On fuzzy soft $ \beta $-continuity and $ \beta $-irresoluteness: some new results

Ibtesam Alshammari; Islam M. Taha; Ibtesam Alshammari; Islam M. Taha

doi:10.3934/math.2024554

AIMS Mathematics

2024, Volume 9, Issue 5: 11304-11319. doi: 10.3934/math.2024554

Previous Article Next Article

Research article

On fuzzy soft $ \beta $-continuity and $ \beta $-irresoluteness: some new results

Ibtesam Alshammari ¹,
Islam M. Taha ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, University of Hafr Al Batin, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Received: 22 January 2024 Revised: 07 March 2024 Accepted: 12 March 2024 Published: 22 March 2024
MSC : 54A05, 54A40, 54C05, 54C10, 54D30

In this paper, we first introduced the concept of $ r $-fuzzy soft $ \beta $-closed sets in fuzzy soft topological spaces based on the sense of Šostak and investigated some properties of them. Also, we defined the closure and interior operators with respect to the classes of $ r $-fuzzy soft $ \beta $-closed and $ r $-fuzzy soft $ \beta $-open sets and studied some of their properties. Moreover, the concept of $ r $-fuzzy soft $ \beta $-connected sets was introduced and characterized with the help of fuzzy soft $ \beta $-closure operators. Thereafter, some properties of a fuzzy soft $ \beta $-continuity were studied. Also, we introduced and studied the concepts of fuzzy soft almost (weakly) $ \beta $-continuous functions, which are weaker forms of a fuzzy soft $ \beta $-continuity. The relationships between these classes of functions were specified with the help of some illustrative examples. Finally, we explored new types of fuzzy soft functions called fuzzy soft $ \beta $-irresolute (strongly $ \beta $-irresolute, $ \beta $-irresolute open, $ \beta $-irresolute closed, and $ \beta $-irresolute homeomorphism) functions and discussed some properties of them. Also, we showed that fuzzy soft strongly $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-continuity, but the converse may not be true.
Citation: Ibtesam Alshammari, Islam M. Taha. On fuzzy soft $ \beta $-continuity and $ \beta $-irresoluteness: some new results[J]. AIMS Mathematics, 2024, 9(5): 11304-11319. doi: 10.3934/math.2024554

Related Papers:

Abstract

In this paper, we first introduced the concept of $ r $-fuzzy soft $ \beta $-closed sets in fuzzy soft topological spaces based on the sense of Šostak and investigated some properties of them. Also, we defined the closure and interior operators with respect to the classes of $ r $-fuzzy soft $ \beta $-closed and $ r $-fuzzy soft $ \beta $-open sets and studied some of their properties. Moreover, the concept of $ r $-fuzzy soft $ \beta $-connected sets was introduced and characterized with the help of fuzzy soft $ \beta $-closure operators. Thereafter, some properties of a fuzzy soft $ \beta $-continuity were studied. Also, we introduced and studied the concepts of fuzzy soft almost (weakly) $ \beta $-continuous functions, which are weaker forms of a fuzzy soft $ \beta $-continuity. The relationships between these classes of functions were specified with the help of some illustrative examples. Finally, we explored new types of fuzzy soft functions called fuzzy soft $ \beta $-irresolute (strongly $ \beta $-irresolute, $ \beta $-irresolute open, $ \beta $-irresolute closed, and $ \beta $-irresolute homeomorphism) functions and discussed some properties of them. Also, we showed that fuzzy soft strongly $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-irresolute $ \Rightarrow $ fuzzy soft $ \beta $-continuity, but the converse may not be true.

References

[1]	D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
[2]	M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006 doi: 10.1016/j.camwa.2011.02.006
[3]	I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 3 (2012), 171–185.
[4]	A. Aygunoglu, H. Aygun, Some notes on soft topological spaces, Neural Comput. Applic., 21 (2012), 113–119. https://doi.org/10.1007/s00521-011-0722-3 doi: 10.1007/s00521-011-0722-3
[5]	M. Terepeta, On separating axioms and similarity of soft topological spaces, Soft Comput., 23 (2019), 1049–1057. https://doi.org/10.1007/s00500-017-2824-z doi: 10.1007/s00500-017-2824-z
[6]	J. C. R. Alcantud, Soft open bases and a novel construction of soft topologies from bases for topologies, Mathematics, 8 (2020), 672. https://doi.org/10.3390/math8050672 doi: 10.3390/math8050672
[7]	Sk. Nazmul, S. K. Samanta, Neighbourhood properties of soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 6 (2013), 1–15.
[8]	H. L. Yang, X. Liao, S. G. Li, On soft continuous mappings and soft connectedness of soft topological spaces, Hacet. J. Math. Stat., 44 (2015), 385–398. https://doi.org/10.15672/HJMS.2015459876 doi: 10.15672/HJMS.2015459876
[9]	S. S. Thakur, A. S. Rajput, Connectedness between soft sets, New Math. Nat. Comput., 14 (2018), 53–71. https://doi.org/10.1142/S1793005718500059 doi: 10.1142/S1793005718500059
[10]	S. Hussain, B. Ahmad, Soft separation axioms in soft topological spaces, Hacet. J. Math. Stat., 44 (2015), 559–568. https://doi.org/10.15672/HJMS.2015449426 doi: 10.15672/HJMS.2015449426
[11]	T. M. Al-Shami, On soft separation axioms and their applications on decision-making problem, Math. Probl. Eng., 2021 (2021), 8876978. https://doi.org/10.1155/2021/8876978 doi: 10.1155/2021/8876978
[12]	M. Akdag, A. Ozkan, Soft $\alpha$-open sets and soft $\alpha$-continuous functions, Abstr. Appl. Anal., 2014 (2014), 891341. https://doi.org/10.1155/2014/891341 doi: 10.1155/2014/891341
[13]	M. Akdag, A. Ozkan, On soft $\beta$-open sets and soft $\beta$-continuous functions, Sci. World J., 2014 (2014), 843456. https://doi.org/10.1155/2014/843456 doi: 10.1155/2014/843456
[14]	S. A. El-Sheikh, R. A. Hosny, A. M. Abd El-latif, Characterizations of $\beta$-soft separation axioms in soft topological spaces, Information Sciences Letters, 4 (2015), 125–133.
[15]	T. M. Al-shami, Soft somewhere dense sets on soft topological spaces, Commun. Korean Math. Soc., 33 (2018), 1341–1356. https://doi.org/10.4134/CKMS.c170378 doi: 10.4134/CKMS.c170378
[16]	S. A. Ghour, Boolean algebra of soft Q-Sets in soft topological spaces, Appl. Comput. Intell. Soft Comput., 2022 (2022), 5200590. https://doi.org/10.1155/2022/5200590 doi: 10.1155/2022/5200590
[17]	T. M. Al-shami, A. Mhemdi, R. Abu-Gdairid, A novel framework for generalizations of soft open sets and its applications via soft topologies, Mathematics, 11 (2023), 840. https://doi.org/10.3390/math11040840 doi: 10.3390/math11040840
[18]	T. M. Al-shami, M. Arar, R. Abu-Gdairi, Z. A. Ameen, On weakly soft $\beta$-open sets and weakly soft $\beta$-continuity, J. Intell. Fuzzy Syst., 45 (2023), 6351–6363. https://doi.org/10.3233/JIFS-230858 doi: 10.3233/JIFS-230858
[19]	S. Kaur, T. M. Al-shami, A. Ozkan, M. Hosny, A new approach to soft continuity, Mathematics, 11 (2023), 3164. https://doi.org/10.3390/math11143164 doi: 10.3390/math11143164
[20]	S. A. Ghour, J. Al-Mufarrij, Between soft complete continuity and soft somewhat-continuity, Symmetry, 15 (2023), 2056. https://doi.org/10.3390/sym15112056 doi: 10.3390/sym15112056
[21]	Z. A. Ameen, R. Abu-Gdairi, T. M. Al-shami, B. A. Asaad, M. Arar, Further properties of soft somewhere dense continuous functions and soft Baire spaces, J. Math. Comput. Sci., 32 (2024), 54–63. http://dx.doi.org/10.22436/jmcs.032.01.05 doi: 10.22436/jmcs.032.01.05
[22]	P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9 (2001), 589–602.
[23]	L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[24]	A. P. Šostak, On a fuzzy topological structure, In: Proceedings of the 13th winter school on abstract analysis, Section of topology, Palermo: Circolo Matematico di Palermo, 1985, 89–103.
[25]	A. Aygünoǧlu, V. Çetkin, H. Aygün, An introduction to fuzzy soft topological spaces, Hacet. J. Math. Stat., 43 (2014), 193–208. https://doi.org/10.15672/HJMS.2015449418 doi: 10.15672/HJMS.2015449418
[26]	V. Çetkin, A. Aygünoǧlu, H. Aygün, On soft fuzzy closure and interior operators, Util. Math., 99 (2016), 341–367.
[27]	V. Çetkin, H. Aygün, Fuzzy soft semiregularization spaces, Ann. Fuzzy Math. Inform., 7 (2014), 687–697.
[28]	I. M. Taha, Compactness on fuzzy soft $r$-minimal spaces, Int. J. Fuzzy Log. Inte., 21 (2021), 251–258. https://doi.org/10.5391/IJFIS.2021.21.3.251 doi: 10.5391/IJFIS.2021.21.3.251
[29]	I. M. Taha, A new approach to separation and regularity axioms via fuzzy soft sets, Annals of Fuzzy Mathematics and Informatics, 20 (2020), 115–123. https://doi.org/10.30948/AFMI.2020.20.2.115 doi: 10.30948/AFMI.2020.20.2.115
[30]	I. M. Taha, Some new separation axioms in fuzzy soft topological spaces, Filomat, 35 (2021), 1775–1783. https://doi.org/10.2298/FIL2106775T doi: 10.2298/FIL2106775T
[31]	B. Ahmad, A. Kharal, On fuzzy soft sets, Adv. Fuzzy Syst., 2009 (2009), 586507. https://doi.org/10.1155/2009/586507 doi: 10.1155/2009/586507
[32]	N. Çaǧman, S. Enginoǧlu, F. Çitak, Fuzzy soft set theory and its applications, Iran. J. Fuzzy Syst., 8 (2011), 137–147. https://doi.org/10.22111/ijfs.2011.292 doi: 10.22111/ijfs.2011.292
[33]	I. M. Taha, Some new results on fuzzy soft $r$-minimal spaces, AIMS Mathematics, 7 (2022), 12458–12470. https://doi.org/10.3934/math.2022691 doi: 10.3934/math.2022691
[34]	S. Mishra, R. Srivastava, Hausdorff fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 9 (2015), 247–260.
[35]	S. Atmaca, I. Zorlutuna, On fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 5 (2013), 377–386.
[36]	I. Alshammari, M. H. Alqahtani, I. M. Taha, On $r$-fuzzy soft $\delta$-open sets and applications via fuzzy soft topologies, Preprints, 2023121240.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)