In this paper, BI-algebras of order 2, 3, and 4 are characterized, and the Cayley tables are presented up to isomorphism. It is shown that there is one BI-algebra of order 2, two BI-algebras of order 3, and eight BI-algebras of order 4 up to isomorphism. Moreover, it is proved that every commutative and right-distributive BI-algebra is a BCK-algebra. In addition, we show which ones are commutative, transitive, and distributive. All ideals of every BI-algebra in this paper are presented.
