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 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 BC...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201010985 |
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| author | Michiro Kondo |
| author_facet | Michiro Kondo |
| author_sort | Michiro Kondo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-cf027510910a4b93a6ff8452a3501821 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-cf027510910a4b93a6ff8452a35018212025-02-03T07:24:50ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0128953554310.1155/S0161171201010985Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroupMichiro Kondo0Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, JapanWe 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.http://dx.doi.org/10.1155/S0161171201010985 |
| spellingShingle | Michiro Kondo Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup International Journal of Mathematics and Mathematical Sciences |
| title | Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup |
| title_full | Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup |
| title_fullStr | Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup |
| title_full_unstemmed | Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup |
| title_short | Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup |
| title_sort | relationship between ideals of bci algebras and order ideals of its adjoint semigroup |
| url | http://dx.doi.org/10.1155/S0161171201010985 |
| work_keys_str_mv | AT michirokondo relationshipbetweenidealsofbcialgebrasandorderidealsofitsadjointsemigroup |