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 is an ideal, then , (2) is the order ideal generated by , (3) if is a BCK-algebra, then for any order ideal of , thus, for each BCK-algebra there is a one-to-one correspondence between the set of all ideals of and the set of all order ideals of it, and (4) the order is an order ideal if and only if 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.