{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:35:53Z","timestamp":1760150153476,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T00:00:00Z","timestamp":1698278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In our ongoing study, we explore the concepts of I3-Cauchy and I3-Cauchy for triple sequences in the context of random 2-normed spaces (RTNS). Moreover, we introduce and analyze the notions of I3-convergence, I3-convergence, I3-limit points, and I3-cluster points for random 2-normed triple sequences. Significantly, we establish a notable finding that elucidates the connection between I3-convergence and I3-convergence within the framework of random 2-normed spaces, highlighting their interrelation. Additionally, we provide an illuminating example that demonstrates how I3-convergence in a random 2-normed space might not necessarily imply I3-convergence. Our observations underscore the importance of condition (AP3) when examining summability using ideals. Furthermore, we thoroughly investigate the relationship between the properties (AP) and (AP3), illustrating through an example how the latter represents a less strict condition compared to the former.<\/jats:p>","DOI":"10.3390\/sym15111983","type":"journal-article","created":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T03:33:51Z","timestamp":1698377631000},"page":"1983","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Regarding the Ideal Convergence of Triple Sequences in Random 2-Normed Spaces"],"prefix":"10.3390","volume":"15","author":[{"given":"Feras","family":"Bani-Ahmad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa 13133, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1073\/pnas.28.12.535","article-title":"Statistical metrics","volume":"28","author":"Menger","year":"1942","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_2","unstructured":"Schweizer, B., and Sklar, A. (2005). Probabilistical Metric Spaces, Dover Publications."},{"key":"ref_3","first-page":"280","article-title":"On the notion of a random normed space","volume":"149","author":"Serstnev","year":"1963","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1002\/mana.19630260109","article-title":"2-metrishe R\u00e4ume und ihr topologishe struktur","volume":"26","year":"1963","journal-title":"Math. Nachr."},{"key":"ref_5","first-page":"107","article-title":"The statistical convergence in 2-Banach spaces","volume":"2","author":"Pehlivan","year":"2004","journal-title":"Thai J. Math."},{"key":"ref_6","first-page":"349","article-title":"On I-Cauchy sequences in 2-normed spaces","volume":"11","year":"2008","journal-title":"Math. Inequal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"517841","DOI":"10.1155\/2011\/517841","article-title":"Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces","volume":"2011","author":"Sarabadan","year":"2011","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_8","first-page":"57","article-title":"I-Convergence and I-Cauchy Sequence of Functions in 2-Normed Spaces","volume":"6","author":"Arslan","year":"2018","journal-title":"Konuralp J. Math."},{"key":"ref_9","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_10","first-page":"149","article-title":"Ideal convergence in 2-fuzzy 2-normed spaces","volume":"46","author":"Rashid","year":"2017","journal-title":"Hacettepe J. Math. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1006\/jmaa.1997.5333","article-title":"Continuity properties of probabilistic norms","volume":"208","author":"Alsina","year":"1997","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","first-page":"95","article-title":"On probabilistic 2-normed spaces","volume":"3","author":"Golet","year":"2006","journal-title":"Novi Sad J. Math."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1080\/00029890.1959.11989303","article-title":"The integrability of certain functions and related summability methods","volume":"66","author":"Schoenberg","year":"1959","journal-title":"Am. Math. Mon."},{"key":"ref_15","first-page":"49","article-title":"Triple sequences and their statistical convergence","volume":"8","author":"Sahiner","year":"2007","journal-title":"Selcuk J. Appl. Math."},{"key":"ref_16","first-page":"9","article-title":"Some i-related properties of triple sequences","volume":"9","author":"Sahiner","year":"2008","journal-title":"Selcuk J. Appl. Math."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","unstructured":"Altaweel, N.H., Rashid, M.H.M., Albalawi, O., Alshehri, M.G., Eljaneid, N.H.E., and Albalawi, R. (2023). On the Ideal Convergent Sequences in Fuzzy Normed Space. Symmetry, 15.","DOI":"10.3390\/sym15040936"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1186\/1687-1847-2012-149","article-title":"Ideal convergence of double sequences in random 2-normed spaces","volume":"2012","author":"Mohiuddine","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"933","DOI":"10.2478\/s12175-011-0059-5","article-title":"On I-convergence in random 2-normed spaces","volume":"61","author":"Mursaleen","year":"2011","journal-title":"Math. Slovaca"},{"key":"ref_21","first-page":"359","article-title":"On ideal convergence of double sequences in probabilistic normed spaces","volume":"12","author":"Mursaleen","year":"2010","journal-title":"Math. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2478\/s12175-011-0071-9","article-title":"On ideal convergence in probabilistic normed spaces","volume":"62","author":"Mursaleen","year":"2012","journal-title":"Math. Slovaca"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Iqbal, M., Malik, M.G.A., Bashir, Y., and Bashir, Z. (2019). The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces. Symmetry, 11.","DOI":"10.3390\/sym11080971"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, M., Cai, G., Bi, S., and Zhang, X. (2023). Improved TV Image Denoising over Inverse Gradient. Symmetry, 15.","DOI":"10.3390\/sym15030678"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Papakonstantinou, P., and Hyun, C.H. (2023). Energy-Density Modeling of Strongly Interacting Matter: Atomic Nuclei and Dense Stars. Symmetry, 15.","DOI":"10.3390\/sym15030683"},{"key":"ref_26","first-page":"549","article-title":"On I-convergent double sequences","volume":"31","author":"Tripathy","year":"2005","journal-title":"Soochow J. Math."},{"key":"ref_27","first-page":"274","article-title":"On some properties of I-convergence","volume":"28","author":"Salat","year":"2004","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_28","first-page":"e01101","article-title":"Wijsman and Wijsman regularly triple ideal convergence sequences of sets","volume":"15","author":"Granados","year":"2022","journal-title":"Sci. Afr."},{"key":"ref_29","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_30","first-page":"326","article-title":"Fuzzy metric and statistical metric spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetika"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/1983\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:12:35Z","timestamp":1760130755000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/1983"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,26]]},"references-count":30,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["sym15111983"],"URL":"https:\/\/doi.org\/10.3390\/sym15111983","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,10,26]]}}}