{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T18:46:19Z","timestamp":1769712379779,"version":"3.49.0"},"reference-count":54,"publisher":"SAGE Publications","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2023,8,1]]},"abstract":"<jats:p>This paper presents the concepts of a complex intuitionistic fuzzy subfield (CIFSF) and the direct product of a complex intuitionistic fuzzy subfield which is generalized from the concept of a complex fuzzy subfield by adding the notion of intuitionistic fuzzy into a complex fuzzy subfield. The main contribution and originality of this research are adding the non-membership term to the definition of a complex fuzzy subfield that assigns for any element a complex-valued grade. We expand the complex fuzzy subfield and obtain a new structure called CIFSF. This new concept is innovative in that it may attain a wider range of values for both membership and non-membership functions where these functions are expanded to the unit disc in the complex plane. Furthermore, we discuss that the direct product of two CIFSFs is CIFSF, and some related properties are investigated. In addition, we present the definition of necessity and possibility operators on the direct product of CIFSF, and some associated theorems are given. Finally, we propose the level subsets of the direct product of two complex intuitionistic fuzzy subsets of a field and prove that the level subset of the direct product of two CIFSFs is a subfield and discuss some related results.<\/jats:p>","DOI":"10.3233\/jifs-230597","type":"journal-article","created":{"date-parts":[[2023,5,23]],"date-time":"2023-05-23T12:23:05Z","timestamp":1684844585000},"page":"2111-2132","source":"Crossref","is-referenced-by-count":0,"title":["A note on direct product of complex intuitionistic fuzzy subfield"],"prefix":"10.1177","volume":"45","author":[{"given":"Adela","family":"Khamis","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia"},{"name":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout, Yemen"}]},{"given":"Abd Ghafur","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia"}]}],"member":"179","reference":[{"key":"10.3233\/JIFS-230597_ref1","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"},{"issue":"3","key":"10.3233\/JIFS-230597_ref2","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":"Journal of Mathematical Analysis and Applications"},{"issue":"3","key":"10.3233\/JIFS-230597_ref3","doi-asserted-by":"crossref","first-page":"252","DOI":"10.3390\/math7030252","article-title":"On (\u03b1,\u03b2)-US Sets in BCK\/BCI-Algebras","volume":"7","author":"Jana","year":"2019","journal-title":"Mathematics"},{"issue":"2","key":"10.3233\/JIFS-230597_ref4","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.3934\/math.2020072","article-title":"Subalgebras of type (\u03b1, \u03b2) based on m-polar fuzzy points in BCK\/BCI-Algebras","volume":"5","author":"Al-Masarwah","year":"2020","journal-title":"AIMS Mathematics"},{"issue":"3","key":"10.3233\/JIFS-230597_ref5","doi-asserted-by":"crossref","first-page":"96","DOI":"10.5899\/2017\/jfsva-00339","article-title":"Fuzzy subgroups on direct product of groups over a t-norm","volume":"2017","author":"Rasuli","year":"2017","journal-title":"Journal of Fuzzy Set Valued Analysis"},{"issue":"3","key":"10.3233\/JIFS-230597_ref6","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1504\/IJISTA.2017.085362","article-title":"Application of (\u03b1,\u03b2)-soft intersectional sets on BCK\/BCI-algebras","volume":"16","author":"Jana","year":"2017","journal-title":"International Journal of Intelligent Systems Technologies and Applications"},{"issue":"1","key":"10.3233\/JIFS-230597_ref7","first-page":"35","article-title":"On generalized direct product of fuzzy multigroups","volume":"3","author":"Ejegwa","year":"2020","journal-title":"Annals of Communications in Mathematics"},{"key":"10.3233\/JIFS-230597_ref8","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"},{"issue":"1","key":"10.3233\/JIFS-230597_ref9","first-page":"819","article-title":"The interaction between a Q-fuzzy normal subgroup and a Q-fuzzy characteristic subgroup","volume":"11","author":"Sailaja","year":"2021","journal-title":"Journal of Mathematical and Computational Science"},{"issue":"1","key":"10.3233\/JIFS-230597_ref10","first-page":"48","article-title":"The kernel of fuzzy and anti-fuzzy groups","volume":"1","author":"Ahmad","year":"2022","journal-title":"Journal of Neutrosophic and Fuzzy Systems"},{"issue":"5","key":"10.3233\/JIFS-230597_ref11","first-page":"752","article-title":"Direct product of finite fuzzy normal subrings over non-associative rings","volume":"17","author":"Kausar","year":"2019","journal-title":"International Journal of Analysis and Applications"},{"issue":"4","key":"10.3233\/JIFS-230597_ref12","doi-asserted-by":"crossref","first-page":"399","DOI":"10.22436\/jmcs.022.04.08","article-title":"Direct product of finite anti fuzzy normal sub-rings over non-associative rings","volume":"22","author":"Kausar","year":"2020","journal-title":"Journal of Mathematics and Computer Science"},{"issue":"2","key":"10.3233\/JIFS-230597_ref13","doi-asserted-by":"crossref","first-page":"139","DOI":"10.35834\/mjms\/1513306827","article-title":"(\u2208, \u2208\u00a0\u2228 q) -bipolar fuzzy BCK\/BCI -algebras","volume":"29","author":"Jana","year":"2017","journal-title":"Missouri Journal of Mathematical Sciences"},{"key":"10.3233\/JIFS-230597_ref14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2021\/6610009","article-title":"Generalized m-polar fuzzy positive implicative ideals of BCK-algebras","volume":"2021","author":"Al-Masarwah","year":"2021","journal-title":"Journal of Mathematics"},{"issue":"3","key":"10.3233\/JIFS-230597_ref15","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0165-0114(90)90034-4","article-title":"Fuzzy subfields","volume":"37","author":"Malik","year":"1990","journal-title":"Fuzzy Sets and Systems"},{"issue":"4","key":"10.3233\/JIFS-230597_ref16","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/02522667.2016.1228316","article-title":"Characterization of Q-fuzzy subrings (anti Q-fuzzy subrings) with respect to a t-norm (t-conorm)","volume":"39","author":"Rasuli","year":"2018","journal-title":"Journal of Information and Optimization Sciences"},{"issue":"1","key":"10.3233\/JIFS-230597_ref17","first-page":"14","article-title":"Some theorems in anti s-fuzzy subfield of a field","volume":"8","author":"Giri","year":"2018","journal-title":"International Journal of New Innovations in Engineering and Technology"},{"issue":"3","key":"10.3233\/JIFS-230597_ref18","first-page":"60","article-title":"A review on Q-fuzzy subgroups in algebra","volume":"14","author":"Hussain","year":"2019","journal-title":"International Journal of Applied Engineering Research"},{"issue":"2","key":"10.3233\/JIFS-230597_ref20","first-page":"19","article-title":"Intuitionistic fuzzy subgroups and subrings","volume":"25","author":"Hur","year":"2003","journal-title":"Honam Mathematical J"},{"issue":"4","key":"10.3233\/JIFS-230597_ref21","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.fiae.2017.05.002","article-title":"Generalized intuitionistic fuzzy ideals of BCK\/BCI-algebras based on 3-valued logic and its computational study","volume":"9","author":"Jana","year":"2017","journal-title":"Fuzzy Information and Engineering"},{"issue":"5","key":"10.3233\/JIFS-230597_ref22","doi-asserted-by":"crossref","first-page":"5217","DOI":"10.3233\/JIFS-169805","article-title":"Intuitionistic fuzzy parameterized fuzzy soft set theory and its application","volume":"35","author":"Joshi","year":"2018","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"issue":"1","key":"10.3233\/JIFS-230597_ref23","doi-asserted-by":"crossref","first-page":"139","DOI":"10.3233\/JIFS-18103","article-title":"(T, S) \u2013Intuitionistic fuzzy algebras","volume":"36","author":"Yilmaz","year":"2019","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"issue":"1","key":"10.3233\/JIFS-230597_ref24","doi-asserted-by":"crossref","first-page":"821","DOI":"10.3233\/JIFS-179451","article-title":"Intuitionistic fuzzy control of twin rotor multiple input multiple output systems","volume":"38","author":"Castillo","year":"2020","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"issue":"9","key":"10.3233\/JIFS-230597_ref25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/math8091594","article-title":"Intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals","volume":"8","author":"Abed Alhaleem","year":"2020","journal-title":"Mathematics"},{"key":"10.3233\/JIFS-230597_ref26","doi-asserted-by":"crossref","first-page":"5231","DOI":"10.3233\/JIFS-191806","article-title":"Fixed point results for intuitionistic fuzzy mappings and an application","volume":"38","author":"Nguyen","year":"2020","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"1","key":"10.3233\/JIFS-230597_ref27","doi-asserted-by":"crossref","first-page":"1176619","DOI":"10.1080\/23311835.2016.1176619","article-title":"Direct product of general intuitionistic fuzzy sets of subtraction algebras","volume":"3","author":"Gulistan","year":"2016","journal-title":"Cogent Mathematics"},{"issue":"1","key":"10.3233\/JIFS-230597_ref28","doi-asserted-by":"crossref","first-page":"563","DOI":"10.3233\/JIFS-219212","article-title":"Intuitionistic fuzzy two-factor variance analysis of movie ticket sales","volume":"42","author":"Traneva","year":"2021","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"3","key":"10.3233\/JIFS-230597_ref29","first-page":"341","article-title":"Intuitionistic fuzzy normal subrings over normed rings","volume":"19","author":"Abed Alhaleem","year":"2021","journal-title":"International Journal of Analysis and Applications"},{"issue":"2","key":"10.3233\/JIFS-230597_ref30","doi-asserted-by":"crossref","first-page":"609","DOI":"10.17576\/jsm-2022-5102-24","article-title":"Intuitionistic anti fuzzy normal subrings over normed rings","volume":"51","author":"Abed Alhaleem","year":"2022","journal-title":"Sains Malaysiana"},{"key":"10.3233\/JIFS-230597_ref31","first-page":"58","article-title":"Intuitionistic fuzzy subfield and its characterizations","volume":"2","author":"Zhang","year":"2010","journal-title":"International Conference on Intelligent Human-Machine Systems and Cybernetics"},{"issue":"4","key":"10.3233\/JIFS-230597_ref32","first-page":"1007","article-title":"Intuitionistic fuzzy subfields of a field with respect to (t, s)-norm","volume":"12","author":"Anandh","year":"2017","journal-title":"Advances in Fuzzy Mathematics"},{"issue":"4","key":"10.3233\/JIFS-230597_ref33","first-page":"1955","article-title":"Some theorems in intuitionistic multi fuzzy subfields of a field","volume":"9","author":"Vasu","year":"2020","journal-title":"Advances in Mathematics: Scientific Journal"},{"issue":"2","key":"10.3233\/JIFS-230597_ref34","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1109\/91.995119","article-title":"Complex fuzzy sets","volume":"10","author":"Ramot","year":"2002","journal-title":"IEEE Transactions on Fuzzy Systems"},{"issue":"3","key":"10.3233\/JIFS-230597_ref35","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.3233\/JIFS-162428","article-title":"Operations on complex multi-fuzzy sets","volume":"33","author":"Al-Qudah","year":"2017","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"key":"10.3233\/JIFS-230597_ref36","first-page":"464","article-title":"Complex intuitionistic fuzzy sets","volume":"1482","author":"Alkouri","year":"2012","journal-title":"International Conference on Fundamental and Applied Sciences"},{"issue":"4","key":"10.3233\/JIFS-230597_ref37","first-page":"563","article-title":"Complex fuzzy subrings","volume":"117","author":"Alsarahead","year":"2017","journal-title":"International Journal of Pure and Applied Mathematics"},{"issue":"3","key":"10.3233\/JIFS-230597_ref38","doi-asserted-by":"crossref","first-page":"455","DOI":"10.12732\/ijpam.v115i3.1","article-title":"Complex intuitionistic fuzzy normal subgroup","volume":"115","author":"Al-Husban","year":"2017","journal-title":"International Journal of Pure and Applied Mathematics"},{"issue":"15","key":"10.3233\/JIFS-230597_ref39","doi-asserted-by":"crossref","first-page":"4959","DOI":"10.1007\/s00500-018-3202-1","article-title":"An advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making","volume":"22","author":"Garg","year":"2018","journal-title":"Soft Computing"},{"issue":"2","key":"10.3233\/JIFS-230597_ref40","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1007\/s10489-018-1290-3","article-title":"A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making","volume":"49","author":"Garg","year":"2019","journal-title":"Applied Intelligence"},{"key":"10.3233\/JIFS-230597_ref41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/9704466","article-title":"A decision-making approach for the evaluation of information security management under complex intuitionistic fuzzy set Environment","volume":"2022","author":"Azam","year":"2022","journal-title":"Journal of Mathematics"},{"key":"10.3233\/JIFS-230597_ref42","first-page":"496","article-title":"Similarity measures on interval-complex neutrosophic soft sets with applications to decision making and medical diagnosis under uncertainty","volume":"51","author":"Al-Sharqi","year":"2022","journal-title":"Neutrosophic Sets and Systems Volume"},{"issue":"2","key":"10.3233\/JIFS-230597_ref43","first-page":"24","article-title":"Complex intuitionistic fuzzy subrings","volume":"38","author":"Alsarahead","year":"2017","journal-title":"Borneo Science"},{"key":"10.3233\/JIFS-230597_ref45","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"},{"issue":"1","key":"10.3233\/JIFS-230597_ref46","doi-asserted-by":"crossref","first-page":"582","DOI":"10.2991\/ijcis.d.210106.001","article-title":"Generalized direct product of complex intuitionistic fuzzy subrings","volume":"14","author":"Gulzar","year":"2021","journal-title":"International Journal of Computational Intelligence Systems"},{"issue":"5","key":"10.3233\/JIFS-230597_ref47","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.14704\/nq.2022.20.5.NQ22245","article-title":"On fundamental algebraic properties of complex anti Q \u2013fuzzy subrings","volume":"20","author":"Rifayathali","year":"2022","journal-title":"NeuroQuantology"},{"issue":"3","key":"10.3233\/JIFS-230597_ref48","doi-asserted-by":"crossref","first-page":"7032","DOI":"10.3934\/math.2023355","article-title":"On fundamental algebraic characterizations of complex intuitionistic Q-fuzzy subfield","volume":"8","author":"Khamis","year":"2023","journal-title":"AIMS Mathematics"},{"issue":"1","key":"10.3233\/JIFS-230597_ref49","first-page":"1","article-title":"Interval-valued neutrosophic soft expert set from real space to complex space","volume":"131","author":"Al-Sharqi","year":"2022","journal-title":"Computer Modeling in Engineering & Sciences"},{"issue":"1","key":"10.3233\/JIFS-230597_ref50","first-page":"149","article-title":"Interval-valued complex neutrosophic soft set and its applications in decision-making","volume":"40","author":"Al-Sharqi","year":"2021","journal-title":"Neutrosophic Sets and Systems"},{"issue":"1","key":"10.3233\/JIFS-230597_ref51","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":"Al-Sharqi","year":"2022","journal-title":"Journal of Intelligent and Fuzzy Systems"},{"issue":"1","key":"10.3233\/JIFS-230597_ref52","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.3233\/JIFS-221579","article-title":"Fuzzy parameterized-interval complex neutrosophic soft sets and their applications under uncertainty","volume":"44","author":"Al-Sharqi","year":"2023","journal-title":"Journal of Intelligent & Fuzzy Systems"},{"issue":"2","key":"10.3233\/JIFS-230597_ref53","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":"Indonesian Journal of Electrical Engineering and Computer Science"},{"issue":"1","key":"10.3233\/JIFS-230597_ref54","first-page":"17","article-title":"Review and new results on intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"2016","journal-title":"International Journal Bioautomation"},{"issue":"8","key":"10.3233\/JIFS-230597_ref55","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1016\/j.ijar.2009.05.010","article-title":"Operation properties and -equalities of complex fuzzy sets","volume":"50","author":"Zhang","year":"2009","journal-title":"International Journal of Approximate Reasoning"},{"issue":"987","key":"10.3233\/JIFS-230597_ref56","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1063\/1.4858782","article-title":"Some operations on complex Atanassov\u2019s intuitionistic fuzzy sets","volume":"1571","author":"Alkouri","year":"2013","journal-title":"AIP Conference Proceedings"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/JIFS-230597","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T07:48:55Z","timestamp":1769672935000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/JIFS-230597"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,1]]},"references-count":54,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.3233\/jifs-230597","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,1]]}}}