Relationship between Ideals of Bci-algebras and Order Ideals of Its Adjoint Semigroup

We consider the relationship between ideals of a BCI-algebra and order ideals of its adjoint semigroup. We show that (1) if I is an ideal, then I =M−1(M(I)), (2) M(M−1(J)) is the order ideal generated by J∩R(X), (3) if X is a BCK-algebra, then J =M(M−1(J)) for any order ideal J of X, thus, for each BCK-algebra X there is a one-to-one correspondence between the set (X) of all ideals of X and the set (X) of all order ideals of it, and (4) the order M(M−1(J)) is an order ideal if and only if M−1(J) is an ideal. These results are the generalization of those denoted by Huang and Wang (1995) and Li (1999). We can answer the open problem of Li affirmatively. 2000 Mathematics Subject Classification. 06F35, 03G25.