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International Journal of Mathematics and Mathematical Sciences
Volume 28, Issue 9, Pages 535-543
http://dx.doi.org/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 Hindawi Publishing Corporation. 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=M1(M(I)), (2) M(M1(J)) is the order ideal generated by JR(X), (3) if X is a BCK-algebra, then J=M(M1(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 𝒪(X) of all order ideals of it, and (4) the order M(M1(J)) is an order ideal if and only if M1(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.