{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:17:45Z","timestamp":1760235465457,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,8,24]],"date-time":"2021-08-24T00:00:00Z","timestamp":1629763200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["MTM 2016-79422-P"],"award-info":[{"award-number":["MTM 2016-79422-P"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["MTM 2016-79422-P"],"award-info":[{"award-number":["MTM 2016-79422-P"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC 2018-098623-B-I00"],"award-info":[{"award-number":["PGC 2018-098623-B-I00"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Minstery of Education of the Czech Republic","award":["GACR 19-09967"],"award-info":[{"award-number":["GACR 19-09967"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets.<\/jats:p>","DOI":"10.3390\/axioms10030201","type":"journal-article","created":{"date-parts":[[2021,8,24]],"date-time":"2021-08-24T22:12:22Z","timestamp":1629843142000},"page":"201","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Self-Aggregations of Min-Subgroups"],"prefix":"10.3390","volume":"10","author":[{"given":"Carlos","family":"Bejines","sequence":"first","affiliation":[{"name":"Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, 121 35 Prague, Czech Republic"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2827-5383","authenticated-orcid":false,"given":"Sergio","family":"Ardanza-Trevijano","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1tica Aplicada, University of Navarra, 31080 Pamplona, Spain"},{"name":"Instituto de Ciencia de los Datos e Inteligencia Artificial, University of Navarra, 31080 Pamplona, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5547-1083","authenticated-orcid":false,"given":"Jorge","family":"Elorza","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1tica Aplicada, University of Navarra, 31080 Pamplona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,24]]},"reference":[{"key":"ref_1","unstructured":"Beliakov, G., Pradera, A., and Calvo, T. (2007). Aggregation Functions: A Guide for Practitioners, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Beliakov, G., Sola, H.B., and S\u00e1nchez, T.C. (2016). A Practical Guide to Averaging Functions, Springer.","DOI":"10.1007\/978-3-319-24753-3"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Calvo, T., Koles\u00e1rov\u00e1, A., Komorn\u00edkov\u00e1, M., and Mesiar, R. (2002). Aggregation Operators, Physica-Verlag.","DOI":"10.1007\/978-3-7908-1787-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/0022-247X(71)90199-5","article-title":"Fuzzy groups","volume":"35","author":"Rosenfeld","year":"1971","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/0022-247X(81)90164-5","article-title":"Fuzzy groups and level subgroups","volume":"84","author":"Das","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/0022-247X(79)90182-3","article-title":"Fuzzy groups redefined","volume":"69","author":"Anthony","year":"1979","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s005000050085","article-title":"Fuzzy subgroups and similarities","volume":"3","author":"Formato","year":"1999","journal-title":"Soft Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.fss.2018.09.002","article-title":"On the relationship between fuzzy subgroups and indistinguishability operators","volume":"373","author":"Boixader","year":"2019","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1016\/0031-3203(94)00312-A","article-title":"Fuzzy mathematical morphologies: A comparative study","volume":"28","author":"Bloch","year":"1995","journal-title":"Pattern Recognit."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.fss.2012.08.010","article-title":"On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems","volume":"218","author":"Elorza","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","unstructured":"Soille, P. (2013). Morphological Image Analysis: Principles and Applications, Springer Science & Business Media."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.fss.2020.05.017","article-title":"Aggregation of fuzzy subgroups","volume":"418","author":"Bejines","year":"2021","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bejines, C., Ardanza-Trevijano, S., Chasco, M., and Elorza, J. (2021). Aggregation of indistinguishability operators. Fuzzy Sets Syst.","DOI":"10.1016\/j.fss.2021.04.023"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","unstructured":"Dudziak, U. (2013, January 11\u201313). Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process. Proceedings of the EUSFLAT Conference, Milano, Italy.","DOI":"10.2991\/eusflat.2013.59"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Fodor, J.C., and Roubens, M. (1994). Fuzzy Preference Modelling and Multicriteria Decision Support, Springer Science & Business Media.","DOI":"10.1007\/978-94-017-1648-2"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.fss.2016.06.017","article-title":"On the TL-transitivity of fuzzy similarity measures","volume":"322","author":"He","year":"2017","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1002\/int.10141","article-title":"Aggregation of T-transitive relations","volume":"18","author":"Jacas","year":"2003","journal-title":"Int. J. Intell. Syst."},{"key":"ref_19","unstructured":"Nebot, A., Binefa, X., and de Mantaras, L. (2016). Preserving T-Transitivity. Artificial Intelligence Research and Development, IOS Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pedraza, T., Rodr\u00edguez-L\u00f3pez, J., and Valero, \u00d3. (2020). Aggregation of fuzzy quasi-metrics. Inf. Sci.","DOI":"10.1016\/j.ins.2020.08.045"},{"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. Fuzziness Knowl. Based Syst."},{"key":"ref_22","unstructured":"Klement, E.P., Mesiar, R., and Pap, E. (2013). Triangular Norms, Springer Science & Business Media."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0165-0114(90)90032-2","article-title":"Level subgroups and union of fuzzy subgroups","volume":"37","author":"Dixit","year":"1990","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/S0165-0114(00)00098-1","article-title":"On an equivalence of fuzzy subgroups I","volume":"123","author":"Murali","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1155\/S0161171203205238","article-title":"On an equivalence of fuzzy subgroups III","volume":"2003","author":"Murali","year":"2003","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0165-0114(92)90219-T","article-title":"Isomorphic fuzzy groups","volume":"50","author":"Ray","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/S0165-0114(99)00102-5","article-title":"Some properties on fuzzy subgroups","volume":"119","author":"Zhang","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bejines, C., Chasco, M.J., Elorza, J., and Montes, S. (2018, January 23\u201326). Equivalence relations on fuzzy subgroups. Proceedings of the Conference of the Spanish Association for Artificial Intelligence, Granada, Spain.","DOI":"10.1007\/978-3-030-00374-6_14"},{"key":"ref_30","first-page":"75","article-title":"Fuzzy subgroups and certain equivalence relations","volume":"3","author":"Jain","year":"2006","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.ins.2011.12.019","article-title":"A generalization of the migrativity property of aggregation functions","volume":"191","author":"Bustince","year":"2012","journal-title":"Inf. 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