{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:40:38Z","timestamp":1760190038240,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,6]],"date-time":"2019-09-06T00:00:00Z","timestamp":1567728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been generalized to others\u2014e.g., intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, or neutrosophic sets\u2014then uninorm generalizations have emerged in those novel frameworks. Neutrosophic sets contain the notion of indeterminacy\u2014which is caused by unknown, contradictory, and paradoxical information\u2014and thus, it includes, aside from the membership and non-membership functions, an indeterminate-membership function. Also, the relationship among them does not satisfy any restriction. Along this line of generalizations, this paper aims to extend uninorms to the framework of neutrosophic offsets, which are called neutrosophic offuninorms. Offsets are neutrosophic sets such that their domains exceed the scope of the interval [0,1]. In the present paper, the definition, properties, and application areas of this new concept are provided. It is necessary to emphasize that the neutrosophic offuninorms are feasible for application in several fields, as we illustrate in this paper.<\/jats:p>","DOI":"10.3390\/sym11091136","type":"journal-article","created":{"date-parts":[[2019,9,9]],"date-time":"2019-09-09T03:14:41Z","timestamp":1567998881000},"page":"1136","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Neutrosophic Offuninorms"],"prefix":"10.3390","volume":"11","author":[{"given":"Erick Gonz\u00e1lez","family":"Caballero","sequence":"first","affiliation":[{"name":"Center of Studies in Mathematic for Technical Sciences, Technological University of Havana, Havana, Havana 19390, Cuba"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5560-5926","authenticated-orcid":false,"given":"Florentin","family":"Smarandache","sequence":"additional","affiliation":[{"name":"Math &amp; Science Department, University of New Mexico, Gallup, NM 87301, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7911-5879","authenticated-orcid":false,"given":"Maikel","family":"Leyva V\u00e1zquez","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Matem\u00e1ticas y F\u00edsicas, University of Guayaquil, Guayas, Guayaquil 090514, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0165-0114(95)00133-6","article-title":"Uninorm aggregation operators","volume":"80","author":"Yager","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1142\/S0218488597000312","article-title":"Structure of uninorms","volume":"5","author":"Fodor","year":"1997","journal-title":"Internat. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_3","first-page":"29","article-title":"Aggregation Operators: Basic Concepts, Issues and Properties","volume":"Volume 97","author":"Calvo","year":"2002","journal-title":"Aggregation Operators: New Trends and Applications"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0165-0114(00)00027-0","article-title":"Uninorms in fuzzy system modeling","volume":"122","author":"Yager","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/S0165-0114(02)00521-3","article-title":"Universal approximation theorem for uninorm-based fuzzy systems modeling","volume":"140","author":"Yager","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jenei, S. (2019, January 16\u201320). A New Class of Uninorm Aggregation Operations for Fuzzy Theory. Proceedings of the International Conference on Artificial Intelligence and Soft Computing, Zakopane, Poland.","DOI":"10.1007\/978-3-030-20912-4_28"},{"key":"ref_7","unstructured":"Fodor, J., Rudas, I.J., and Bede, B. (2006, January 29\u201330). Uninorms and absorbing norms with applications to Image Processing. Proceedings of the 4th Serbian-Hungarian Joint Symposium on Intelligent Systems, SISY, Subotica, Serbia."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Xie, A. (2017). On Extended Representable Uninorms and Their Extended Fuzzy Implications (Coimplications). Symmetry, 8.","DOI":"10.3390\/sym9080160"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.knosys.2012.07.025","article-title":"Information aggregation in intelligent systems: An application oriented approach","volume":"38","author":"Rudas","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_10","unstructured":"Depaire, B., Vanhoof, K., and Wets, G. (2016, January 12\u201314). The Application of Uninorms in Importance-Performance Analysis. Proceedings of the 7th WSEAS International Conference on Fuzzy Systems, Cavtat, Croatia."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1002\/int.22039","article-title":"Application of uninorms to market basket analysis","volume":"34","author":"Moodley","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.knosys.2017.02.028","article-title":"Cross-ratio uninorms as an effective aggregation mechanism in Sentiment Analysis","volume":"124","author":"Appel","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.knosys.2018.05.039","article-title":"Geo-uninorm consistency control module for preference similarity network hierarchical clustering based consensus model","volume":"162","author":"Kamis","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.knosys.2016.01.004","article-title":"Uninorm trust propagation and aggregation methods for group decision making in social network with four tuple information","volume":"96","author":"Wu","year":"2016","journal-title":"Knowl.-Based Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.ins.2018.11.037","article-title":"A review on trust propagation and opinion dynamics in social networks and group decision making frameworks","volume":"478","author":"Kou","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.neucom.2013.04.047","article-title":"Uninorm based evolving neural networks and approximation capabilities","volume":"127","author":"Bordignon","year":"2014","journal-title":"Neurocomputing"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wang, S. (2019). A Proof of the Standard Completeness for the Involutive Uninorm Logic. Symmetry, 11.","DOI":"10.3390\/sym11040445"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Yang, E. (2019). Fixpointed Idempotent Uninorm (Based) Logics. Mathematics, 7.","DOI":"10.3390\/math7010107"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez-Hidalgo, M., Massanet, S., Mir, A., and Ruiz-Aguilera, D. (2014, January 15\u201319). A new edge detector based on uninorms. Proceedings of the 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Montpellier, France.","DOI":"10.1007\/978-3-319-08855-6_19"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.ins.2018.01.042","article-title":"Interval-valued fuzzy morphological associative memories: Some theoretical aspects and applications","volume":"438","author":"Sussner","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0165-0114(80)90062-7","article-title":"Latent connectives in human decision making","volume":"4","author":"Zimmermann","year":"1980","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1177\/0193841X16662421","article-title":"Fuzzy analysis of students\u2019 ratings","volume":"40","author":"Marasini","year":"2016","journal-title":"Eval. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.fss.2003.12.006","article-title":"Uninorms in L*-fuzzy set theory","volume":"148","author":"Deschrijver","year":"2004","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","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_25","unstructured":"Gonz\u00e1lez, E., Leyva, M., and Smarandache, F. On Neutrosophic Uninorms, Neutrosophic Sets Syst., Under review."},{"key":"ref_26","unstructured":"Smarandache, F. (2005). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy Set, Neutrosophic Probability and Statistics, American Research Press. [4th ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/j.patcog.2008.10.002","article-title":"New neutrosophic approach to image segmentation","volume":"42","author":"Guo","year":"2009","journal-title":"Pattern Recognit."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2710","DOI":"10.1016\/j.patcog.2015.02.018","article-title":"NCM: Neutrosophic c-means clustering algorithm","volume":"48","author":"Guo","year":"2015","journal-title":"Pattern Recognit."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Abdel-Basset, M., Mohamed, M., and Smarandache, F. (2018). An extension of neutrosophic AHP\u2013SWOT analysis for strategic planning and decision-making. Symmetry, 10.","DOI":"10.3390\/sym10040116"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Abdel-Basset, M., Mohamed, M., and Smarandache, F. (2018). A hybrid neutrosophic group ANP-TOPSIS framework for supplier selection problems. Symmetry, 10.","DOI":"10.3390\/sym10060226"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Smarandache, F. (2016). Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset:Similarly for Neutrosophic Over-\/Under-\/Off-Logic, Probability, and Statistics, Pons Editions.","DOI":"10.5958\/2320-3226.2016.00007.2"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shortliffe, E.H. (1976). Computer-Based Medical Consultations: MYCIN, Elsevier.","DOI":"10.1016\/B978-0-444-00179-5.50009-3"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0165-0114(96)00185-6","article-title":"The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons","volume":"93","author":"Tsadiras","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/S0165-0114(98)00265-6","article-title":"Van Melle\u2019s combining function in MYCIN is a representable uninorm: An alternative proof","volume":"104","author":"Fodor","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5","DOI":"10.5120\/20580-2998","article-title":"Fuzzy Logic based Assessment Model Proposal for Online Problem-based Learning","volume":"117","year":"2015","journal-title":"IJCA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2845","DOI":"10.1016\/j.neucom.2008.07.017","article-title":"Binary classification using ensemble neural networks and interval neutrosophic sets","volume":"72","author":"Kraipeerapun","year":"2009","journal-title":"Neurocomputing"},{"key":"ref_37","first-page":"31","article-title":"Neutrosophic Logic for Mental Model Elicitation and Analysis","volume":"2","year":"2014","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_38","first-page":"410","article-title":"Single valued neutrosophic sets","volume":"4","author":"Wang","year":"2010","journal-title":"Multispace Multistruct."},{"key":"ref_39","unstructured":"Smarandache, F. (2007). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, InfoLearnQuest. [6th ed.]."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1109\/TSMC.1979.4310095","article-title":"Symmetric Summation: A Class of Operations on Fuzzy Sets","volume":"9","author":"Silvert","year":"1979","journal-title":"IEEE Trans. Syst. Man Cybernet"},{"key":"ref_41","unstructured":"Baczynski, M., and Jayaram, B. (2008). Fuzzy Implications, Springer."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Roth, A.E. (1988). Introduction to the Shapley value. The Shapley Value: Essays in Honor of Lloyd S. Shapley, Cambridge University Press.","DOI":"10.1017\/CBO9780511528446"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1002\/int.21755","article-title":"Games Played under Fuzzy Constraints","volume":"31","author":"Zhang","year":"2016","journal-title":"Int. J. Intell. Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1136\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:17:36Z","timestamp":1760188656000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1136"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,6]]},"references-count":43,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091136"],"URL":"https:\/\/doi.org\/10.3390\/sym11091136","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,9,6]]}}}