{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:51:06Z","timestamp":1777704666945,"version":"3.51.4"},"reference-count":49,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2018,11,8]],"date-time":"2018-11-08T00:00:00Z","timestamp":1541635200000},"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":[[2018,12,24]]},"abstract":"<jats:p>\n                    The main motivation behind this paper is to study some structural properties of a non-associative structure Abel Grassmann\u2019s groupoid (AG-groupoid) in terms of double-framed soft sets (DFS sets) as it hasn\u2019t attracted much attention compared to associative structures. An AG-groupoid can be referred to as a non-associative semigroup, as the main difference between a semigroup and an AG-groupoid is the switching of an associative law. In this paper, we introduce the concept of (\n                    <jats:italic>M<\/jats:italic>\n                    ,\n                    <jats:italic>N<\/jats:italic>\n                    )-double-framed soft ideals (briefly, (\n                    <jats:italic>M<\/jats:italic>\n                    ,\n                    <jats:italic>N<\/jats:italic>\n                    )-DFS ideal) of AG-groupoids and investigate some properties of these notions. We have shown that every (\n                    <jats:italic>M<\/jats:italic>\n                    ,\n                    <jats:italic>N<\/jats:italic>\n                    )-DFS ideal is (\n                    <jats:italic>M<\/jats:italic>\n                    ,\n                    <jats:italic>N<\/jats:italic>\n                    )-DFS AG-groupoid but the converse is not true. This is shown with the help of an example. We also discuss the properties of (\n                    <jats:italic>M<\/jats:italic>\n                    ,\n                    <jats:italic>N<\/jats:italic>\n                    )-DFS ideals in regular AG-groupoids. Moreover a decision making algorithm based on DFS-sets is given.\n                  <\/jats:p>","DOI":"10.3233\/jifs-181119","type":"journal-article","created":{"date-parts":[[2018,11,9]],"date-time":"2018-11-09T15:21:42Z","timestamp":1541776902000},"page":"6313-6327","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":2,"title":["(\n                    <i>M<\/i>\n                    ,\n                    <i>N<\/i>\n                    )-Double framed soft ideals of Abel Grassmann\u2019s groupoids"],"prefix":"10.1177","volume":"35","author":[{"given":"Muhammad","family":"Izhar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan, KP, Pakistan"}]},{"given":"Asghar","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan, KP, 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