Commutative ideals of BCK-algebras based on makgeolli structures

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