The young structure based on N -structure and its application in BCI/BCK-algebras

Abstract

A new structure called a young structure based on the ${\mathcal N}$-structure is introduced and applied to BCK/BCI-algebra. The concepts of young subalgebra, $\in$-subset, $q$-subset, and $O$-subset are introduced, and their related properties are studied. The relationship between ${\mathcal N}$-subalgebra and young subalgebra is discussed. It is observed that every ${\mathcal N}$-subalgebra is a young subalgebra, and an example is give to show that the converse does not hold. The characterization of young subalgebra is considered, and conditions are given to show that a young structure based on ${\mathcal N}$-structure is a (strong) young subalgebra. Conditions under which $\in$-subset, $q$-subset, and $O$-subset can be subalgebra are explored.