{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,6]],"date-time":"2026-07-06T21:37:17Z","timestamp":1783373837252,"version":"3.54.6"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T00:00:00Z","timestamp":1723507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MCIN\/AEI\/10.13039\/501100011033\/FEDER, UE","award":["PID2022-136627NB-I00"],"award-info":[{"award-number":["PID2022-136627NB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We deal with two apparently disparate theories. One of them studies extensions of orderings from a set to its power set. The other one defines suitable scores on hesitant fuzzy elements. We show that both theories have the same mathematical substrate. Thus, important possibility\/impossibility results concerning criteria for extensions can be transferred to new results on scores. And conversely, conditions imposed a priori on scores can give rise to new extension criteria. This enhances and enriches both theories. We show examples of translations of classical results on extensions in the context of scores. Also, we state new results concerning the impossibility of finding a utility function representing some kind of extension order if some restrictions are imposed on the utility function considered as a score.<\/jats:p>","DOI":"10.3390\/axioms13080549","type":"journal-article","created":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T10:54:54Z","timestamp":1723546494000},"page":"549","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extensions of Orders to a Power Set vs. Scores of Hesitant Fuzzy Elements: Points in Common of Two Parallel Theories"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1511-5658","authenticated-orcid":false,"given":"Esteban","family":"Indur\u00e1in","sequence":"first","affiliation":[{"name":"InaMat2 (Institute for Advanced Materials and Mathematics), Departamento de Estad\u00edstica, Inform\u00e1tica y Matem\u00e1ticas, Universidad P\u00fablica de Navarra, 31006 Pamplona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4064-1289","authenticated-orcid":false,"given":"Ana","family":"Mun\u00e1rriz","sequence":"additional","affiliation":[{"name":"Inarbe (Institute for Advanced Research in Business and Economics), Departamento de Estad\u00edstica, Inform\u00e1tica y Matem\u00e1ticas, Universidad P\u00fablica de Navarra, 31006 Pamplona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-3084-3511","authenticated-orcid":false,"given":"M. Sergio","family":"Sara","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica, Inform\u00e1tica y Matem\u00e1ticas, Universidad P\u00fablica de Navarra, 31006 Pamplona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Barber\u00e0, S., Hammond, P., and Seidl, C. (2004). Ranking sets of Objects. Chapter 17 of Handbook of Utility Theory: Volume 2 Extensions, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4020-7964-1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/0022-0531(84)90080-2","article-title":"A note on the extension of an order on a set to the power set","volume":"32","author":"Kannai","year":"1984","journal-title":"J. Econ. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"119500","DOI":"10.1016\/j.ins.2023.119500","article-title":"Scores of hesitant fuzzy elements revisited: \u201cWas sind und was sollen\u201d","volume":"648","author":"Alcantud","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0022-0531(76)90016-8","article-title":"Manipulation of social choice functions","volume":"13","year":"1976","journal-title":"J. Econ. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.inffus.2018.05.005","article-title":"Necessary and possible hesitant fuzzy sets: A novel model for group decision making","volume":"46","author":"Alcantud","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_6","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_7","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","article-title":"Hesitant fuzzy information aggregation in decision making","volume":"52","author":"Xia","year":"2011","journal-title":"Int. J. Approx. Reason."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1109\/TFUZZ.2015.2451692","article-title":"A historical account of types of fuzzy sets and their relationships","volume":"24","author":"Bustince","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Narukawa, Y., and Torra, V. (2021, January 11\u201314). An score index for hesitant fuzzy sets based on the Choquet integral. Proceedings of the 2021 IEEE International Conference on Fuzzy Systems, Luxembourg.","DOI":"10.1109\/FUZZ45933.2021.9494347"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Torra, V., and Narukawa, Y. (2009, January 20\u201324). On hesitant fuzzy sets and decision. Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Republic of Korea.","DOI":"10.1109\/FUZZY.2009.5276884"},{"key":"ref_12","first-page":"10","article-title":"Type-2 fuzzy sets: Some questions and answers","volume":"1","author":"Mendel","year":"2003","journal-title":"IEEE Connect. Newsl. IEEE Neural Netw. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/91.995115","article-title":"Type-2 fuzzy sets made simple","volume":"10","author":"Mendel","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.ins.2013.08.024","article-title":"Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms","volume":"255","author":"Bedregal","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_16","first-page":"83","article-title":"Criterios de extensi\u00f3n al conjunto potencia de ordenaciones sobre un conjunto finito","volume":"94","author":"Goicoechea","year":"2000","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Madr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"439","DOI":"10.2307\/1911220","article-title":"Strategy proofness and social choice functions without singlevaluedness","volume":"45","author":"Kelly","year":"1977","journal-title":"Econometrica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0022-0531(84)90083-8","article-title":"Extending an order on a set to the power set: Some remarks on Kannai and Peleg\u2019s approach","volume":"32","author":"Pattanaik","year":"1984","journal-title":"J. Econ. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1016\/j.apm.2012.04.031","article-title":"Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis","volume":"37","author":"Chen","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1002\/int.21600","article-title":"A novel method of ranking hesitant fuzzy values for multiple attribute decision-making problems","volume":"28","author":"Farhadinia","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.ins.2014.02.009","article-title":"A series of score functions for hesitant fuzzy sets","volume":"277","author":"Farhadinia","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.inffus.2015.11.004","article-title":"A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. Towards high quality progress","volume":"29","author":"Bedregal","year":"2016","journal-title":"Inf. Fusion"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1109\/TFUZZ.2011.2170076","article-title":"Hesitant fuzzy linguistic term sets for decision making","volume":"20","author":"Herrera","year":"2012","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1002\/int.21654","article-title":"Hesitant fuzzy sets: State of the art and future directions","volume":"29","author":"Torra","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1007\/s40815-019-00722-x","article-title":"Deviation degree: A perspective on score functions in hesitant fuzzy Sets","volume":"21","author":"Wang","year":"2019","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Xu, Z. (2014). Hesitant Fuzzy Sets Theory, Springer International Publishing. Volume 314 of Studies in Fuzziness and Soft Computing.","DOI":"10.1007\/978-3-319-04711-9"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1016\/j.ins.2011.01.028","article-title":"Distance and similarity measures for hesitant fuzzy sets","volume":"181","author":"Xu","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_28","first-page":"12","article-title":"An axiomatic approach to finite means","volume":"457","author":"Candeal","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/BF01082056","article-title":"Preference extension rules for ranking sets of alternatives with a fixed cardinality","volume":"39","author":"Bossert","year":"1995","journal-title":"Theory Decis."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Maly, J., and Woltran, S. (2017). Ranking Specific Sets of Objects. Lecture Notes in Informatics (LNI), Gesellschaft f\u00fcr Informatik, Springer.","DOI":"10.1007\/s13222-017-0264-7"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Bridges, D.S., and Mehta, G.B. (1995). Representation of Preference Orderings, Springer.","DOI":"10.1007\/978-3-642-51495-1"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/549\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:35:56Z","timestamp":1760110556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/549"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,13]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080549"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080549","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,13]]}}}