. Pure Appl. Math., 30 (1999), 581-593.
[41] K. Kannan, K. C. Rao, Pasting lemmas for some continuous functions, Thai J. Math., 12 (2014), 245-249.
[42] D. Vidhya, R. Parimelazhagan, g b-continuous maps and pasting lemma in topological spaces, Int. J. Math. Anal., 6 (2012), 2307-2315.
[43] S. Hussain, B. Ahmad, Some properties of soft topological space, Comput. Math. Appl., 62 (2011), 4058-4067.
[44] A. Kharal, B. Ahmad, Mappings on soft classes, New Math. Nat. Comput., 7(03) (2011), 471-481.