{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:12:28Z","timestamp":1777702348620,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"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":[[2013,7]]},"abstract":"<jats:p>\n                    Using the notions of belonging (\u2208) and quasi-k-coincidence (q\n                    <jats:sub>k<\/jats:sub>\n                    ) of a fuzzy point with a fuzzy set, we define the concepts of $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy normal subgroups and $(\\overline{\\in }, \\overline{\\in } \\vee \\overline{q_{k}})$-fuzzy cosets which is a generalization of fuzzy normal subgroups, fuzzy coset, $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q})$-fuzzy normal subgroups and $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q})$-fuzzy cosets. We give characterizations of an $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy normal subgroup and $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy coset, and deal with several related properties. The important achievement of the study with an $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy normal subgroup and $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy cosets is the generalization of that the notions of fuzzy normal subgroups, fuzzy coset, $(\\overline{\\in} ,\\overline{\\in} \\vee \\overline{q})$-fuzzy normal subgroups and $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q})$-fuzzy cosets. We prove that the set of all $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$-fuzzy cosets of G is a group, where the multiplication is defined by $\\overleftarrow{f_{x}}\\cdot \\overleftarrow{f_{y}} = \\overleftarrow{f_{xy}}$ for all $x,y\\in G.$ If $\\widetilde{f}:F \\rightarrow [0,1]$ is defined by $\\widetilde{f}(\\overleftarrow{f_{x}}) = f(x) $ for all $x\\in G.$ Then $\\widetilde{f}$ is a fuzzy normal subgroup of F.\n                  <\/jats:p>","DOI":"10.3233\/ifs-2012-0612","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T17:51:49Z","timestamp":1575309109000},"page":"37-47","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":16,"title":["A new type of fuzzy normal subgroups and fuzzy cosets"],"prefix":"10.1177","volume":"25","author":[{"given":"Saleem","family":"Abdullah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Aslam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tazeem Ahmed","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Naeem","sequence":"additional","affiliation":[{"name":"Deanship of Preparatory Years, Umm al Qurrra University, Makkah, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2013,1]]},"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-2012-0612","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-2012-0612","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:35:37Z","timestamp":1777455337000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/IFS-2012-0612"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,7]]}},"alternative-id":["10.3233\/IFS-2012-0612"],"URL":"https:\/\/doi.org\/10.3233\/ifs-2012-0612","relation":{"is-cited-by":[{"id-type":"doi","id":"10.1155\/2014\/437324","asserted-by":"object"}]},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}