{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T02:05:30Z","timestamp":1767665130453,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,14]],"date-time":"2022-04-14T00:00:00Z","timestamp":1649894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object\u2019s symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal.<\/jats:p>","DOI":"10.3390\/sym14040815","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T00:22:43Z","timestamp":1650414163000},"page":"815","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Bipolar Fuzzy Set Theory Applied to the Certain Ideals in BCI-Algebras"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5529-0565","authenticated-orcid":false,"given":"N.","family":"Abughazalah","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5841","authenticated-orcid":false,"given":"G.","family":"Muhiuddin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed E. A.","family":"Elnair","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"},{"name":"Department of Mathematics and Physics, Gezira University, P.O. Box 20, Wad Medani 2667, Sudan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Mahboob","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle 517325, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,14]]},"reference":[{"key":"ref_1","first-page":"19","article-title":"On axiom systems of propositional calculi","volume":"42","author":"Imai","year":"1966","journal-title":"Proc. Jpn. Acad."},{"key":"ref_2","first-page":"26","article-title":"An algebra related with a propositional calculus","volume":"42","year":"1966","journal-title":"Proc. Jpn. 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