As a generalization of the self-distributive BE-algebra, the JNB-algebra is introduced, and its basic properties are investigated. This could play various roles in the study of logical algebra, including BCK-algebra. First, examples are presented showing that the three axioms of JNB-algebra are independent of each other. The basic properties of JNB-algebras that will be needed to study various theories about JNB-algebras are explored. Upper sets based on one and two elements are introduced and its associated properties are examined. Two concepts so called JNB-deductive system and JNB-filter are introduced, and their properties are investigated. Characterizations of the JNB-deductive system and the JNB-filter are discussed. It is finally confirmed that the JNB-deductive system matches the JNB-filter.
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