{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T17:39:53Z","timestamp":1776361193733,"version":"3.51.2"},"reference-count":41,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["New Math. and Nat. Computation"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p> The primary purpose of this paper is to introduce the concept of the fuzzy module of fractions and investigate a relationship between fuzzy modules of fractions and rough sets (RSs). In this respect, first, the concept of the soft module of fractions is introduced. Then, the idea of fuzzy approximations of the FS of a module of fractions is defined by using a soft module of fractions and obtaining a new hybrid model called multi-granulation soft rough FSs (MGSR-FSs) in a module of fractions. Moreover, the idea of a fuzzy upper rough module of fractions is introduced. However, it is important to note that the fuzzy lower approximation of a fuzzy module of fractions is not a fuzzy lower rough module of fractions. Several examples are provided to elaborate on the proposed notions. <\/jats:p>","DOI":"10.1142\/s1793005724500145","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T08:23:59Z","timestamp":1680251039000},"page":"247-271","source":"Crossref","is-referenced-by-count":6,"title":["A Study of Modules of Fractions Under Fuzzy Sets and Soft Sets"],"prefix":"10.1142","volume":"20","author":[{"given":"Saba","family":"Ayub","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44230, Pakistan"}]},{"given":"Waqas","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44230, Pakistan"}]},{"given":"Muhammad","family":"Shabir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44230, Pakistan"}]},{"given":"Rizwan","family":"Gul","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44230, Pakistan"}]}],"member":"219","published-online":{"date-parts":[[2023,7,8]]},"reference":[{"key":"S1793005724500145BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.03.034"},{"key":"S1793005724500145BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2006.12.008"},{"key":"S1793005724500145BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.11.009"},{"key":"S1793005724500145BIB004","doi-asserted-by":"publisher","DOI":"10.11650\/twjm\/1500406424"},{"issue":"4","key":"S1793005724500145BIB005","first-page":"177","volume":"1","author":"Aslam M.","year":"2012","journal-title":"Annals of Fuzzy Mathematics and Informatics"},{"key":"S1793005724500145BIB006","unstructured":"M. F. Atiyah and  I. G. Macdonald ,  Introduction to Commutative Algebra  (Westview Press Incorporated,  1969) pp.  139."},{"key":"S1793005724500145BIB007","doi-asserted-by":"publisher","DOI":"10.30538\/oms2018.0027"},{"issue":"196","key":"S1793005724500145BIB008","first-page":"1","volume":"8","author":"Ayub S.","year":"2019","journal-title":"Computational and Applied Mathematics"},{"key":"S1793005724500145BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s40314-020-1098-8"},{"key":"S1793005724500145BIB010","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2022.3218695"},{"key":"S1793005724500145BIB011","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-011-0752-x"},{"key":"S1793005724500145BIB012","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2019.2927317"},{"key":"S1793005724500145BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-010-0633-8"},{"key":"S1793005724500145BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2003.10.001"},{"issue":"3","key":"S1793005724500145BIB015","first-page":"59","volume":"1","author":"Davvaz B.","year":"2006","journal-title":"Iranian Journal of Fuzzy Systems"},{"key":"S1793005724500145BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2006.02.014"},{"key":"S1793005724500145BIB017","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-009-0465-6"},{"key":"S1793005724500145BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(95)00282-0"},{"key":"S1793005724500145BIB019","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(91)90037-U"},{"key":"S1793005724500145BIB020","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-016-2077-2"},{"key":"S1793005724500145BIB021","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(90)90043-A"},{"key":"S1793005724500145BIB022","doi-asserted-by":"publisher","DOI":"10.3233\/JIFS-162177"},{"key":"S1793005724500145BIB023","doi-asserted-by":"publisher","DOI":"10.3233\/JIFS-171037"},{"key":"S1793005724500145BIB024","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(03)00016-6"},{"key":"S1793005724500145BIB025","doi-asserted-by":"publisher","DOI":"10.2298\/FIL1305789M"},{"key":"S1793005724500145BIB026","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(99)00056-5"},{"key":"S1793005724500145BIB027","doi-asserted-by":"publisher","DOI":"10.1142\/3929"},{"key":"S1793005724500145BIB028","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5921-9"},{"key":"S1793005724500145BIB029","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(87)90156-4"},{"key":"S1793005724500145BIB030","doi-asserted-by":"publisher","DOI":"10.1007\/BF01001956"},{"key":"S1793005724500145BIB031","volume-title":"Rough sets-Theoretical Aspects of Reasoning anout Data","author":"Pawlak Z.","year":"1991"},{"key":"S1793005724500145BIB032","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2009.11.023"},{"key":"S1793005724500145BIB033","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(71)90199-5"},{"key":"S1793005724500145BIB034","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2011.05.050"},{"key":"S1793005724500145BIB035","doi-asserted-by":"publisher","DOI":"10.1155\/IJMMS.2005.163"},{"key":"S1793005724500145BIB036","first-page":"1786","volume":"61","author":"Shabir M.","year":"2011","journal-title":"Computers and Mathematics"},{"key":"S1793005724500145BIB037","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2012.11.012"},{"key":"S1793005724500145BIB038","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-79721-0_56"},{"key":"S1793005724500145BIB039","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-78311-4"},{"key":"S1793005724500145BIB040","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(98)00012-7"},{"key":"S1793005724500145BIB041","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"}],"container-title":["New Mathematics and Natural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793005724500145","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,4]],"date-time":"2024-04-04T01:53:19Z","timestamp":1712195599000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1793005724500145"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,8]]},"references-count":41,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["10.1142\/S1793005724500145"],"URL":"https:\/\/doi.org\/10.1142\/s1793005724500145","relation":{},"ISSN":["1793-0057","1793-7027"],"issn-type":[{"value":"1793-0057","type":"print"},{"value":"1793-7027","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,8]]}}}