{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:37:01Z","timestamp":1777703821044,"version":"3.51.4"},"reference-count":25,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T00:00:00Z","timestamp":1531353600000},"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 & Fuzzy Systems"],"published-print":{"date-parts":[[2018,7,27]]},"abstract":"In this paper, we introduce the notion of reference point, lower and upper approximation with respect to reference point by a Lie algebra. We are concerned with some important properties of them.<\/jats:p>\n \n For a fuzzy Lie subalgebra\n \u03bc<\/jats:italic>\n of a Lie algebra\n L<\/jats:italic>\n ,\n t<\/jats:italic>\n -level relation\n U<\/jats:italic>\n \u00a0(\n \u03bc<\/jats:italic>\n ,\n t<\/jats:italic>\n )\u00a0:\u00a0=\u00a0{(\n x<\/jats:italic>\n ,\n y<\/jats:italic>\n )\u00a0\u2208\u00a0\n L<\/jats:italic>\n \u00a0\u00d7\u00a0\n L<\/jats:italic>\n \u00a0:\u00a0\n \u03bc<\/jats:italic>\n \u00a0(\n x<\/jats:italic>\n \u00a0-\u00a0\n y<\/jats:italic>\n )\u00a0\u2265\u00a0\n t<\/jats:italic>\n ,\n \u03bc<\/jats:italic>\n \u00a0([\n x<\/jats:italic>\n ,\n y<\/jats:italic>\n ])\u00a0\u2265\u00a0\n t<\/jats:italic>\n } on\n L<\/jats:italic>\n , and\n t<\/jats:italic>\n -level relation with respect to the reference point\n a<\/jats:italic>\n ,\n U<\/jats:italic>\n \n e<\/jats:italic>\n <\/jats:sub>\n \u00a0(\n \u03bc<\/jats:italic>\n ,\n t<\/jats:italic>\n ,\n a<\/jats:italic>\n )\u00a0:\u00a0=\u00a0{(\n x<\/jats:italic>\n ,\n y<\/jats:italic>\n )\u00a0\u2208\u00a0\n L<\/jats:italic>\n \u00a0\u00d7\u00a0\n L<\/jats:italic>\n \u00a0:\u00a0\n \u03bc<\/jats:italic>\n \u00a0([\n a<\/jats:italic>\n ,\n x<\/jats:italic>\n \u00a0-\u00a0\n y<\/jats:italic>\n ])\u00a0\u2265\u00a0\n t<\/jats:italic>\n }, are equivalence relations on\n L<\/jats:italic>\n , for every\n t<\/jats:italic>\n \u00a0\u2208\u00a0[0, 1] and every\n a<\/jats:italic>\n \u00a0\u2208\u00a0\n L<\/jats:italic>\n . We study lower and upper approximation with respect to the equivalence relations\n U<\/jats:italic>\n \u00a0(\n \u03bc<\/jats:italic>\n ,\n t<\/jats:italic>\n ) and\n U<\/jats:italic>\n \n e<\/jats:italic>\n <\/jats:sub>\n \u00a0(\n \u03bc<\/jats:italic>\n ,\n t<\/jats:italic>\n ,\n a<\/jats:italic>\n ) on a Lie algebra\n L<\/jats:italic>\n , for every\n t<\/jats:italic>\n \u00a0\u2208\u00a0[0, 1] and every\n a<\/jats:italic>\n \u00a0\u2208\u00a0\n L<\/jats:italic>\n . Furthermore, we show that if\n \u03bc<\/jats:italic>\n is a fuzzy Lie subalgebra of a Lie algebra\n L<\/jats:italic>\n ,\n a<\/jats:italic>\n \u00a0\u2208\u00a0\n L<\/jats:italic>\n and\n t<\/jats:italic>\n \u00a0\u2208\u00a0[0, 1] then\n \n \n \n Fix<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n (<\/mml:mo>\n U<\/mml:mi>\n (<\/mml:mo>\n \u03bc<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n ,\n \n \n \n Fix<\/mml:mi>\n _<\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n U<\/mml:mi>\n (<\/mml:mo>\n \u03bc<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n ,\n \n \n \n Fix<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n (<\/mml:mo>\n \n U<\/mml:mi>\n e<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n \u03bc<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n and\n \n \n \n Fix<\/mml:mi>\n _<\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n \n U<\/mml:mi>\n e<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n \u03bc<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n are distributive complete lattices with respect to inclusion, where\n \n \n \n Fix<\/mml:mi>\n _<\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n \u03b8<\/mml:mi>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n (\n \n \n \n Fix<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n (<\/mml:mo>\n \u03b8<\/mml:mi>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n ) stand for the set of fixed points of upper (lower) equivalence relation\n \u03b8<\/jats:italic>\n . Also we obtain some relationship between ideals of a Lie algebra\n L<\/jats:italic>\n and rough ideals with respect to the equivalence relations\n U<\/jats:italic>\n \n e<\/jats:italic>\n <\/jats:sub>\n \u00a0(\n \u03bc<\/jats:italic>\n ,\n t<\/jats:italic>\n ,\n a<\/jats:italic>\n ) on\n L<\/jats:italic>\n .\n <\/jats:p>","DOI":"10.3233\/jifs-171620","type":"journal-article","created":{"date-parts":[[2018,7,13]],"date-time":"2018-07-13T13:35:37Z","timestamp":1531488937000},"page":"887-899","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":2,"title":["Fuzzy roughness in Lie algebra by reference point"],"prefix":"10.1177","volume":"35","author":[{"given":"Hossien","family":"Eghdami","sequence":"first","affiliation":[{"name":"Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran"}]},{"given":"Ali Akbar","family":"Estaji","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran"}]},{"given":"Toktam","family":"Haghdadi","sequence":"additional","affiliation":[{"name":"Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran"}]}],"member":"179","published-online":{"date-parts":[[2018,7,12]]},"reference":[{"key":"e_1_3_2_2_2","first-page":"251","article-title":"Rough groups and rough subgroups","volume":"42","author":"Biswas R.","year":"1994","unstructured":"BiswasR. and NandaS., Rough groups and rough subgroups, Bull Polish Acad Sic Math 42 (1994), 251\u2013254.","journal-title":"Bull Polish Acad Sic Math"},{"key":"e_1_3_2_3_2","volume-title":"Honorary Volume dedicated to prof Emeritus J Mittas","author":"Corsini P.","year":"1999","unstructured":"CorsiniP., Rough sets, fuzzy sets and join spaces, Honorary Volume dedicated to prof Emeritus J Mittas, Aristotle Univ of Thessaloniki, 1999-2000."},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02871859"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2005.10.001"},{"issue":"2","key":"e_1_3_2_6_2","first-page":"49","article-title":"Rough sets in a fundamental ring","volume":"24","author":"Davvaz B.","year":"1998","unstructured":"DavvazB., Rough sets in a fundamental ring, Bull Iranian Math Soc 24 (2) (1998), 49\u201361.","journal-title":"Bull Iranian Math Soc"},{"issue":"6","key":"e_1_3_2_7_2","first-page":"109","article-title":"Roughness in modules by using the notation of reference points","volume":"10","author":"Davvaz B.","year":"2013","unstructured":"DavvazB. and MalekzadeA., Roughness in modules by using the notation of reference points, Iraninan Journal of Fuzzy System 10 (6) (2013), 109\u2013124.","journal-title":"Iraninan Journal of Fuzzy System"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1080\/03081079008935107"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2011.04.043"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.02.060"},{"key":"e_1_3_2_11_2","unstructured":"FultonB. Gonzalez Lecture Notes on Lie algebras 2007."},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.22436\/jmcs.013.04.02"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2010.04.023"},{"key":"e_1_3_2_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(96)00230-8"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2005.01.001"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(96)00274-5"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(95)00282-0"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10462-016-9490-x"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01001956"},{"key":"e_1_3_2_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(95)00109-3"},{"key":"e_1_3_2_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"e_1_3_2_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2017.03.038"},{"key":"e_1_3_2_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2015.04.039"},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-014-1528-x"},{"key":"e_1_3_2_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2016.09.012"},{"key":"e_1_3_2_26_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-016-2119-9"}],"container-title":["Journal of Intelligent & Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-171620","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-171620","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-171620","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:39:42Z","timestamp":1777455582000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-171620"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,12]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,7,27]]}},"alternative-id":["10.3233\/JIFS-171620"],"URL":"https:\/\/doi.org\/10.3233\/jifs-171620","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,12]]}}}