Yazd University Algebraic Structures and Their Applications 2382-9761 10 2 2023 08 01 Commutative True-False ideals in BCI/BCK-algebras 155 172 3027 10.22034/as.2023.3027 EN Mohammad Mohseni Takallo Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran. Rajab Ali Borzooei Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran. 0000-0001-7538-7885 Mona Aaly Kologani Hatef Higher Education Institute, Zahedan, Iran. Young Bae Jun Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea. 0000-0002-0181-8969 Journal Article 2021 08 18 The notion of a (limited) commutative $T\&F$-ideal in BCK-algebras and BCI-algebras is introduced, and their properties are investigated. A relationship between a $T\&F$-ideal and a commutative $T\&F$-ideal in BCK-algebras and BCI-algebras is established, and examples to show that any $T\&F$-ideal may not be commutative are given. Proper conditions for a $T\&F$-ideal to be commutative are provided. Using a commutative ideal of a BCK-algebra and a BCI-algebra, a commutative $T\&F$-ideal is established. The closed $T\&F$-ideal in a BCI-algebra is introduced, and a condition for a closed $T\&F$-ideal to be commutative is discussed. Characterization of a commutative $T\&F$-ideal in a BCI-algebra is considered.<br />  https://as.yazd.ac.ir/article_3027_61185cda9cf4f3ff734f5f77d8332e71.pdf