Commutative True-False ideals in BCI/BCK-algebras

Document Type : Research Paper


1 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran.

2 Hatef Higher Education Institute, Zahedan, Iran.

3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea.


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.


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