{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:41:50Z","timestamp":1777704110115,"version":"3.51.4"},"reference-count":36,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2018,7,31]],"date-time":"2018-07-31T00:00:00Z","timestamp":1532995200000},"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":[[2018,8,26]]},"abstract":"<jats:p>In this paper, we consider a left quantale module represented by Q-module as a universal set and we introduce the notion of rough Q-submodule with respect to a Q-submodule of Q-module. Some results about homomorphic images of rough Q-submodules are also generalized and improved in the field of Q-modules. Next, the concepts of set-valued homomorphism and strong set-valued homomorphism of Q-modules are introduced, and related properties are investigated. The notions of generalized lower and upper approximation operators, by means of a set-valued mapping are defined.<\/jats:p>","DOI":"10.3233\/jifs-17886","type":"journal-article","created":{"date-parts":[[2018,8,5]],"date-time":"2018-08-05T06:40:18Z","timestamp":1533451218000},"page":"2359-2372","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":22,"title":["Roughness in quantale modules"],"prefix":"10.1177","volume":"35","author":[{"given":"Saqib Mazher","family":"Qurashi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad, Pakistan"},{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"}]},{"given":"Muhammad","family":"Shabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"}]}],"member":"179","published-online":{"date-parts":[[2018,7,31]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500000189"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-011-0757-5"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4418-9"},{"key":"e_1_3_1_5_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, Bulletin of the Polish Academy of Sciences Mathematics 42 (1994), 251\u2013254.","journal-title":"Bulletin of the Polish Academy of Sciences Mathematics"},{"key":"e_1_3_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.03.009"},{"key":"e_1_3_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2005.05.009"},{"key":"e_1_3_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2003.10.001"},{"key":"e_1_3_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2008.03.014"},{"key":"e_1_3_1_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.02.060"},{"key":"e_1_3_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2011.03.030"},{"key":"e_1_3_1_12_2","first-page":"1","article-title":"An extension of the Galois theory of Grothendieck","volume":"309","author":"Joyal A.","year":"1984","unstructured":"JoyalA. and TierneyM., An extension of the Galois theory of Grothendieck, Mem Am Math Soc 309 (1984), 1\u201371.","journal-title":"Mem Am Math Soc"},{"key":"e_1_3_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(96)00274-5"},{"issue":"1","key":"e_1_3_1_14_2","first-page":"183","article-title":"Structure of rough sets and rough groups","volume":"5","author":"Kuroki N.","year":"1997","unstructured":"KurokiN. and MordesonJ.N., Structure of rough sets and rough groups, J Fuzzy Math 5(1) (1997), 183\u2013191.","journal-title":"J Fuzzy Math"},{"key":"e_1_3_1_15_2","doi-asserted-by":"publisher","DOI":"10.3233\/JIFS-161681"},{"issue":"2","key":"e_1_3_1_16_2","first-page":"99","volume":"12","author":"Mulvey C.J.","year":"1986","unstructured":"MulveyC.J., Rendiconti del Circolo Matematico di Palermo 12(2) (1986), 99\u2013104.","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"e_1_3_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01001956"},{"key":"e_1_3_1_18_2","doi-asserted-by":"crossref","unstructured":"QurashiS.M. and ShabirM. Generalized rough fuzzy ideals in quantales Discrete Dynamics in Nature and Society (2018) 11. Article ID 1085201.","DOI":"10.1155\/2018\/1085201"},{"issue":"4","key":"e_1_3_1_19_2","first-page":"329","article-title":"Modules over a quantale and models for the operator ! in linear logic","volume":"35","author":"Rosenthal K.I.","year":"1994","unstructured":"RosenthalK.I., Modules over a quantale and models for the operator ! in linear logic, Cah Topologie Geom Differ Categ 35(4) (1994), 329\u2013333.","journal-title":"Cah Topologie Geom Differ Categ"},{"key":"e_1_3_1_20_2","unstructured":"RosenthalK.I. The Theory of Quantaloids 1996\u2013Longman Essex."},{"issue":"2","key":"e_1_3_1_21_2","first-page":"405","article-title":"Quantales and their modules: Projective objects, ideals, and congruences","volume":"2","author":"Russo C.","year":"2016","unstructured":"RussoC., Quantales and their modules: Projective objects, ideals, and congruences, South American Journal of Logic 2(2) (2016), 405\u2013424.","journal-title":"South American Journal of Logic"},{"key":"e_1_3_1_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2012.11.012"},{"key":"e_1_3_1_23_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-011-0801-5"},{"key":"e_1_3_1_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2004.12.010"},{"key":"e_1_3_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(98)00012-7"},{"key":"e_1_3_1_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2012.07.042"},{"key":"e_1_3_1_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2009.12.026"},{"key":"e_1_3_1_28_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141214"},{"key":"e_1_3_1_29_2","doi-asserted-by":"publisher","DOI":"10.2298\/FIL1713167Y"},{"issue":"1","key":"e_1_3_1_30_2","first-page":"119","article-title":"On rough (m,n) bi-\u0393-hyperideals in \u0393-semihypergroups","volume":"75","author":"Yaqoob N.","year":"2013","unstructured":"YaqoobN., AslamM., DavvazB. and SaeidA.B., On rough (m,n) bi-\u0393-hyperideals in \u0393-semihypergroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 75(1) (2013), 119\u2013128.","journal-title":"UPB Scientific Bulletin, Series A: Applied Mathematics and Physics"},{"key":"e_1_3_1_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-012-0809-5"},{"key":"e_1_3_1_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2017.03.038"},{"key":"e_1_3_1_33_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2016.09.012"},{"issue":"2","key":"e_1_3_1_34_2","first-page":"153","article-title":"Roughness in nonassociative po-semihyprgroups based on pseudohyperorder relations","volume":"28","author":"Zhan J.","year":"2017","unstructured":"ZhanJ., YaqoobN. and KhanM., Roughness in nonassociative po-semihyprgroups based on pseudohyperorder relations, Journal of Multiple-Valued Logic and Soft Computing 28(2-3) (2017), 153\u2013177.","journal-title":"Journal of Multiple-Valued Logic and Soft Computing"},{"key":"e_1_3_1_35_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-016-2119-9"},{"key":"e_1_3_1_36_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2007.05.037"},{"key":"e_1_3_1_37_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2009.02.013"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-17886","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-17886","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-17886","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:40:27Z","timestamp":1777455627000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-17886"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,31]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,8,26]]}},"alternative-id":["10.3233\/JIFS-17886"],"URL":"https:\/\/doi.org\/10.3233\/jifs-17886","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,7,31]]}}}