{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:13:50Z","timestamp":1760710430485,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T00:00:00Z","timestamp":1609286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003093","name":"Ministry of Higher Education, Malaysia","doi-asserted-by":"publisher","award":["FRGS\/1\/2019\/STG06\/UPM\/02\/6"],"award-info":[{"award-number":["FRGS\/1\/2019\/STG06\/UPM\/02\/6"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, T-orderings, defined based on a t-norm T and infimum operator (for infinite case) or minimum operator (for finite case), have been applied as a generalization of the notion of crisp orderings to fuzzy setting. When this concept is extending to m-polar fuzzy data, it is questioned whether the generalized definition can be expanded for any aggregation function, not necessarily the minimum operator, or not. To answer this question, the present study focuses on constructing m-polar T-orderings based on aggregation functions A, in particular, m-polar T-preorderings (which are reflexive and transitive m-polar fuzzy relations w.r.t T and A) and m-polar T-equivalences (which are symmetric m-polar T-preorderings). Moreover, the construction results for generating crisp preference relations based on m-polar T-orderings are obtained. Two algorithms for solving ranking problem in decision-making are proposed and validated by an illustrative example.<\/jats:p>","DOI":"10.3390\/sym13010051","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T20:13:41Z","timestamp":1609359221000},"page":"51","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["m-Polar Generalization of Fuzzy T-Ordering Relations: An Approach to Group Decision Making"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3205-5212","authenticated-orcid":false,"given":"Azadeh","family":"Zahedi Khameneh","sequence":"first","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1217-963X","authenticated-orcid":false,"given":"Adem","family":"Kilicman","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0020-0255(71)80005-1","article-title":"Similarity relations and fuzzy orderings","volume":"3","author":"Zadeh","year":"1971","journal-title":"Inf. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/0165-0114(84)90032-0","article-title":"Fuzzy preference orderings in group decision-making","volume":"12","author":"Tanino","year":"1984","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0165-0114(91)90259-S","article-title":"Strict preference relations based on weak t-norms","volume":"43","author":"Fodor","year":"1991","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0165-0114(92)90134-P","article-title":"Fuzzy ordered sets","volume":"46","author":"Venugopalan","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/S0165-0114(98)00287-5","article-title":"Transitivity of fuzzy preference relations\u2014An empirical study","volume":"118","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.ijar.2007.06.001","article-title":"Fuzzy orders and fuzzifying topologies","volume":"48","author":"Fang","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.knosys.2014.03.003","article-title":"Weak transitivity of interval-valued fuzzy relations","volume":"63","author":"Xu","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.ins.2017.10.002","article-title":"Aggregation of diverse types of fuzzy orders for decision-making problems","volume":"424","author":"Bentkowska","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Fuster-Parra, P., Mart\u00edn, J., Recasens, J., and Valero, \u00d3. (2020). T-Equivalences: The Metric Behavior Revisited. Mathematics, 8.","DOI":"10.3390\/math8040495"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1142\/S0218488500000411","article-title":"A similarity-based generalization of fuzzy orderings preserving the classical axioms","volume":"8","author":"Bodenhofer","year":"2000","journal-title":"Int. J. Uncertain. Fuzzy"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0165-0114(02)00436-0","article-title":"Representations and constructions of similarity-based fuzzy orderings","volume":"137","author":"Bodenhofer","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fodor, J.C., and Roubens, M.R. (1994). Fuzzy Preference Modelling and Multicriteria Decision Support, Springer Science & Business Media Dordrecht. eBook ISBN 978-94-017-1648-2.","DOI":"10.1007\/978-94-017-1648-2"},{"key":"ref_13","unstructured":"Gottwald, S. (1993). Fuzzy Sets and Fuzzy Logic: The Foundations of Application\u2014From a Mathematical Point of View, Vieweg and Teubner Verlag. eBook ISBN 978-3-322-86812-1."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0165-0114(85)90096-X","article-title":"On the structure of F-indistinguishability operators","volume":"17","author":"Valverde","year":"1985","journal-title":"Fuzzy Sets Syst."},{"key":"ref_15","unstructured":"Zhang, W.R. (1994). Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent decision analysis. NAFIPS\/IFIS\/NASA\u201994. Proceedings of the First, International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige, IEEE."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chen, J., Li, S., Ma, S., and Wang, X. (2014). m-polar fuzzy sets: An extension of bipolar fuzzy sets. Sci. World J., 416530.","DOI":"10.1155\/2014\/416530"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.engappai.2017.09.011","article-title":"m-polar fuzzy graph representation of concept lattice","volume":"67","author":"Singh","year":"2018","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Calvo, T., Mayor, G., and Mesiar, R. (2002). Aggregation operators: Properties, classes and construction methods. Aggregation Operators. Studies in Fuzziness and Soft Computing, Physica.","DOI":"10.1007\/978-3-7908-1787-4"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.ins.2010.08.040","article-title":"Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes","volume":"181","author":"Grabisch","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zahedi Khameneh, A., and Kilicman, A. (2020). Some Construction Methods of Aggregation Operators in Decision-Making Problems: An Overview. Symmetry, 12.","DOI":"10.20944\/preprints202001.0388.v1"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1142\/S0218488502001806","article-title":"Domination of aggregation operators and preservation of transitivity","volume":"10","author":"Saminger","year":"2002","journal-title":"Int. J. Uncertain. Fuzzy"},{"key":"ref_22","unstructured":"Klement, E.P., Mesiar, R., and Pap, E. (2000). Triangular Norms, Springer Science & Business Media Dordrecht. eBook ISBN 978-94-015-9540-7."},{"key":"ref_23","first-page":"115","article-title":"Preservation of properties of fuzzy relations during aggregation processes","volume":"43","author":"Drewniak","year":"2007","journal-title":"Kybernetika"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.fss.2015.06.001","article-title":"Preservation of fuzzy relation properties based on fuzzy conjunctions and disjunctions during aggregation process","volume":"291","author":"Bentkowska","year":"2016","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.ins.2014.12.060","article-title":"Multi-polar t-conorms and uninorms","volume":"301","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.ins.2019.07.090","article-title":"A fuzzy majority-based construction method for composed aggregation functions by using combination operator","volume":"505","author":"Kilicman","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.fss.2011.01.015","article-title":"Aggregation functions on bounded partially ordered sets and their classification","volume":"175","author":"Mesiar","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Zahedi Khameneh, A., and Kilicman, A. (2018). m-polar fuzzy soft weighted aggregation operators and their applications in group decision-making. Symmetry, 10.","DOI":"10.3390\/sym10110636"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/51\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:48:06Z","timestamp":1760179686000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/51"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,30]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010051"],"URL":"https:\/\/doi.org\/10.3390\/sym13010051","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,12,30]]}}}