{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:08:45Z","timestamp":1776845325028,"version":"3.51.2"},"reference-count":50,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:00:00Z","timestamp":1627603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (\u03b1,\u03b2)-complex fuzzy sets and then define \u03b1,\u03b2-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (\u03b1,\u03b2)-complex fuzzy subgroup and define (\u03b1,\u03b2)-complex fuzzy normal subgroups of given group. We extend this ideology to define (\u03b1,\u03b2)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (\u03b1,\u03b2)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (\u03b1,\u03b2)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (\u03b1,\u03b2)-complex fuzzy subgroup of the classical quotient group and show that the set of all (\u03b1,\u03b2)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of \u03b1,\u03b2-complex fuzzy subgroups and investigate the (\u03b1,\u03b2)-complex fuzzification of Lagrange\u2019s theorem analog to Lagrange\u2019 theorem of classical group theory.<\/jats:p>","DOI":"10.3390\/e23080992","type":"journal-article","created":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T12:59:24Z","timestamp":1627649964000},"page":"992","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["A Novel Algebraic Structure of (\u03b1,\u03b2)-Complex Fuzzy Subgroups"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7845-7751","authenticated-orcid":false,"given":"Hanan","family":"Alolaiyan","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2371-4287","authenticated-orcid":false,"given":"Halimah A.","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, King Saud University, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5097-8099","authenticated-orcid":false,"given":"Muhammad Haris","family":"Mateen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-1942","authenticated-orcid":false,"given":"Dragan","family":"Pamucar","sequence":"additional","affiliation":[{"name":"Department of Logistics, Military Academy, University of Defence in Belgrade, 11000 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6794-7942","authenticated-orcid":false,"given":"Muhammad","family":"Gulzar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,30]]},"reference":[{"key":"ref_1","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_2","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_3","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/0165-0114(82)90003-3","article-title":"Fuzzy invariant subgroups and fuzzy ideals","volume":"8","author":"Liu","year":"1982","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","first-page":"1005","article-title":"Direct product of fuzzy groups and fuzzy rings","volume":"6","author":"Yetkin","year":"2011","journal-title":"Int. Math. Forum."},{"key":"ref_5","first-page":"349","article-title":"Anti fuzzy ideal of ring","volume":"25","author":"Azam","year":"2013","journal-title":"Annal. Fuzzy Math. Inform."},{"key":"ref_6","first-page":"187","article-title":"Sum and product of fuzzy ideals of ring","volume":"13","author":"Kellil","year":"2018","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0165-0114(89)90122-X","article-title":"fuzzy complex numbers","volume":"33","author":"Buckley","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0020-0255(94)90084-1","article-title":"Fuzzy orders relative to fuzzy subgroups","volume":"80","author":"Kim","year":"1994","journal-title":"Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/0165-0114(94)90175-9","article-title":"Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups","volume":"61","author":"Ajmal","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0165-0114(98)00411-4","article-title":"On product of fuzzy subgroups","volume":"105","author":"Ray","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1109\/91.995119","article-title":"Complex fuzzy sets","volume":"10","author":"Ramot","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1109\/TFUZZ.2003.814832","article-title":"Complex fuzzy logic","volume":"11","author":"Ramot","year":"2003","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1016\/j.ijar.2009.05.010","article-title":"Operation properties and \u03b4-equalities of complex fuzzy sets","volume":"50","author":"Zhang","year":"2009","journal-title":"Int. J. Approx. Reason."},{"key":"ref_15","first-page":"5","article-title":"Application of complex fuzzy sets","volume":"6","author":"Thirunavukarasu","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_16","first-page":"251","article-title":"Engel fuzzy subgroups","volume":"34","author":"Ameri","year":"2015","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1016\/j.asoc.2016.05.055","article-title":"Relations between complex vague soft sets","volume":"47","author":"Selvachandran","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_18","first-page":"1433","article-title":"Complex fuzzy group based on complex fuzzy space","volume":"12","author":"Salleh","year":"2016","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3227","DOI":"10.3233\/IFS-152065","article-title":"A novel framework of complex valued fuzzy metric spaces and fixed point theorems","volume":"164","author":"Singh","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_20","first-page":"243","article-title":"Energy of complex fuzzy graph","volume":"10","author":"Thirunavukarasu","year":"2016","journal-title":"Int. J. Math. Sci. Eng. Appl."},{"key":"ref_21","first-page":"2011","article-title":"Complex fuzzy subgroups","volume":"11","author":"Alsarahead","year":"2017","journal-title":"Appl. Math. Sci."},{"key":"ref_22","first-page":"563","article-title":"Complex fuzzy subrings","volume":"117","author":"Alsarahead","year":"2017","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_23","first-page":"17","article-title":"Complex fuzzy soft subgroups","volume":"13","author":"Alsarahead","year":"2017","journal-title":"J. Qual. Meas. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"979","DOI":"10.3233\/JIFS-16228","article-title":"A class of fuzzy subgroups of finite reflection groups","volume":"33","author":"Makamba","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_25","first-page":"1","article-title":"Results on Engel fuzzy subgroups","volume":"4","author":"Mohamadzadeh","year":"2017","journal-title":"Algeb. Struc. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Mohamadzadeh, E., and Borzouei, R.A. (2018). Nilpotent fuzzy subgroups. Mathematics, 6.","DOI":"10.3390\/math6020027"},{"key":"ref_27","first-page":"373","article-title":"Fuzzy homomorphism theorem on groups","volume":"26","author":"Addis","year":"2018","journal-title":"Korean J. Math"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Emniyet, A., and Sahin, M. (2018). Fuzzy normed rings. Symmetry, 10.","DOI":"10.3390\/sym10100515"},{"key":"ref_29","first-page":"1","article-title":"A study of (\u03b1, \u03b2)-complex fuzzy hyperideals in non-associative hyperrings","volume":"36","author":"Gulistan","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2419","DOI":"10.3233\/JIFS-182725","article-title":"M-hazy lattices and its induced fuzzifying convexities","volume":"37","author":"Liu","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","first-page":"265","article-title":"Complex fuzzy sets with application in BCK\/BCI-Algebras","volume":"48","author":"Yun","year":"2019","journal-title":"Bull. Sect. Log."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2017.01.010","article-title":"A systematic review of complex fuzzy sets and logic","volume":"338","author":"Yazdanbakhsh","year":"2018","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6343","DOI":"10.3233\/JIFS-181131","article-title":"The approximate parallelity of complex fuzzy sets","volume":"6","author":"Hu","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2247","DOI":"10.3233\/JIFS-172264","article-title":"Distances of complex fuzzy sets and continuity of complex fuzzy operations","volume":"35","author":"Hu","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Al-Qudah, Y., Hassan, M., and Hassan, N. (2019). Fuzzy parameterized complex multi-fuzzy soft expert set theory and its application in decision making. Symmetry, 11.","DOI":"10.3390\/sym11030358"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3351","DOI":"10.3233\/JIFS-191718","article-title":"The distance measures and cross-entropy based on complex fuzzy sets and their application in decision making","volume":"39","author":"Liu","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.ins.2019.12.008","article-title":"Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information","volume":"514","author":"Alcantud","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"41615","DOI":"10.1109\/ACCESS.2020.2976731","article-title":"The Soft sets and fuzzy sets based neural networks and application","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1048","DOI":"10.11591\/ijeecs.v21.i2.pp1048-1056","article-title":"A note on complex fuzzy subfield","volume":"21","author":"Gulzar","year":"2021","journal-title":"Indones. J. Electr. Comput. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"295","DOI":"10.22436\/jmcs.022.03.08","article-title":"On some characterization of Q complex fuzzy subrings","volume":"22","author":"Gulzar","year":"2021","journal-title":"J. Math. Comput. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.fss.2018.11.001","article-title":"The annihilator of fuzzy subgroups","volume":"369","author":"Trevijano","year":"2019","journal-title":"Fuzzy Sets Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"196075","DOI":"10.1109\/ACCESS.2020.3034626","article-title":"A Novel Applications of Complex Intuitionistic Fuzzy Sets in Group Theory","volume":"8","author":"Gulzar","year":"2020","journal-title":"IEEE Access"},{"key":"ref_43","first-page":"265","article-title":"Algebraic properties of \u03c9-Q-fuzzy subgroup","volume":"15","author":"Gulzar","year":"2020","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"163260","DOI":"10.1109\/ACCESS.2020.3020366","article-title":"A Certain Class of t-Intuitionistic Fuzzy Subgroups","volume":"8","author":"Gulzar","year":"2020","journal-title":"IEEE Access"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Imtiaz, A., Shuaib, U., Razaq, A., and Gulistan, M. (2021). Image development in the framework of \u03b6\u2013complex fuzzy morphisms. J. Intell. Fuzzy Syst., 1\u201313. Preprint.","DOI":"10.3233\/JIFS-201261"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2038724","DOI":"10.1155\/2020\/2038724","article-title":"On Structural Properties of -Complex Fuzzy Sets and Their Applications","volume":"2020","author":"Imtiaz","year":"2020","journal-title":"Complexity"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Altassan, A., Mateen, M.H., and Pamucar, D. (2021). On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product. Symmetry, 13.","DOI":"10.3390\/sym13060998"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3463","DOI":"10.1007\/BF02355446","article-title":"Monomial algebras","volume":"87","author":"Belov","year":"1997","journal-title":"J. Math. Sci."},{"key":"ref_49","unstructured":"Latyshev, V.N. (1987). Combinatorial Ring Theory. Complexity of Algebraic Algorithms, Moscow State University Publishing."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1134\/S0005117921050106","article-title":"Sharpness Estimation of Combinatorial Generalization Ability Bounds for Threshold Decision Rules","volume":"82","author":"Ishkina","year":"2021","journal-title":"Autom. Remote Control"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/992\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:37:56Z","timestamp":1760164676000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/992"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,30]]},"references-count":50,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23080992"],"URL":"https:\/\/doi.org\/10.3390\/e23080992","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,30]]}}}