International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 9, Pages 535-543
doi:10.1155/S0161171201010985
Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
Received 4 October 2000; Revised 6 February 2001
Copyright © 2001 Michiro Kondo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 ℐ(X) of all ideals of X and the set 𝒪(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.