To contribute to soft topology, we originate the notion of soft bioperators $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $. Then, we apply them to analyze soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open sets and study main properties. We also prove that every soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set is soft open; however, the converse is true only when the soft topological space is soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-regular. After that, we define and study two classes of soft closures namely $ Cl_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Cl $ operators, and two classes of soft interior namely $ Int_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Int $ operators. Moreover, we introduce the notions of soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $.closed sets and soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ T_{\frac{1}{2}} $ spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.
Citation: Baravan A. Asaad, Tareq M. Al-shami, Abdelwaheb Mhemdi. Bioperators on soft topological spaces[J]. AIMS Mathematics, 2021, 6(11): 12471-12490. doi: 10.3934/math.2021720
To contribute to soft topology, we originate the notion of soft bioperators $ \tilde{\gamma} $ and $ {\tilde{\gamma}}^{'} $. Then, we apply them to analyze soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open sets and study main properties. We also prove that every soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-open set is soft open; however, the converse is true only when the soft topological space is soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-regular. After that, we define and study two classes of soft closures namely $ Cl_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Cl $ operators, and two classes of soft interior namely $ Int_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $ and $ \tilde{\tau}_{(\tilde{\gamma}, {\tilde{\gamma}}^{'})} $-$ Int $ operators. Moreover, we introduce the notions of soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ g $.closed sets and soft $ (\tilde{\gamma}, {\tilde{\gamma}}^{'}) $-$ T_{\frac{1}{2}} $ spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.
[1] | 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. doi: 10.1016/j.camwa.2008.11.009 |
[2] | T. M. Al-shami, Comments on "Soft mappings spaces", Sci. World J., 2019 (2019), 6903809. |
[3] | T. M. Al-shami, Investigation and corrigendum to some results related to $g$-soft equality and $gf$-soft equality relations, Filomat, 33 (2019), 3375–3383. doi: 10.2298/FIL1911375A |
[4] | T. M. Al-shami, Comments on some results related to soft separation axioms, Afrika Mat., 31 (2020), 1105–1119. doi: 10.1007/s13370-020-00783-4 |
[5] | T. M. Al-shami, Compactness on soft topological ordered spaces and its application on the information system, J. Math., 2021 (2021), 6699092. |
[6] | T. M. Al-shami, On soft separation axioms and their applications on decision-making problem, Math. Probl. Eng., 2021 (2021), 8876978. |
[7] | T. M. Al-shami, E. A. Abo-Tabl, Soft $\alpha$-separation axioms and $\alpha$-fixed soft points, AIMS Math., 6 (2021), 5675–5694. doi: 10.3934/math.2021335 |
[8] | T. M. Al-shami, M. E. El-Shafei, $T$-soft equality relation, Turk. J. Math., 44 (2020), 1427–1441. doi: 10.3906/mat-2005-117 |
[9] | T. M. Al-shami, M. E. El-Shafei, M. Abo-Elhamayel, On soft topological ordered spaces, J. King Saud Univ. Sci., 31 (2019), 556–566. doi: 10.1016/j.jksus.2018.06.005 |
[10] | T. M. Al-shami, L. D. R. Kočinac, The equivalence between the enriched and extended soft topologies, Appl. Comput. Math., 18 (2019), 149–162. |
[11] | S. S. Benchalli, P. G. Patil, N. S. Kabbur, On soft $\gamma$-operations in soft topological spaces, J. New Theory, 6 (2015), 20–32. |
[12] | B. Chen, The parametrization reduction of soft sets and its applications, Comput. Math. Appl., 49 (2005), 757–763. doi: 10.1016/j.camwa.2004.10.036 |
[13] | S. Das, S. K. Samanta, Soft metric, Ann. Fuzzy Math. Inform., 6 (2013), 77–94. |
[14] | M. E. El-Shafei, T. M. Al-shami, Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem, Comput. Appl. Math., 39 (2020), 138. doi: 10.1007/s40314-020-01161-3 |
[15] | M. E. El-Shafei, M. Abo-Elhamayel, T. M. Al-shami, Partial soft separation axioms and soft compact spaces, Filomat, 32 (2018), 4755–4771. doi: 10.2298/FIL1813755E |
[16] | F. Feng, Y. B. Jun, X. Zhao, Soft semirings, Comput. Math. Appl., 56 (2008), 2621–2628. |
[17] | F. Feng, Y. M. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14 (2010), 899–911. doi: 10.1007/s00500-009-0465-6 |
[18] | D. N. Georgiou, A. C. Megaritis, V. I. Petropoulos, On soft topological spaces, Appl. Math. Inform. Sci., 7 (2013), 1889–1901. |
[19] | S. Hussain, B. Ahmad, Some properties of soft topological spaces, Comput. Math. Appl., 62 (2011), 4058–4067. doi: 10.1016/j.camwa.2011.09.051 |
[20] | D. S. Jankovic, On functions with $\alpha$-closed graphs, Glasnik Mat., 18 (1983), 141–148. |
[21] | N. Kalaivani, D. Saravanakumar, G. S. S. Krishnan, On $\gamma$-operations in soft topological spaces, Far East J. Math. Sci., 101 (2017), 2067–2077. |
[22] | A. Kalavathia, Studies on generalizations of soft closed sets and their operation approaches in soft topological spaces, PhD Thesis, Anna University, 2017. |
[23] | A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. Abd El-latif, Soft semi compactness via soft ideals, Appl. Math. Inform. Sci., 8 (2014), 2297–2306. doi: 10.12785/amis/080524 |
[24] | S. Kasahara, Operation compact spaces, Math. Jpn., 24 (1979), 97–105. |
[25] | A. Kharal, B. Ahmad, Mappings on soft classes, New Math. Nat. Comput., 7 (2011), 471–481. |
[26] | Z. Kong, L. Gao, L. Wong, S. Li, The normal parameter reduction of soft sets and its algorithm, Comput. Math. Appl., 56 (2008), 3029–3037. doi: 10.1016/j.camwa.2008.07.013 |
[27] | P. K. Maji, R. Biswas, R. Roy, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. doi: 10.1016/S0898-1221(02)00216-X |
[28] | P. K. Maji, R. Biswas, R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. |
[29] | D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. |
[30] | H. Ogata, Operation on topological spaces and associated topology, Math. Jpn., 36 (1991), 175–184. |
[31] | D. Pie, D. Miao, From soft sets to information systems, 2005 IEEE International Conference on Granular Computing, 617–621. |
[32] | M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. |
[33] | J. Umehara, H. Maki, T. Noiri, Bioperations on topological spaces and some separation axioms, Mem. Fac. Sci. Kochi Univ. Ser. A (Math.), 13 (1992), 45–59. |
[34] | Y. Yumak, A. K. Kaymakci, Soft $\beta$-open sets and their applications, J. New Theory, 4 (2015), 80–89. |
[35] | I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., 3 (2012), 171–185. |
[36] | I. Zorlutuna, H. Cakir, On continuity of soft mappings, Appl. Math. Inform. Sci., 9 (2015), 403–409. doi: 10.12785/amis/090147 |