{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,15]],"date-time":"2025-03-15T04:07:37Z","timestamp":1742011657672,"version":"3.38.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:00:00Z","timestamp":1736467200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:00:00Z","timestamp":1736467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12301595"],"award-info":[{"award-number":["12301595"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1007\/s40314-024-03052-3","type":"journal-article","created":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T11:48:01Z","timestamp":1736509681000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The degree of reflexivity of fuzzy relations in terms of left-continuous t-norms"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1946-4182","authenticated-orcid":false,"given":"Yexing","family":"Dan","sequence":"first","affiliation":[]},{"given":"Xiaodong","family":"Pan","sequence":"additional","affiliation":[]},{"given":"Junsheng","family":"Qiao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,10]]},"reference":[{"key":"3052_CR1","volume-title":"Fuzzy Implications, Studies in Fuzziness and Soft Computing","author":"M Baczy\u0144ski","year":"2008","unstructured":"Baczy\u0144ski M, Jayaram B (2008) Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol 231. Springer, Berlin, Heidelberg"},{"key":"3052_CR2","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1016\/j.fss.2007.10.012","volume":"159","author":"L B\u011bhounek","year":"2008","unstructured":"B\u011bhounek L, Bodenhofer U, Cintula P (2008) Relations in fuzzy class theory: initial steps. Fuzzy Sets and Systems 159:1729\u20131772","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR3","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.fss.2004.12.010","volume":"154","author":"L B\u011bhounek","year":"2005","unstructured":"B\u011bhounek L, Cintula P (2005) Fuzzy class theory. Fuzzy Sets and Systems 154:34\u201355","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0633-1","volume-title":"Fuzzy Relational Systems","author":"R B\u011blohl\u00e1vek","year":"2002","unstructured":"B\u011blohl\u00e1vek R (2002) Fuzzy Relational Systems. Kluwer America\/Plenum Publishers, New York, Foundations and Principles"},{"key":"3052_CR5","doi-asserted-by":"crossref","unstructured":"Boixader D, Recasens J (2009) Approximate fuzzy preorders and equivalences, IEEE International Conference on Fuzzy Systems, pp.\u00a0564\u2013568, Korea","DOI":"10.1109\/FUZZY.2009.5277247"},{"key":"3052_CR6","doi-asserted-by":"crossref","unstructured":"Boixader D, Recasens J (2010) Approximate fuzzy preorders and equivalences, A similarity based approach, IEEE International Conference on Fuzzy Systems, pp.\u00a0788\u2013793, Spain","DOI":"10.1109\/FUZZY.2010.5584180"},{"key":"3052_CR7","doi-asserted-by":"crossref","unstructured":"Boixader D, Recasens J (2010) Introducing strong $$T$$-transitivity in approximate fuzzy preorders and equivalences, Annual Meeting of the North American Fuzzy Information Processing Society, pp.\u00a01\u20135, Canada","DOI":"10.1109\/NAFIPS.2010.5548261"},{"key":"3052_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.fss.2018.05.020","volume":"360","author":"Y Dan","year":"2019","unstructured":"Dan Y, Hu BQ, Qiao J (2019) Some results on the degree of symmetry of fuzzy relations. Fuzzy Sets and Systems 360:1\u201332","journal-title":"Fuzzy Sets and Systems"},{"issue":"7","key":"3052_CR9","first-page":"15","volume":"13","author":"B De Baets","year":"2016","unstructured":"De Baets B, Bouremel H, Zedam L (2016) On the compatibility of a crisp relation with a fuzzy equivalence relation. Iranian Journal of Fuzzy Systems 13(7):15\u201331","journal-title":"Iranian Journal of Fuzzy Systems"},{"key":"3052_CR10","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/0165-0114(95)00142-5","volume":"79","author":"B De Baets","year":"1996","unstructured":"De Baets B, Van de Walle B (1996) Weak and strong fuzzy interval orders. Fuzzy Sets and Systems 79:213\u2013225","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-1648-2","volume-title":"Fuzzy Preference Modelling and Multicriteria Decision Support","author":"JC Fodor","year":"1994","unstructured":"Fodor JC, Roubens M (1994) Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht"},{"key":"3052_CR12","volume-title":"Fuzzy Choice Functions: A Revealed Preference Approach","author":"I Georgescu","year":"2007","unstructured":"Georgescu I (2007) Fuzzy Choice Functions: A Revealed Preference Approach. Springer Publishing Company, Incorporated"},{"key":"3052_CR13","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1142\/S1793005712400133","volume":"8","author":"I Georgescu","year":"2012","unstructured":"Georgescu I (2012) New revealed preference indicators of fuzzy choice functions. New Mathematics and Natural Computation 8:239\u2013256","journal-title":"New Mathematics and Natural Computation"},{"key":"3052_CR14","first-page":"553","volume":"9","author":"I Georgescu","year":"2015","unstructured":"Georgescu I (2015) On indicators of fuzzy relations. Annals of Fuzzy Mathematics and Informatics 9:553\u2013571","journal-title":"Annals of Fuzzy Mathematics and Informatics"},{"key":"3052_CR15","unstructured":"Gottwald S (1991) Fuzzified fuzzy relations, in: R.\u00a0Lowen, M.\u00a0Roubens (Editors.), Proceedings of 4th IFSA Congress, In: P.\u00a0Wuyts (Editor.), Mathematics, Brussels, pp. 82\u201386"},{"key":"3052_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-86812-1","volume-title":"Fuzzy Sets and Fuzzy Logic: Foundations of Application-From a Mathematical Point of View","author":"S Gottwald","year":"1993","unstructured":"Gottwald S (1993) Fuzzy Sets and Fuzzy Logic: Foundations of Application-From a Mathematical Point of View. Vieweg, Wiesbaden"},{"issue":"3","key":"3052_CR17","first-page":"79","volume":"15","author":"YL Han","year":"2018","unstructured":"Han YL, Shi FG (2018) A new way to extend fuzzy implications. Iranian Journal of Fuzzy Systems 15(3):79\u201397","journal-title":"Iranian Journal of Fuzzy Systems"},{"key":"3052_CR18","unstructured":"H\u00f6hle U (1993) Fuzzy equalities and indistinguishability. In Proceedings EUFIT \u201993, vol.\u00a01, pp. 358\u2013363, Aachen"},{"key":"3052_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-9540-7","volume-title":"Triangular Norms","author":"EP Klement","year":"2000","unstructured":"Klement EP, Mesiar R, Pap E (2000) Triangular Norms. Kluwer Academic Publishers, Dordrecht"},{"key":"3052_CR20","doi-asserted-by":"publisher","first-page":"1865","DOI":"10.1016\/j.fss.2006.02.013","volume":"157","author":"H Lai","year":"2006","unstructured":"Lai H, Zhang D (2006) Fuzzy preorder and fuzzy topology. Fuzzy Sets and Systems 157:1865\u20131885","journal-title":"Fuzzy Sets and Systems"},{"issue":"107","key":"3052_CR21","first-page":"1","volume":"42","author":"W Li","year":"2023","unstructured":"Li W, Yang B, Qiao J (2023) $$(O, G)$$-granular variable precision fuzzy rough sets based on overlap and grouping functions. Computational and Applied Mathematics 42(107):1\u201330","journal-title":"Computational and Applied Mathematics"},{"key":"3052_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8741-9","volume-title":"Fuzzy Set Theory","author":"R Lowen","year":"1996","unstructured":"Lowen R (1996) Fuzzy Set Theory. Techniques and Bibliography, Springer, Netherlands Publishers, Basic Concepts"},{"key":"3052_CR23","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/0165-0114(91)90048-U","volume":"40","author":"SV Ovchinnikov","year":"1991","unstructured":"Ovchinnikov SV (1991) Similarity relations, fuzzy partitions, and fuzzy ordering. Fuzzy Sets and Systems 40:107\u2013126","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR24","unstructured":"Recasens J (2011) Indistinguishability Operators. Springer-Verlag, Berlin Heidelberg Publishers, Spain, Modelling Fuzzy Equalities and Fuzzy Equivalence Relations"},{"key":"3052_CR25","volume-title":"Quantales and Their Applications","author":"KI Rosenthal","year":"1990","unstructured":"Rosenthal KI (1990) Quantales and Their Applications. Longman, Essex"},{"issue":"110","key":"3052_CR26","first-page":"1","volume":"39","author":"F Sun","year":"2020","unstructured":"Sun F, Qu X-B, Huang X-J (2020) Characterizations of residual implications derived from uni-nullnorms. Computational and Applied Mathematics 39(110):1\u201318","journal-title":"Computational and Applied Mathematics"},{"key":"3052_CR27","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/0165-0114(85)90096-X","volume":"17","author":"L Valverde","year":"1985","unstructured":"Valverde L (1985) On the structure of $$F$$-indistinguishability operators. Fuzzy Sets and Systems 17:313\u2013328","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR28","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/j.fss.2012.07.013","volume":"216","author":"X Wang","year":"2013","unstructured":"Wang X, Cao X, Wu C, Chen J (2013) Indicators of fuzzy relations. Fuzzy Sets and Systems 216:91\u2013107","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR29","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.fss.2014.01.008","volume":"246","author":"X Wang","year":"2014","unstructured":"Wang X, Xue Y (2014) Traces and property indicators of fuzzy relations. Fuzzy Sets and Systems 246:78\u201390","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR30","doi-asserted-by":"crossref","unstructured":"Wu C, Wang L (2012a) Some results on the relationships between transitivity-related indicators of fuzzy relations, 2012 International Conference on Uncertainty Reasoning and Knowledge Engineering, Vol.\u00a014\u201315, pp. 235\u2013239, Jakarta, Indonesia","DOI":"10.1109\/URKE.2012.6319554"},{"key":"3052_CR31","doi-asserted-by":"crossref","unstructured":"Wu C, Wang X (2012b) A further study on transitivity-related property indicators of fuzzy relations. Advanced Science Letters 7:630\u2013633","DOI":"10.1166\/asl.2012.2685"},{"key":"3052_CR32","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/j.fss.2017.04.002","volume":"343","author":"X-P Yang","year":"2018","unstructured":"Yang X-P, Lin H-T, Zhou X-G, Cao B-Y (2018) Addition-min fuzzy relation inequalities with application inBitTorrent-like Peer-to-Peer file sharing system. Fuzzy Sets and Systems 343:126\u2013140","journal-title":"Fuzzy Sets and Systems"},{"key":"3052_CR33","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.ins.2016.04.014","volume":"358\u2013359","author":"X-P Yang","year":"2016","unstructured":"Yang X-P, Zhou X-G, Cao B-Y (2016) Latticized linear programming subject to max-product fuzzy relation inequalities with application in wireless communication. Information Sciences 358\u2013359:44\u201355","journal-title":"Information Sciences"},{"key":"3052_CR34","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA (1965) Fuzzy sets. Information and Control 8:338\u2013353","journal-title":"Information and Control"},{"key":"3052_CR35","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/S0020-0255(71)80005-1","volume":"3","author":"LA Zadeh","year":"1971","unstructured":"Zadeh LA (1971) Similarity relations and fuzzy orderings. Information Sciences 3:177\u2013200","journal-title":"Information Sciences"},{"issue":"134","key":"3052_CR36","first-page":"1","volume":"42","author":"S Zhang","year":"2023","unstructured":"Zhang S, Pang B (2023) Overlap function-based amongness spaces. Computational and Applied Mathematics 42(134):1\u201325","journal-title":"Computational and Applied Mathematics"},{"key":"3052_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-0646-0","volume-title":"Fuzzy Set Theory and Its Applications","author":"HJ Zimmermann","year":"2001","unstructured":"Zimmermann HJ (2001) Fuzzy Set Theory and Its Applications. Kluwer Academic Publishers, Aachen"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03052-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-03052-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03052-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T06:06:33Z","timestamp":1741932393000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-03052-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,10]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["3052"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-03052-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,1,10]]},"assertion":[{"value":"16 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no Conflict of interest in the manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"114"}}