{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:16:09Z","timestamp":1760228169700,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,7]],"date-time":"2022-05-07T00:00:00Z","timestamp":1651881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shandong Province","award":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"],"award-info":[{"award-number":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"]}]},{"name":"Ministry of Education Funded Project for Humanities and Social Sciences Research","award":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"],"award-info":[{"award-number":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"]}]},{"name":"Shandong Province Higher Educational Youth Innovation Team Development Program","award":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"],"award-info":[{"award-number":["ZR2018BG008","ZR2019PG009","21YJC630088","19YJC630059","2021RW020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min\/max and commonly used increasing\/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified.<\/jats:p>","DOI":"10.3390\/sym14050958","type":"journal-article","created":{"date-parts":[[2022,5,8]],"date-time":"2022-05-08T23:27:25Z","timestamp":1652052445000},"page":"958","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Aggregation of Comonotone or Countermonotone Fuzzy Relations"],"prefix":"10.3390","volume":"14","author":[{"given":"Yuanyuan","family":"Liu","sequence":"first","affiliation":[{"name":"School of Management Science and Engineering, Shangdong University of Finance and Economics, Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fan","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Management Science and Engineering, Shangdong University of Finance and Economics, Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s11750-010-0134-z","article-title":"On the Ferrers property of valued interval orders","volume":"19","author":"Montes","year":"2011","journal-title":"Top"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0165-0114(94)00379-9","article-title":"Fuzzy preference structures without incomparability","volume":"76","author":"Kerre","year":"1995","journal-title":"Fuzzy Sets Syst."},{"doi-asserted-by":"crossref","unstructured":"Fodor, J., and Roubens, M. 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