International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences / 2001 / Article

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Volume 28 |Article ID 454570 | 9 pages | https://doi.org/10.1155/S0161171201010985

Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup

Michiro Kondo1

1Department of Mathematics and Computer Science, Shimane University, Japan

Received04 Oct 2000
Revised06 Feb 2001

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.

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.

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