{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T10:44:37Z","timestamp":1782384277764,"version":"3.54.5"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T00:00:00Z","timestamp":1681862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009391","name":"University of Tabuk, Saudi Arabia","doi-asserted-by":"publisher","award":["S-0239-1443"],"award-info":[{"award-number":["S-0239-1443"]}],"id":[{"id":"10.13039\/100009391","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.<\/jats:p>","DOI":"10.3390\/sym15040936","type":"journal-article","created":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T03:25:11Z","timestamp":1681961111000},"page":"936","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On the Ideal Convergent Sequences in Fuzzy Normed Space"],"prefix":"10.3390","volume":"15","author":[{"given":"Nifeen H.","family":"Altaweel","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3816-5287","authenticated-orcid":false,"given":"Mohammad H. M.","family":"Rashid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mu\u2019tah University, P.O. Box 7, Al-Karak 61710, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olayan","family":"Albalawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maryam G.","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nidal H. E.","family":"Eljaneid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Razan","family":"Albalawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"241","DOI":"10.4064\/cm-2-3-4-241-244","article-title":"Sur la convergence statistique","volume":"2","author":"Fast","year":"1951","journal-title":"Colloq. Math."},{"key":"ref_2","first-page":"73","article-title":"Sur la convergence ordinaire et la convergence asymptotique","volume":"2","author":"Steinhaus","year":"1951","journal-title":"Colloq. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.topol.2008.01.015","article-title":"Statistical convergence in topology","volume":"156","year":"2008","journal-title":"Topol. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1524\/anly.1985.5.4.301","article-title":"On statistical convergence","volume":"5","author":"Fridy","year":"1985","journal-title":"Analysis"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.jmaa.2006.05.040","article-title":"Statistical convergence and ideal convergence for sequences of functions","volume":"328","author":"Balcerzak","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1155\/2011\/420419","article-title":"Statistical convergence in function spaces","volume":"2011","author":"Caserta","year":"2011","journal-title":"Abstr. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"185","DOI":"10.4064\/fm-66-2-185-193","article-title":"A new kind of compactness for topological spaces","volume":"66","author":"Bernstein","year":"1970","journal-title":"Fund. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"669","DOI":"10.2307\/44154069","article-title":"I-Convergence","volume":"26","author":"Kostyrko","year":"2000","journal-title":"Real Anal. Exch."},{"key":"ref_9","first-page":"443","article-title":"I-Convergence and Extremal I-Limit Points","volume":"55","author":"Kostyrko","year":"2005","journal-title":"Math. Slovaca."},{"key":"ref_10","first-page":"97","article-title":"On ideal convergence of double sequences in the topology induced by a fuzzy 2-norm","volume":"8","author":"Rashid","year":"2017","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_11","first-page":"149","article-title":"Ideal convergence in 2-fuzzy 2-normed spaces","volume":"46","author":"Rashid","year":"2017","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0165-0114(84)90034-4","article-title":"Fuzzy topological vector spaces","volume":"12","author":"Katsaras","year":"1984","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0165-0114(92)90338-5","article-title":"Finite dimensional fuzzy normed linear spaces","volume":"48","author":"Felbin","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0165-0114(84)90069-1","article-title":"On fuzzy metric spaces","volume":"12","author":"Kaleva","year":"1984","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","first-page":"429","article-title":"Fuzzy linear operators and fuzzy normed linear spaces","volume":"86","author":"Cheng","year":"1994","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_16","first-page":"687","article-title":"Finite dimensional fuzzy normed linear spaces","volume":"11","author":"Bag","year":"2003","journal-title":"J. Fuzzy Math."},{"key":"ref_17","first-page":"326","article-title":"Fuzzy Metric and Statistical Metric Spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetika"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.fss.2004.05.004","article-title":"Fuzzy bounded linear operators","volume":"151","author":"Bag","year":"2005","journal-title":"Fuzzy Sets Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2910","DOI":"10.1016\/j.ins.2005.07.013","article-title":"Fixed point theorems on fuzzy normed linear spaces","volume":"176","author":"Bag","year":"2006","journal-title":"Inf. Sci."},{"key":"ref_20","first-page":"274","article-title":"On some properties of I\u2013convergence","volume":"28","author":"Salat","year":"2004","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.aml.2010.08.007","article-title":"Further investigations of ideal summability in 2-normed spaces","volume":"24","author":"Das","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_22","unstructured":"Schweizer, B., and Sklar, A. (2005). Probabilistical Metric Spaces, Dover Publications."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.14321\/realanalexch.30.1.0123","article-title":"On I-Cauchy sequences","volume":"30","author":"Dems","year":"2004","journal-title":"Real Anal. Exch."},{"key":"ref_24","first-page":"569","article-title":"On I-Canchy sequences","volume":"11","author":"Nabin","year":"2007","journal-title":"Taiwan J. Math."},{"key":"ref_25","unstructured":"Kelley, J.L. (1955). General Topology, Springer."},{"key":"ref_26","first-page":"431","article-title":"I and I*-Convergence of Nets","volume":"33","author":"Das","year":"2007","journal-title":"Real Anal. Exch."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"191","DOI":"10.21136\/MB.2001.133923","article-title":"Statistical convergence of subsequences of a given sequence","volume":"126","author":"Macaj","year":"2001","journal-title":"Math. Bohem."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2597","DOI":"10.1016\/j.camwa.2010.01.027","article-title":"Some further results on I-Cauchy sequences and condition (AP)","volume":"59","author":"Das","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.fss.2022.07.008","article-title":"Parallel fuzzy relation matrix factorization towards algebraic formulation, universal approximation and interpretability of MIMO hierarchical fuzzy systems","volume":"450","author":"Sun","year":"2022","journal-title":"Fuzzy Sets Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/936\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:19:02Z","timestamp":1760123942000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/936"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,19]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040936"],"URL":"https:\/\/doi.org\/10.3390\/sym15040936","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,19]]}}}