Dokdo commutative ideals of BCK-algebras

Abstract

The purpose of this paper is to study by applying Dokdo structure to commutative ideal in $BCK$-algebras. The notion of Dokdo commutative ideal is introduced, and their properties are investigated. The relationship between Dokdo ideal and Dokdo commutative ideal is discussed. Example to show that a Dokdo ideal may not be a Dokdo commutative ideal is provided, and then the conditions under which a Dokdo ideal can be a Dokdo commutative ideal are explored. Conditions for a Dokdo structure to be a Dokdo commutative ideal are provided, and characterizations of a Dokdo commutative ideal are displayed. Finally, the extension property for a Dokdo commutative ideal is established.