{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T23:02:01Z","timestamp":1776812521453,"version":"3.51.2"},"reference-count":65,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"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>This research aims to introduce and explore the theory of neutrosophic soft hyperalgebras (NSHAs), focusing on their core principles and potential applications in decision-making under uncertainty. By defining key operations such as intersection and union, we clarify the foundational characteristics of NSHAs and their relationship to soft hyperalgebras. The concepts of \u03be\u03b2-identity NSHA and \u03be-absolute NSHA are also examined to better understand their properties. The practical relevance of NSHA is demonstrated through applications in various fields, highlighting its adaptability in addressing complex decision-making scenarios. This approach offers a novel, more precise method for navigating uncertainty in areas such as project methodology selection, sensitivity analysis, and AI chatbot selection.<\/jats:p>","DOI":"10.3390\/sym16101288","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T05:26:33Z","timestamp":1727760393000},"page":"1288","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Note on Neutrosophic Soft Set over Hyperalgebras"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3084-7694","authenticated-orcid":false,"given":"Serkan","family":"Onar","sequence":"first","affiliation":[{"name":"Department of Mathematical Engineering, Yildiz Technical University, \u0130stanbul 34220, T\u00fcrkiye"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,1]]},"reference":[{"key":"ref_1","unstructured":"Marty, F. (1934, January 14\u201318). Sur une generalization de la notion de group. Proceedings of the 8th Congres Mathematicians Scandinaves, Stockholm, Sweden."},{"key":"ref_2","first-page":"721","article-title":"A new characterization of fundamental relation on hyperrings","volume":"5","author":"Ameri","year":"2010","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_3","unstructured":"Corsini, P. (1993). Prolegomena of hypergroup theory. Supplement to Rivista di Matematica Pura ed Applicata, Aviani Editore."},{"key":"ref_4","unstructured":"Davvaz, B., and Leoreanu-Fotea, V. (2007). Hyperring Theory and Applications, International Academic Press."},{"key":"ref_5","first-page":"165","article-title":"Hypergroupes canoniques","volume":"2","author":"Mittas","year":"1972","journal-title":"Math. Balk."},{"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":"Inform. Control"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Mordeson, J.N., and Malik, M.S. (1998). Fuzzy Commutative Algebra, World Scientific Publishing Co. Pvt. Ltd.","DOI":"10.1142\/9789812815897"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0898-1221(99)00056-5","article-title":"Soft set theory-first results","volume":"37","author":"Molodtsov","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"11373","DOI":"10.3934\/math.2023576","article-title":"A weak form of soft \u03b1-open sets and its applications via soft topologies","volume":"8","author":"Mhemdi","year":"2023","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.3906\/mat-2005-117","article-title":"T-soft equality relation","volume":"44","year":"2020","journal-title":"Turk. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Al-Shami, T., Mhemdi, A., and Abu-Gdairi, R. (2023). A Novel framework for generalizations of soft open sets and its applications via soft topologies. Mathematics, 11.","DOI":"10.3390\/math11040840"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"35","DOI":"10.61356\/j.nswa.2023.41","article-title":"New Types of Soft Sets HyperSoft Set, IndetermSoft Set, IndetermHyperSoft Set, and TreeSoft Set: An Improved Version","volume":"8","author":"Smarandache","year":"2023","journal-title":"Neutrosophic Syst. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3308","DOI":"10.1016\/j.camwa.2010.03.015","article-title":"Soft matrix theory and its decision making","volume":"59","author":"Cagman","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_14","first-page":"109","article-title":"Fuzzy soft matrix theory and its applications in decision making","volume":"9","author":"Cagman","year":"2012","journal-title":"Iran. J. Fuzzy Systems"},{"key":"ref_15","first-page":"1100","article-title":"Intuitionistic Fuzzy Soft Matrix Theory and Its Application in Decision Making","volume":"2","author":"Rajarajeswari","year":"2013","journal-title":"Int. J. Eng. Res. Technol. (IJERT)"},{"key":"ref_16","unstructured":"Smarandache, F. (1998). A Unifying Eld in Logics. Neutrosophy\/Neutrosophic Probability, Set and Logic, American Research Press."},{"key":"ref_17","unstructured":"Kandasamy, W.B.V., and Smarandache, F. (2006). Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, Hexis."},{"key":"ref_18","first-page":"13","article-title":"Soft neutrosophic group","volume":"1","author":"Shabir","year":"2013","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_19","unstructured":"Wang, H., Smarandache, F., Zhang, Y., and Sunderraman, R. (2005, January 21\u201326). Single valued neutrosophic sets. Proceedings of the 10th International Conference on Fuzzy Theory and Technology, Salt Lake City, UT, USA."},{"key":"ref_20","first-page":"791","article-title":"On neutrosophic submodules of a module","volume":"46","author":"Cetkin","year":"2017","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1245","DOI":"10.3233\/IFS-130810","article-title":"On similarity and entropy of neutrosophic sets","volume":"26","author":"Majumdar","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"31","DOI":"10.9790\/5728-0343135","article-title":"Neutrosophic set and neutrosophic topological spaces","volume":"3","author":"Salama","year":"2012","journal-title":"IOSR J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"87","DOI":"10.54216\/IJNS.070201","article-title":"On refined neutrosophic r-module","volume":"7","author":"Olgun","year":"2020","journal-title":"Int. J. Neutrosophic Sci. (IJNS)"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"132","DOI":"10.32323\/ujma.383115","article-title":"An approach to neutrosophic ideals","volume":"1","author":"Cetkin","year":"2018","journal-title":"Univers. J. Math. Appl."},{"key":"ref_25","first-page":"157","article-title":"Neutrosophic soft set","volume":"5","author":"Maji","year":"2013","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_26","first-page":"169","article-title":"Neutrosophic soft relations and some properties","volume":"9","author":"Deli","year":"2015","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_27","first-page":"1","article-title":"Relations on interval valued neutrosophic soft sets","volume":"3","author":"Broumi","year":"2014","journal-title":"J. New Results Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4077","DOI":"10.3233\/JIFS-17999","article-title":"Bipolar neutrosophic soft sets and applications in decision making","volume":"33","author":"Ali","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_29","first-page":"481","article-title":"A new approach to operations on neutrosophic soft sets and to neutrosophic soft topological spaces","volume":"10","author":"Ozturk","year":"2019","journal-title":"Commu. Math. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s12190-020-01495-5","article-title":"Relations on neutrosophic soft set and their application in decision making","volume":"67","year":"2021","journal-title":"J. Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1007\/s12530-018-9247-7","article-title":"Neutrosophic soft set decision making for stock trending analysis","volume":"10","author":"Jha","year":"2019","journal-title":"Evol. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Afzal, U., Alrveili, H., Ahamd, N., and Aslam, M. (2021). Neutrosophic statistical analysis of resistance depending on the temperature variance of conducting material. Sci. Rep., 11.","DOI":"10.1038\/s41598-021-03347-z"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sherwani, R.A.K., Aslam, M., Shakeel, H., Abbas, K., and Jamal, F. (2021). Neutrosophic Statistics for Grouped Data: Theory and Applications. Neutrosophic Operational Research: Methods and Applications, Springer.","DOI":"10.1007\/978-3-030-57197-9_14"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"17","DOI":"10.52547\/HATEF.JAHLA.3.2.2","article-title":"Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra","volume":"3","author":"Smarandache","year":"2022","journal-title":"J. Algebr. Hyperstruct. Log. Algebr."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"548","DOI":"10.29020\/nybg.ejpam.v16i1.4637","article-title":"Single-valued Neutrosophic Soft sets in Hyper UP-Algebra","volume":"16","author":"Cano","year":"2023","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_36","first-page":"313","article-title":"A neutrosophic soft set approach to a decision making problem","volume":"3","author":"Maji","year":"2012","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"955","DOI":"10.3233\/JIFS-161548","article-title":"Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set","volume":"32","author":"Peng","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1007\/s13042-015-0461-3","article-title":"Interval-valued neutrosophic soft sets and its decision making","volume":"8","author":"Deli","year":"2017","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2233","DOI":"10.3233\/IFS-141505","article-title":"Neutrosophic soft matrices and NSM decision making","volume":"28","author":"Deli","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.asoc.2016.07.013","article-title":"Possibility neutrosophic soft sets and PNS-decision making method","volume":"54","author":"Karaaslan","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ulu\u00e7ay, V., \u015eahin, M., and Hassan, N. (2018). Generalized neutrosophic soft expert set for multiple-criteria decision-making. Symmetry, 10.","DOI":"10.3390\/sym10100437"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Abu Qamar, M., and Hassan, N. (2019). An approach toward a Q-neutrosophic soft set and its application in decision making. Symmetry, 11.","DOI":"10.3390\/sym11020139"},{"key":"ref_43","first-page":"183","article-title":"Multi-polar neutrosophic soft sets with application in medical diagnosis andDecision-making","volume":"33","author":"Saeed","year":"2020","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Acharjya, D.P., and Arya, D. (2021). Multicriteria decision-making using interval valued neutrosophic soft set. Artificial Intelligence and Global Society, Chapman and Hall\/CRC.","DOI":"10.1201\/9781003006602-8"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2007","DOI":"10.1007\/s13042-021-01289-6","article-title":"Distance, similarity and entropy measures of dynamic interval-valued neutrosophic soft sets and their application in decision making","volume":"12","author":"Dong","year":"2021","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2036","DOI":"10.1007\/s40815-022-01257-4","article-title":"The sequence of neutrosophic soft sets and a decision-making problem in medical diagnosis","volume":"24","author":"Bui","year":"2022","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"48","DOI":"10.54216\/IJNS.190103","article-title":"Review of Generalized Neutrosophic Soft Set in Solving Multiple Expert Decision Making Problems","volume":"19","author":"Priyadarsini","year":"2022","journal-title":"Int. J. Neutrosophic Sci. (IJNS)"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"745","DOI":"10.3233\/JIFS-212422","article-title":"Interval complex neutrosophic soft relations and their application in decision-making","volume":"43","author":"Ahmad","year":"2022","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"22127","DOI":"10.3934\/math.20231128","article-title":"An innovative decision-making framework for supplier selection based on a hybrid interval-valued neutrosophic soft expert set","volume":"8","author":"Ihsan","year":"2023","journal-title":"AIMS Math."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"62","DOI":"10.56578\/josa010203","article-title":"Optimizing hard disk selection via a fuzzy parameterized single-valued neutrosophic soft set approach","volume":"1","author":"Ihsan","year":"2023","journal-title":"J. Oper. Strateg. Anal."},{"key":"ref_51","first-page":"159","article-title":"A New Approach to Neutrosophic Soft Sets and their Application in Decision Making","volume":"60","author":"Mohanty","year":"2023","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1007\/s13370-023-01101-4","article-title":"Effective neutrosophic soft set theory and its application to decision-making","volume":"34","author":"Karatas","year":"2023","journal-title":"Afr. Mat."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1007\/s10462-022-10207-3","article-title":"A novel perspective for Q-neutrosophic soft relations and their application in decision making","volume":"56","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"ref_54","first-page":"22","article-title":"Decision-making techniques based on similarity measures of possibility neutrosophic soft expert sets","volume":"55","author":"Alotaibi","year":"2023","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1","DOI":"10.22436\/jmcs.031.01.01","article-title":"Applications of neutrosophic soft open sets in decision mak-ing via operation approach","volume":"31","author":"Rabunia","year":"2023","journal-title":"J. Math. Comput. Sci. (JMCS)"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"e35059","DOI":"10.1016\/j.heliyon.2024.e35059","article-title":"Utilizing aggregation operators based on q-rung orthopair neutrosophic soft sets and their applications in multi-attributes decision making problems","volume":"10","author":"Ali","year":"2024","journal-title":"Heliyon"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Zhang, H., Wang, L., Wang, D., Huang, Z., Yu, D., and Peng, Y. (2024). A Parameter Reduction-Based Decision-Making Method with Interval-Valued Neutrosophic Soft Sets for the Selection of Bionic Thin-Wall Structures. Biomimetics, 9.","DOI":"10.3390\/biomimetics9040208"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"e2024ss0110","DOI":"10.31893\/multiscience.2024ss0110","article-title":"Neutrosophic soft set matrix and their applications based on multi criteria decision-making problems","volume":"6","author":"Jayaprakash","year":"2024","journal-title":"Multidiscip. Sci. J."},{"key":"ref_59","first-page":"75","article-title":"Approach to Multi-Criteria Decision-Making in a Neutro sophic Picture Hyper-Soft Set Environment using Generalized Neutrosophic TOPSIS","volume":"67","author":"Kaur","year":"2024","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.camwa.2010.08.059","article-title":"Fuzzy hyperalgebras","volume":"61","author":"Ameri","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_61","first-page":"21","article-title":"Hyperalgebraic systems","volume":"6","author":"Ameri","year":"1999","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"5179","DOI":"10.3233\/JIFS-190016","article-title":"Fuzzy soft hyperalgebras","volume":"37","author":"Nozari","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_63","first-page":"118","article-title":"Introduction to neutrosophic soft groups","volume":"13","author":"Bera","year":"2016","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"5742288","DOI":"10.1155\/2021\/5742288","article-title":"A Study of Bipolar Fuzzy Soft Sets and Its Application in Decision-Making Problems","volume":"2021","author":"Mustafa","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_65","first-page":"643785","article-title":"Hesitant Fuzzy Soft Set and Its Applications in Multicriteria Decision Making","volume":"2014","author":"Wang","year":"2014","journal-title":"J. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1288\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:08:38Z","timestamp":1760112518000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1288"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,1]]},"references-count":65,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101288"],"URL":"https:\/\/doi.org\/10.3390\/sym16101288","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,1]]}}}