In this paper, we first introduced and studied a new class of fuzzy open sets, called $ j $-fuzzy $ \gamma\mathcal{I} $-open ($ j $-F$ \gamma\mathcal{I} $-open) sets on fuzzy ideal topological spaces ($ \mathcal{FITS}s $). The class of $ j $-F$ \gamma\mathcal{I} $-open sets is contained in the class of $ j $-fuzzy strong $ \beta $-$ \mathcal{I} $-open ($ j $-FS$ \beta\mathcal{I} $-open) sets and contains all $ j $-fuzzy pre-$ \mathcal{I} $-open ($ j $-FP$ \mathcal{I} $-open) sets and $ j $-fuzzy semi-$ \mathcal{I} $-open ($ j $-FS$ \mathcal{I} $-open) sets. We also defined and investigated the closure and interior operators with respect to the classes of $ j $-F$ \gamma\mathcal{I} $-closed sets and $ j $-F$ \gamma\mathcal{I} $-open sets. However, we explored and discussed novel types of fuzzy $ \mathcal{I} $-separation axioms using $ j $-F$ \gamma\mathcal{I} $-closed sets, called $ j $-F$ \gamma\mathcal{I} $-regular spaces and $ j $-F$ \gamma\mathcal{I} $-normal spaces. Thereafter, we displayed and investigated the concept of fuzzy $ \gamma\mathcal{I} $-continuity (F$ \gamma\mathcal{I} $-continuity) using $ j $-F$ \gamma\mathcal{I} $-open sets. Also, we presented and characterized the concepts of fuzzy weak $ \gamma\mathcal{I} $-continuity (FW$ \gamma\mathcal{I} $-continuity) and fuzzy almost $ \gamma\mathcal{I} $-continuity (FA$ \gamma\mathcal{I} $-continuity), which are weaker forms of F$ \gamma\mathcal{I} $-continuity. Moreover, we showed that F$ \gamma $$ \mathcal{I} $-continuity $ \Longrightarrow $ FA$ \gamma $$ \mathcal{I} $-continuity $ \Longrightarrow $ FW$ \gamma $$ \mathcal{I} $-continuity, but the converse may not be true. Finally, we defined and studied some new fuzzy $ \gamma\mathcal{I} $-mappings via $ j $-F$ \gamma\mathcal{I} $-open sets and $ j $-F$ \gamma\mathcal{I} $-closed sets, called F$ \gamma\mathcal{I} $-open mappings, F$ \gamma\mathcal{I} $-closed mappings, F$ \gamma\mathcal{I} $-irresolute mappings, F$ \gamma\mathcal{I} $-irresolute open mappings, and F$ \gamma\mathcal{I} $-irresolute closed mappings. The relationships between these classes of mappings were discussed with the help of some examples.
Citation: Fahad Alsharari, Jawaher Al-Mufarrij, Islam M. Taha. On $ j $-fuzzy $ \gamma\mathcal{I} $-open sets with some applications[J]. AIMS Mathematics, 2025, 10(9): 20550-20570. doi: 10.3934/math.2025917
In this paper, we first introduced and studied a new class of fuzzy open sets, called $ j $-fuzzy $ \gamma\mathcal{I} $-open ($ j $-F$ \gamma\mathcal{I} $-open) sets on fuzzy ideal topological spaces ($ \mathcal{FITS}s $). The class of $ j $-F$ \gamma\mathcal{I} $-open sets is contained in the class of $ j $-fuzzy strong $ \beta $-$ \mathcal{I} $-open ($ j $-FS$ \beta\mathcal{I} $-open) sets and contains all $ j $-fuzzy pre-$ \mathcal{I} $-open ($ j $-FP$ \mathcal{I} $-open) sets and $ j $-fuzzy semi-$ \mathcal{I} $-open ($ j $-FS$ \mathcal{I} $-open) sets. We also defined and investigated the closure and interior operators with respect to the classes of $ j $-F$ \gamma\mathcal{I} $-closed sets and $ j $-F$ \gamma\mathcal{I} $-open sets. However, we explored and discussed novel types of fuzzy $ \mathcal{I} $-separation axioms using $ j $-F$ \gamma\mathcal{I} $-closed sets, called $ j $-F$ \gamma\mathcal{I} $-regular spaces and $ j $-F$ \gamma\mathcal{I} $-normal spaces. Thereafter, we displayed and investigated the concept of fuzzy $ \gamma\mathcal{I} $-continuity (F$ \gamma\mathcal{I} $-continuity) using $ j $-F$ \gamma\mathcal{I} $-open sets. Also, we presented and characterized the concepts of fuzzy weak $ \gamma\mathcal{I} $-continuity (FW$ \gamma\mathcal{I} $-continuity) and fuzzy almost $ \gamma\mathcal{I} $-continuity (FA$ \gamma\mathcal{I} $-continuity), which are weaker forms of F$ \gamma\mathcal{I} $-continuity. Moreover, we showed that F$ \gamma $$ \mathcal{I} $-continuity $ \Longrightarrow $ FA$ \gamma $$ \mathcal{I} $-continuity $ \Longrightarrow $ FW$ \gamma $$ \mathcal{I} $-continuity, but the converse may not be true. Finally, we defined and studied some new fuzzy $ \gamma\mathcal{I} $-mappings via $ j $-F$ \gamma\mathcal{I} $-open sets and $ j $-F$ \gamma\mathcal{I} $-closed sets, called F$ \gamma\mathcal{I} $-open mappings, F$ \gamma\mathcal{I} $-closed mappings, F$ \gamma\mathcal{I} $-irresolute mappings, F$ \gamma\mathcal{I} $-irresolute open mappings, and F$ \gamma\mathcal{I} $-irresolute closed mappings. The relationships between these classes of mappings were discussed with the help of some examples.
| [1] | L. A. Zadeh, Fuzzy sets, Information and control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X |
| [2] | H.-J. Zimmermann, Fuzzy set theory — and its applications, 2 Eds., Dordrecht: Springer, 1991. https://doi.org/10.1007/978-94-015-7949-0 |
| [3] |
Y. Gupta, A. Saini, A. K. Saxena, A new fuzzy logic based ranking function for efficient information retrieval system, Expert Syst. Appl., 42 (2015), 1223–1234. https://doi.org/10.1016/j.eswa.2014.09.009 doi: 10.1016/j.eswa.2014.09.009
|
| [4] | R. R. Yager, L. A. Zadeh, An introduction to fuzzy logic applications in intelligent systems, New York: Springer, 1992. https://doi.org/10.1007/978-1-4615-3640-6 |
| [5] | H.-C. Wu, Mathematical foundations of fuzzy sets, John Wiley & Sons, 2023. https://doi.org/10.1002/9781119981558 |
| [6] | I. Bloch, A. Ralescu, Fuzzy sets methods in image processing and understanding, Cham: Springer, 2023. https://doi.org/10.1007/978-3-031-19425-2 |
| [7] |
H. Wu, Z. Xu, Fuzzy logic in decision support: methods, applications and future trends, International Journal of Computers Communications and Control, 16 (2021), 4044. https://doi.org/10.15837/ijccc.2021.1.4044 doi: 10.15837/ijccc.2021.1.4044
|
| [8] |
A. Aleksic, D. Tadic, Industrial and management applications of type-2 multi-attribute decision-making techniques extended with type-2 fuzzy sets from 2013 to 2022, Mathematics, 11 (2023), 2249. https://doi.org/10.3390/math11102249 doi: 10.3390/math11102249
|
| [9] |
J. Kannan, V. Jayakumar, N. Kausar, D. Pamucar, V. Simic, Enhancing decision-making with linear diophantine multi-fuzzy set: application of novel information measures in medical and engineering fields, Sci. Rep., 14 (2024), 28537. https://doi.org/10.1038/s41598-024-79725-0 doi: 10.1038/s41598-024-79725-0
|
| [10] | B. Ahmad, A. Kharal, On fuzzy soft sets, Adv. Fuzzy Syst., 2009 (2009), 586507. https://doi.org/10.1155/2009/586507 |
| [11] |
N. Cagman, S. Enginoglu, F. Citak, Fuzzy soft set theory and its application, Iran. J. Fuzzy Syst., 8 (2011), 137–147. https://doi.org/10.22111/ijfs.2011.292 doi: 10.22111/ijfs.2011.292
|
| [12] |
M. Atef, M. I. Ali, T. M. Al-shami, Fuzzy soft covering based multi-granulation fuzzy rough sets and their applications, Comput. Appl. Math., 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x doi: 10.1007/s40314-021-01501-x
|
| [13] | C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–190. https://doi.org/10.1016/0022-247X(68)90057-7 |
| [14] |
S. Saleh, T. M. Al-shami, A. A. Azzam, M. Hosny, Stronger forms of fuzzy pre-separation and regularity axioms via fuzzy topology, Mathematics, 11 (2023), 4801. https://doi.org/10.3390/math11234801 doi: 10.3390/math11234801
|
| [15] |
S. A. M. Saleh, F. M. Birkea, T. Al-shami, M. Arar, M. Omran, Novel categories of spaces in the frame of generalized fuzzy topologies via fuzzy g$\mu$-closed sets, Eur. J. Pure Appl. Math., 18 (2025), 5856. https://doi.org/10.29020/nybg.ejpam.v18i1.5856 doi: 10.29020/nybg.ejpam.v18i1.5856
|
| [16] | A. P. Šostak, On a fuzzy topological structure, In: Proceedings of the 13th winter school on abstract analysis, Palermo: Circolo Matematico di Palermo, Srni, Czech Republic, 27 January–3 February, 1985, 89–103. |
| [17] | A. A. Ramadan, Smooth topological spaces, Fuzzy Set. Syst., 48 (1992), 371–375. https://doi.org/10.1016/0165-0114(92)90352-5 |
| [18] |
K. C. Chattopadhyay, S. K. Samanta, Fuzzy topology: fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy Set. Syst., 54 (1993), 207–212. https://doi.org/10.1016/0165-0114(93)90277-O doi: 10.1016/0165-0114(93)90277-O
|
| [19] |
M. K. El-Gayyar, E. E. Kerre, A. A. Ramadan, Almost compactness and near compactness in smooth topological spaces, Fuzzy Set. Syst., 62 (1994), 193–202. https://doi.org/10.1016/0165-0114(94)90059-0 doi: 10.1016/0165-0114(94)90059-0
|
| [20] |
U. Höhle, A. P. Šostak, A general theory of fuzzy topological spaces, Fuzzy Set. Syst., 73 (1995), 131–149. https://doi.org/10.1016/0165-0114(94)00368-H doi: 10.1016/0165-0114(94)00368-H
|
| [21] | Y. C. Kim, A. A. Ramadan, S. E. Abbas, Weaker forms of continuity in Šostak's fuzzy topology, Indian J. Pure Appl. Math., 34 (2003), 311–333. |
| [22] |
S. E. Abbas, Fuzzy $\beta$-irresolute functions, Appl. Math. Comput., 157 (2004), 369–380. https://doi.org/10.1016/j.amc.2003.08.040 doi: 10.1016/j.amc.2003.08.040
|
| [23] | Y. C. Kim, S. E. Abbas, On several types of $R$-fuzzy compactness, J. Fuzzy Math., 12 (2004), 827–844. |
| [24] |
F. Alsharari, H. Y. Saleh, I. M. Taha, Some characterizations of $k$-fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity with further selected topics, Symmetry, 17 (2025), 678. https://doi.org/10.3390/sym17050678 doi: 10.3390/sym17050678
|
| [25] |
I. M. Taha, S. E. Abbas, A new notion of fuzzy local function and some applications, Adv. Fuzzy Sys., 2022 (2022), 8954163. https://doi.org/10.1155/2022/8954163 doi: 10.1155/2022/8954163
|
| [26] |
I. M. Taha, On $r$-fuzzy $\ell$-open sets and continuity of fuzzy multifunctions via fuzzy Ideals, J. Math. Comput. Sci., 10 (2020), 2613–2633. https://doi.org/10.28919/jmcs/4944 doi: 10.28919/jmcs/4944
|
| [27] |
I. M. Taha, On $r$-generalized fuzzy $\ell$-closed sets: properties and applications, J. Math., 2021 (2021), 4483481. https://doi.org/10.1155/2021/4483481 doi: 10.1155/2021/4483481
|
| [28] |
I. M. Taha, $r$-fuzzy $\delta$-$\ell$-open sets and fuzzy upper (lower) $\delta$-$\ell$-continuity via fuzzy idealization, J. Math. Comput. Sci., 25 (2022), 1–9. https://doi.org/10.22436/jmcs.025.01.01 doi: 10.22436/jmcs.025.01.01
|
| [29] | A. A. Ramadan, Y. C. Kim, M. K. Ei-Gayyar, Smooth ideals, Journal of the Korean Institute of Intelligent Systems, 12 (2002), 90–95. https://doi.org/10.5391/JKIIS.2002.12.1.090 |
| [30] |
A. Kandil, M. E. El-Shafei, Regularity axioms in fuzzy topological spaces and FRi-proximities, Fuzzy Set. Syst., 27 (1988), 217–231. https://doi.org/10.1016/0165-0114(88)90151-0 doi: 10.1016/0165-0114(88)90151-0
|
| [31] |
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
|
| [32] |
I. M. Taha, Compactness on fuzzy soft $r$-minimal spaces, Int. J. Fuzzy Log. Intell. Syst., 21 (2021), 251–258. https://doi.org/10.5391/IJFIS.2021.21.3.251 doi: 10.5391/IJFIS.2021.21.3.251
|
| [33] |
I. M. Taha, Some properties of $(r, s)$-generalized fuzzy semi-closed sets and some applications, J. Math. Comput. Sci., 27 (2022), 164–175. https://doi.org/10.22436/jmcs.027.02.06 doi: 10.22436/jmcs.027.02.06
|
Fahad Alsharari, Jawaher Al-Mufarrij, Islam M. Taha. On $ j $-fuzzy $ \gamma\mathcal{I} $-open sets with some applications[J]. AIMS Mathematics, 2025, 10(9): 20550-20570. doi: 10.3934/math.2025917
DownLoad: