{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:11:47Z","timestamp":1760231507671,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,26]],"date-time":"2022-09-26T00:00:00Z","timestamp":1664150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University Researchers Supporting Project","award":["TURSP-2020\/246"],"award-info":[{"award-number":["TURSP-2020\/246"]}]}],"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 order to provide a mathematical tool for dealing with negative information, a negative-valued function came into existence along with N-structures. In the present analysis, the notion of N-structures is applied to the ideals, especially the commutative ideals of BCI-algebras. Firstly, several properties of N-subalgebras and N-ideals in BCI-algebras are investigated. Furthermore, the notion of a commutative N-ideal is defined, and related properties are investigated. In addition, useful characterizations of commutative N-ideals are established. A condition for a closed N-ideal to be a commutative N-ideal is provided. Finally, it is proved that in a commutative BCI-algebra, every closed N-ideal is a commutative N-ideal.<\/jats:p>","DOI":"10.3390\/sym14102015","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T01:23:16Z","timestamp":1664414596000},"page":"2015","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["\ud835\udca9-Structures Applied to Commutative Ideals of BCI-Algebras"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5841","authenticated-orcid":false,"given":"Ghulam","family":"Muhiuddin","sequence":"first","affiliation":[{"name":"Algebra, Discrete Mathematics and Number Theory Research Group, Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}]},{"given":"Mohamed E.","family":"Elnair","sequence":"additional","affiliation":[{"name":"Algebra, Discrete Mathematics and Number Theory Research Group, 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"}]},{"given":"Deena","family":"Al-Kadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistic, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,26]]},"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. 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