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

Michiro Kondo

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 28, 1-9

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.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:454570

DOI: 10.1155/S0161171201010985

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