{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:37:32Z","timestamp":1777703852367,"version":"3.51.4"},"reference-count":36,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2019,7,2]],"date-time":"2019-07-02T00:00:00Z","timestamp":1562025600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2019,10,25]]},"abstract":"<jats:p>\n                    \u00a0The generalization of different types of fuzzy sets is on the way since the discovery of fuzzy sets by Zadeh 1965 based on mathematical logic. Some of them are, the interval-valued fuzzy sets,\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets and interval-valued\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets, where\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets are used to the co-domain [\u22121, 0] for the membership function. Cubic sets are the combination of interval-valued fuzzy sets and fuzzy sets presented in 2012 by Jun et al. Since the cubic sets considered only the positive aspects of a certain thing in many physical problems and ignore the negative aspects totally. So to cover the negative part the need was felt to extend the concept of cubic sets along the co-domain [\u22121, 0]. Thus in this paper, we initiate a new theory by mixing the cubic sets with N-fuzzy sets and define the new notion of\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic sets. We are inspired by the hybrid nature of the cubic sets which can handle positive characteristics of a certain thing in a much better way as compared to the fuzzy sets. Whereas, our case will be more helpful in order to characterize the negative characteristics of a certain thing in a more affective way as compared to\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets. Motivating from the idea of cubic sets, we initiate to combine the interval-valued\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -fuzzy sets to produce a new class of cubic sets known as\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic sets based on mathematical logics. We define\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic sets and define many different operations like P-union (resp., P-intersection) R-union (resp., R-intersection) and complement. The notions of external and internal\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic sets and related operations and properties are investigated. We discuss some algebraic properties of\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic sets and define some aggregation operators which will be helpful in order to find more algebraic and practical applications in the future. At the end we present a\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"script\">N<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    -cubic-decision making Problem by using the developed procedure.\n                  <\/jats:p>","DOI":"10.3233\/jifs-182595","type":"journal-article","created":{"date-parts":[[2019,7,5]],"date-time":"2019-07-05T10:53:59Z","timestamp":1562324039000},"page":"5009-5023","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":10,"title":["N-Cubic sets and aggregation operators"],"prefix":"10.1177","volume":"37","author":[{"given":"Sheikh","family":"Rashid","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Gulistan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Young Bae","family":"Jun","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Gyeongsang National University, Jinju, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Salma","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seifedine","family":"Kadry","sequence":"additional","affiliation":[{"name":"Department of Mathematics &amp; Computer Science, Beirut Arab University, Beirut, Lebanon"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2019,7,2]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(75)90036-5"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(86)80034-3"},{"key":"e_1_3_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(89)90205-4"},{"key":"e_1_3_1_6_2","doi-asserted-by":"publisher","DOI":"10.4134\/CKMS.2010.25.2.173"},{"key":"e_1_3_1_7_2","first-page":"31","article-title":"Generlized N-ideals of subtraction algebra","volume":"09","author":"Williams P.","year":"2013","unstructured":"P.Williams and A.B.Saeid, Generlized N-ideals of subtraction algebra, The Journal of Uncertain System 09 (2013), 31\u201348.","journal-title":"The Journal of Uncertain System"},{"key":"e_1_3_1_8_2","first-page":"239","article-title":"Cubic sub-algebras and ideals of BCK\/BCI-algebras","volume":"44","author":"Jun Y.B.","year":"2010","unstructured":"Y.B.Jun, C.S.Kim and M.S.Kang, Cubic sub-algebras and ideals of BCK\/BCI-algebras, Far East Journal of Mathematical Sciences 44 (2010a), 239\u2013250.","journal-title":"Far East Journal of Mathematical Sciences"},{"issue":"1","key":"e_1_3_1_9_2","first-page":"83","article-title":"Cubic sets","volume":"4","author":"Jun Y.B.","year":"2012","unstructured":"Y.B.Jun, C.S.Kim and K.O.Yang, Cubic sets, Annals of Fuzzy Mathematics and Informatics 4(1) (2012), 83\u201398.","journal-title":"Annals of Fuzzy Mathematics and Informatics"},{"issue":"9","key":"e_1_3_1_10_2","first-page":"3334","article-title":"Cubic structures applied to ideals of BCI-algebras","volume":"62","author":"Jun Y.B.","year":"2011","unstructured":"Y.B.Jun, K.J.Lee and M.S.Kang, Cubic structures applied to ideals of BCI-algebras, ComputersandMathematicswith Applications 62(9) (2011), 3334\u20133342.","journal-title":"ComputersandMathematicswith Applications"},{"issue":"1","key":"e_1_3_1_11_2","first-page":"25","article-title":"Cubic q-ideals of BCI-algebras","volume":"1","author":"Jun Y.B.","year":"2011","unstructured":"Y.B.Jun, C.S.Kim and J.G.Kang. 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