{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T18:47:11Z","timestamp":1770749231910,"version":"3.50.0"},"reference-count":36,"publisher":"SAGE Publications","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2015,10,7]]},"DOI":"10.3233\/ifs-151720","type":"journal-article","created":{"date-parts":[[2015,10,13]],"date-time":"2015-10-13T19:18:05Z","timestamp":1444763885000},"page":"1065-1077","source":"Crossref","is-referenced-by-count":1,"title":["Functional equations related to fuzzy sets and\u00a0representable orderings1"],"prefix":"10.1177","volume":"29","author":[{"given":"Mar\u00eda Jes\u00fas","family":"Campi\u00f3n","sequence":"first","affiliation":[{"name":"INAMAT and Departamento de Matem\u00e1ticas, Universidad P\u00fablica de Navarra, E-31006 Pamplona, Spain"}]},{"given":"Raquel G.","family":"Catal\u00e1n","sequence":"additional","affiliation":[{"name":"INAMAT and Departamento de Matem\u00e1ticas, Universidad P\u00fablica de Navarra, E-31006 Pamplona, Spain"}]},{"given":"Jos\u00e9","family":"Garay","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad de Zaragoza, E-50009 Zaragoza, Spain"}]},{"given":"Esteban","family":"Indur\u00e1in","sequence":"additional","affiliation":[{"name":"INAMAT and Departamento de Matem\u00e1ticas, Universidad P\u00fablica de Navarra, E-31006 Pamplona, Spain"}]}],"member":"179","reference":[{"issue":"1-2","key":"10.3233\/IFS-151720_ref1","doi-asserted-by":"crossref","first-page":"557","DOI":"10.5486\/PMD.2011.4784","article-title":"Generalized Abel functional equations and numerical representability of semiorders","volume":"78","author":"Abr\u00edsqueta","year":"2011","journal-title":"Publ Math Debrecen"},{"key":"10.3233\/IFS-151720_ref2","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-3749-9","volume-title":"A Short Course on Functional Equations Based upon Recent Applications to the Social and Behavioral Sciences","author":"Acz\u00e9l","year":"1987"},{"key":"10.3233\/IFS-151720_ref3","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1063\/1.1751367","article-title":"The associativity equation re-revisited","volume":"707","author":"Acz\u00e9l","year":"2004","journal-title":"AIP Conf Proc"},{"key":"10.3233\/IFS-151720_ref4","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF02020630","article-title":"Generalized associativity and bisymmetry on quasigroups","volume":"11","author":"Acz\u00e9l","year":"1960","journal-title":"Acta Math Acad Sci Hungar"},{"issue":"6","key":"10.3233\/IFS-151720_ref5","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1080\/18756891.2012.747653","article-title":"Numerical representability of fuzzy total preorders","volume":"5","author":"Agud","year":"2012","journal-title":"Int J Comput Int Sys"},{"key":"10.3233\/IFS-151720_ref6","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S0165-0114(02)00247-6","article-title":"Topological interpretation of fuzzy sets and intervals","volume":"135","author":"Albrecht","year":"2003","journal-title":"Fuzzy Sets and Systems"},{"key":"10.3233\/IFS-151720_ref8","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.knosys.2013.09.008","article-title":"Topological interpretations of fuzzy subsets. A unified approach for fuzzy thresholding algorithms","volume":"54","author":"Barrenechea","year":"2013","journal-title":"Knowledge-Based Systems"},{"key":"10.3233\/IFS-151720_ref9","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/S0304-4068(02)00003-4","article-title":"The non-existence of a utility function and the structure of non-representable preference relations","volume":"37","author":"Beardon","year":"2002","journal-title":"J Math Econom"},{"key":"10.3233\/IFS-151720_ref10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-247X(66)90071-0","article-title":"Abstraction and pattern classification","volume":"13","author":"Bellman","year":"1966","journal-title":"J Math Anal Appl"},{"key":"10.3233\/IFS-151720_ref11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1023\/A:1011974420295","article-title":"Numerical representations of interval orders","volume":"18","author":"Bosi","year":"2001","journal-title":"Order"},{"key":"10.3233\/IFS-151720_ref12","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s11083-009-9124-x","article-title":"Unified representability of total preorders and interval orders through a single function: The lattice approach","volume":"26","author":"Bosi","year":"2009","journal-title":"Order"},{"key":"10.3233\/IFS-151720_ref13","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-51495-1","volume-title":"Representations of Preference Orderings","author":"Bridges","year":"1995"},{"key":"10.3233\/IFS-151720_ref14","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s00010-013-0225-7","article-title":"New trends on the permutability equation","volume":"88","author":"Bustince","year":"2014","journal-title":"Aequat Math"},{"issue":"1","key":"10.3233\/IFS-151720_ref15","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1142\/S0218488511006903","article-title":"Aggregation of preferences in crisp and fuzzy settings: Functional equations leading to possibility results","volume":"19","author":"Campi\u00f3n","year":"2011","journal-title":"Internat J Uncertain Fuzziness Knowledge-Based Systems"},{"key":"10.3233\/IFS-151720_ref16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2005.06.018","article-title":"Representability of binary relations through fuzzy numbers","volume":"157","author":"Campi\u00f3n","year":"2006","journal-title":"Fuzzy Sets and Systems"},{"key":"10.3233\/IFS-151720_ref17","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s11083-006-9052-y","article-title":"Order embeddingswith irrational codomain:Debreu properties of real subsets","volume":"23","author":"Campi\u00f3n","year":"2006","journal-title":"Order"},{"key":"10.3233\/IFS-151720_ref18","doi-asserted-by":"crossref","first-page":"367","DOI":"10.3233\/IFS-131005","article-title":"Reinterpreting a fuzzy subset by means of a Sincov\u2019s functional equation","volume":"27","author":"Campi\u00f3n","year":"2014","journal-title":"J Intell Fuzzy Systems"},{"key":"10.3233\/IFS-151720_ref20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.5486\/PMD.1997.1818","article-title":"Associativity equation revisited","volume":"51","author":"Candeal","year":"1997","journal-title":"Publ Math Debrecen"},{"issue":"2","key":"10.3233\/IFS-151720_ref21","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1142\/S0218488509005814","article-title":"Universal codomains to represent interval orders","volume":"17","author":"Candeal","year":"2009","journal-title":"Internat J Uncertain Fuzziness Knowledge-Based Systems"},{"key":"10.3233\/IFS-151720_ref22","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s00010-012-0128-z","article-title":"Bivariate functional equations around associativity","volume":"84","author":"Candeal","year":"2012","journal-title":"Aequat Math"},{"key":"10.3233\/IFS-151720_ref23","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/978-3-642-39165-1_24","article-title":"The consensus functional equation in agreement theory, Aggregation functions in theory and practise","volume":"228","author":"Candeal","year":"2013","journal-title":"Advances in Intelligent Systems and Computing"},{"key":"10.3233\/IFS-151720_ref24","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/S0165-4896(01)00082-8","article-title":"Numerical representability of semiorders","volume":"43","author":"Candeal","year":"2002","journal-title":"Math Social Sci"},{"key":"10.3233\/IFS-151720_ref25","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.fss.2007.07.005","article-title":"A note on the convex combinations of triangular norms","volume":"159","author":"Durante","year":"2008","journal-title":"Fuzzy Sets and Systems"},{"key":"10.3233\/IFS-151720_ref26","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/0022-2496(70)90062-3","article-title":"Intransitive indifference with unequal indifference intervals","volume":"7","author":"Fishburn","year":"1970","journal-title":"J Math Psych"},{"key":"10.3233\/IFS-151720_ref27","first-page":"881","article-title":"Topological spaces and soft sets","volume":"13","author":"Ge","year":"2011","journal-title":"J Comput Anal Appl"},{"key":"10.3233\/IFS-151720_ref28","first-page":"1","article-title":"A remark on Sincov\u2019s functional equation","volume":"31","author":"Gronau","year":"2000","journal-title":"Not S Afr Math Soc"},{"key":"10.3233\/IFS-151720_ref29","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.knosys.2012.07.011","article-title":"Topological analysis of knowledge maps","volume":"36","author":"Liu","year":"2012","journal-title":"Knowledge-Based Systems"},{"key":"10.3233\/IFS-151720_ref30","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.ins.2014.02.119","article-title":"Bimigrativity of binary aggregation functions","volume":"274","author":"L\u00f3pez-Molina","year":"2014","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151720_ref31","doi-asserted-by":"crossref","first-page":"178","DOI":"10.2307\/1905751","article-title":"Semiorders and a theory of utility discrimination","volume":"24","author":"Luce","year":"1956","journal-title":"Econometrica"},{"issue":"2","key":"10.3233\/IFS-151720_ref32","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s00233-011-9295-9","article-title":"A description of n-ary semigroups polynomial-derived from integral domains","volume":"83","author":"Marichal","year":"2011","journal-title":"Semigroup Forum"},{"issue":"1","key":"10.3233\/IFS-151720_ref34","first-page":"39","article-title":"Foundations of fuzzy theory for soft computing","volume":"8","author":"Nguyen","year":"2006","journal-title":"Int J Fuzzy Syst"},{"key":"10.3233\/IFS-151720_ref35","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1006\/jeth.1997.2346","article-title":"Representability of interval orders","volume":"78","author":"Ol\u00f3riz","year":"1998","journal-title":"J Econom Theory"},{"key":"10.3233\/IFS-151720_ref36","doi-asserted-by":"crossref","first-page":"113","DOI":"10.2307\/2964389","article-title":"Foundational aspects of theories of measurement","volume":"23","author":"Scott","year":"1958","journal-title":"J Symbolic Logic"},{"issue":"1","key":"10.3233\/IFS-151720_ref37","first-page":"93","article-title":"Associative n-dimensional copulas","volume":"47","author":"Stup\u0148anov\u00e1","year":"2011","journal-title":"Kybernetika"},{"key":"10.3233\/IFS-151720_ref38","first-page":"441","article-title":"Contribution to the theory of relative position","volume":"17","author":"Wiener","year":"1914","journal-title":"Proceedings of the Cambridge Philosophical Society"},{"key":"10.3233\/IFS-151720_ref40","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":"Information and Control"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/IFS-151720","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T08:22:54Z","timestamp":1770711774000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/IFS-151720"}},"subtitle":[],"editor":[{"given":"Reza","family":"Langari","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2015,10,7]]},"references-count":36,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.3233\/ifs-151720","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,10,7]]}}}