On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers

Şükrü Tortop; Funda Türegün

doi:10.33401/fujma.1952238

On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers

Abstract

This paper is devoted to the investigation of ideal-type level-wise convergence for sequences of fuzzy numbers. By using $\alpha$-level sets, we define level-wise $\mathcal I$-convergence, uniformly level-wise $\mathcal I$-convergence, and almost everywhere level-wise $\mathcal I$-convergence. A characterization of level-wise $\mathcal I$-convergence is obtained via the endpoint functions of the $\alpha$-cuts, and uniqueness of the limit is established. It is shown that uniformly level-wise $\mathcal I$-convergence implies level-wise $\mathcal I$-convergence, whereas the converse fails in general. Furthermore, we prove that level-wise $\mathcal I$-convergence yields almost everywhere level-wise $\mathcal I$-convergence, whereas the converse implication does not hold in general. Several examples are included to illustrate the strictness of these implications.

Keywords

Supporting Institution

This research received no external funding.

Ethical Statement

This article does not contain any studies with human or animal subjects. It is declared that during the preparation process of this study, scientific and ethical principles were followed and all the studies benefited from are stated in the bibliography.

Thanks

The author would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

[1] L.A. Zadeh, Fuzzy sets, Inf. Control, 8(3) (1965), 338–353. $ \href{https://doi.org/10.1016/S0019-9958(65)90241-X}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/34248666540?origin=resultslist}{\mbox{[Scopus]}} $
[2] J.X. Fang and H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy Sets Syst., 147(3) (2004), 417–435. $ \href{https://doi.org/10.1016/j.fss.2003.08.001}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/4544240400?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000224286200005}{\mbox{[Web of Science]}} $
[3] B. Arslan, S. Jalali and S. Enginoğlu, Statistical Riesz and Nörlund convergence for sequences of fuzzy numbers, J. Amasya Univ. Inst. Sci. Tech.,4(2) (2023), 100–107. $ \href{https://doi.org/10.54559/jauist.1400386}{\mbox{[CrossRef]}} $
[4] Tanushri, A. Ahmad and A. Eşi, A framework for $\mathcal I^{*}$-statistical convergence of fuzzy numbers, Axioms, 13(9) (2024), 639. $ \href{https://doi.org/10.3390/axioms13090639}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001326346700001}{\mbox{[Web of Science]}} $
[5] F. Babaarslan and M.E. Çayar, Rough ideal convergence of double sequences of fuzzy numbers, Bozok J. Sci., 2(1), 30-37. $ \href{https://izlik.org/JA44CC28EP}{\mbox{[Web]}} $
[6] E. Nuray Yıldırım and F. Nuray, Cesaro summable and statistically convergent sequences in Polya’s sense, Fundam. J. Math. Appl., 8(4) (2025), 203–211. $ \href{https://doi.org/10.33401/fujma.1649410}{\mbox{[CrossRef]}} $
[7] İ. Osmanoğlu and E. Dündar, A note on statistical continuity of functions, Fundam. J. Math. Appl., 7(4) (2024), 212–217. $ \href{https://doi.org/10.33401/fujma.1443574}{\mbox{[CrossRef]}} $
[8] Ö. Kişi, On $I_{\theta}$-convergence in neutrosophic normed spaces, Fundam. J. Math. Appl., 4(2) (2021), 67–76. $ \href{https://doi.org/10.33401/fujma.873029}{\mbox{[CrossRef]}} $

[9] P. Kostyrko, T. Salat and W. Wilczynski, $\mathcal I$-convergence, Real Anal. Exch., 26(2) (2000/2001), 669–686. $\href{https://bishtref.com/articles/10.2307/44154069}{\mbox{[Web]}} $
[10] P. Kostyrko, M. Macaj, T. Salat and M. Sleziak, $\mathcal I$-convergence and extremal $\mathcal I$-limit points, Math. Slovaca, 55(4) (2005), 443–464. $ \href{https://dml.cz/bitstream/handle/10338.dmlcz/132906/MathSlov_55-2005-4_6.pdf}{\mbox{[Web]}} $
[11] K. Demirci, $\mathcal I$-limit superior and $\mathcal I$-limit inferior, Math. Commun., 6 (2001), 165–172. $\href{https://hrcak.srce.hr/clanak/1589}{\mbox{[Web]}} $
[12] V. Kumar and K. Kumar, On the ideal convergence of sequences of fuzzy numbers, Inf. Sci., 178(24) (2008), 4670–4678. $ \href{https://doi.org/10.1016/j.ins.2008.08.013}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/54049145834?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000261445800006}{\mbox{[Web of Science]}} $
[13] B. Hazarika, Pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy-valued functions, J. Intell. Fuzzy Syst., 32(3) (2017), 2665–2677. $ \href{https://doi.org/10.3233/JIFS-16662}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85014281050?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000395904400082}{\mbox{[Web of Science]}} $
[14] U. Ulusu, F. Nuray and E. Dündar, Lacunary $\mathcal I$-invariant convergence, Cumhuriyet Sci. J., 41(3) (2020), 617–624. $ \href{https://doi.org/10.17776/csj.689877}{\mbox{[CrossRef]}} $
[15] U. Ulusu and F. Nuray, $\mathcal I$-limit and $\mathcal I$-cluster points for functions defined on amenable semigroups, Fundam. J. Math. Appl., 4(2) (2021), 45–48. $ \href{https://doi.org/10.33401/fujma.842104}{\mbox{[CrossRef]}} $
[16] Ş. Tortop, Ideal epi-convergence of sequences of functions, Filomat, 38(4) (2024), 1357–1366. $ \href{https://doi.org/10.2298/FIL2404357T}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85179339439?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001122661100001}{\mbox{[Web of Science]}} $
[17] U. Yamancı, E. Savaş and M. Gürdal, $\mathcal I$-localized sequence in two normed spaces, Malays. J. Math. Sci., 14(3) (2020), 491–503. $ \href{https://www.scopus.com/pages/publications/85100151619?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000610402800010}{\mbox{[Web of Science]}} $
[18] E. Dündar, U. Ulusu and F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inform, 27(2) (2018), 161–169. $ \href{https://doi.org/10.37193/CMI.2018.02.08}{\mbox{[CrossRef]}} $
[19] E. Gülle, E. Dündar and U. Ulusu, Ideal convergence in partial metric spaces, Soft Comput., 27(19) (2023), 13789–13795. $ \href{https://doi.org/10.1007/s00500-023-08994-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85165903443?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001038581900002}{\mbox{[Web of Science]}} $
[20] S. Aytar and S. Pehlivan, Levelwise statistical cluster and limit points of a sequence of fuzzy numbers, Int. J. Gen. Syst, 37(5) (2008), 573–584. $ \href{https://doi.org/10.1080/03081070701607342}{\mbox{[CrossRef]}} $
[21] S. Aytar, A characterization theorem for levelwise statistical convergence, Filomat, 25(1) (2011), 133–143. $\href{https://doi.org/10.2298/FIL1101133A}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/79960789645?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000288755700010}{\mbox{[Web of Science]}} $
[22] Ö. Talo, On the almost everywhere statistical convergence of sequences of fuzzy numbers, Filomat, 33(9) (2019), 2683–2693. $ \href{https://doi.org/10.2298/FIL1909683T}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85079439412?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000499102700012}{\mbox{[Web of Science]}} $

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Şükrü Tortop ^*
0000-0001-5342-7612
Türkiye

Funda Türegün This is me
0009-0008-6552-8945
Türkiye

Publication Date

September 30, 2026

Submission Date

May 15, 2026

Acceptance Date

September 9, 2026

Published in Issue

Year 2026 Volume: 9 Number: 3

DOI

https://doi.org/10.33401/fujma.1952238

IZ

https://izlik.org/JA58FR74KE

Cite

RIS / Bibtex

APA

Tortop, Ş., & Türegün, F. (2026). On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers. Fundamental Journal of Mathematics and Applications, 9(3), 152-162. https://doi.org/10.33401/fujma.1952238

AMA

1.Tortop Ş, Türegün F. On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers. Fundam. J. Math. Appl. 2026;9(3):152-162. doi:10.33401/fujma.1952238

Chicago

Tortop, Şükrü, and Funda Türegün. 2026. “On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers”. Fundamental Journal of Mathematics and Applications 9 (3): 152-62. https://doi.org/10.33401/fujma.1952238.

EndNote

Tortop Ş, Türegün F (September 1, 2026) On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers. Fundamental Journal of Mathematics and Applications 9 3 152–162.

IEEE

[1]Ş. Tortop and F. Türegün, “On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers”, Fundam. J. Math. Appl., vol. 9, no. 3, pp. 152–162, Sept. 2026, doi: 10.33401/fujma.1952238.

ISNAD

Tortop, Şükrü - Türegün, Funda. “On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers”. Fundamental Journal of Mathematics and Applications 9/3 (September 1, 2026): 152-162. https://doi.org/10.33401/fujma.1952238.

JAMA

1.Tortop Ş, Türegün F. On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers. Fundam. J. Math. Appl. 2026;9:152–162.

MLA

Tortop, Şükrü, and Funda Türegün. “On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers”. Fundamental Journal of Mathematics and Applications, vol. 9, no. 3, Sept. 2026, pp. 152-6, doi:10.33401/fujma.1952238.

Vancouver

1.Şükrü Tortop, Funda Türegün. On Level-Wise Ideal Convergence of Sequences of Fuzzy Numbers. Fundam. J. Math. Appl. 2026 Sep. 1;9(3):152-6. doi:10.33401/fujma.1952238