1. Introduction
In 1991, O. G. Xi [1] applied the concept of fuzzy sets to BCK-algebras. After that, concept of fuzzy sets to BCK-algebras has been considered by a number of authors, amongst them, Y. B. Jun [2]. E. Y. Deeba (1980) [3] introduced the concept of lattice filters. Y. B. Jun, S. M. Hong and J. Meng (1998) [4] introduced the concept of fuzzy BCK-Filter. For the general development of BCK-algebras, the filter theory plays an important role as well as ideal theory. The purpose of this paper is to give the fuzzification of BCK-algebras and we study relationships between BCK-filters and lattice filters in BCK-algebras.
2. Preliminaries
(For more details of BCK/BCI-algebras we refer to [5] [6] and [7].)
An algebra
of type (2, 0) is said to be a BCK-algebra if it satisfies the following
(I)
(II)
(III)
(IV)
(V)
and
imply
A BCK-algebras can be (partially) ordered by
if and only if
. The following statements are true in any BCK-algebras: for all
,
(p1)
is a partially ordered set.
(p2)
.
(p3)
(p4)
(p5)
implies
and
BCK-algebra X satisfying the identity
where
for all
is said to be commutative. If there is an element
of BCK-algebra X satisfying
for all x in X, the element
is called the unit of X. A BCK-algebra with the unit is called to be bounded. In a bounded BCK-algebra, we denote
by
for every
. In bounded commutative BCK-algebra denote
.
(p6) In bounded BCK-algebra we have
) [8].
3. Fuzzy BCK-Algebra
Definition 3.1. Zadeh [9] A fuzzy subset of a BCK-algebra X is a function
.
Definition 3.2. [10] Let X be a BCK-algebra. A fuzzy set
in X is called a fuzzy subalgebra of X if, for all
,
Definition 3.3. [11] Let X be a BCK-algebra and let I be a non empty subset of X. Then I it is called to be an ideal of X if, for all
in X.
(a)
(b)
and
imply
.
Definition 3.4. [12]. Let
be a fuzzy set of X. For
, the set
is called a level subset of
.
Theorem 3.5. Let X be a BCK-algebra and let
be an arbitrary fuzzy subalgebra of X, then for every
,
is an ideal of X, when
.
Proof. For any element
we have
, where
,
so that
. Let
and
, and for any
in X, denote
Then by the hypothesis
is a fuzzy subalgebra of X, we have
.
If
, then
so that
. Or
which implies
. Hence
is an ideal of X.
Definition 3.6. [13] A partially ordered set
is said to be a lower semilattice if every pair of elements in X has a greatest lower bound (meet (
)); it is called to be an upper semilattice if every pair of elements in X has least upper bound (Join (
)). If
is both an upper and a lower semilattice, then it is called a lattice.
Definition 3.7. Let X be a bounded BCK-algebra,
be a fuzzy set in X. Then
is called a fuzzy BCK-filter of X if, for all
(FF1)
(FF2)
Theorem 3.8. A BCK-algebra
is commutative if and only if it is lower semilattice with respect to BCK-order
.
Theorem 3.9. Let
be a fuzzy BCK-filter of bounded commutative BCK-algebra. Then for every
,
is an upper semilattice with respect to BCK-order
, when
Proof. For bounded commutative BCK-algebra
for all
(by Y.B.Jun) and
(by Theorem 3.7), then we have
and
for all
This shows that
is a common upper-bounded of x and y. Next if
, then
. (by Theorem 3.7) we obtain
thus
. Hence
is a least upper bound of x and y, so it remains to prove
, from (p6) we have
, and so
.
Hence
, so that
. Hence
is upper semilattice. This completes the proof. □
Theorem 3.10. Let
be a fuzzy BCK-filter of a bounded commutative BCK-algebra X. If
for all
, then for every
,
is a lower semilattice with respect to BCK-order
, when
Proof. Let
. By (II) we know that
. Let z be any element of X such that
. Then
, so
.
By the same reason, we have
, hence
.
This says that for all
in
,
it is the greatest lower bound, so it is remain to prove
. Then from (p6) we have
, and so
.
Hence
so that
. Hence
is lower semilattice. This completes the proof. □
Theorem 3.11. Let
be a fuzzy BCK-filter of a bounded commutative BCK-algebra X. If
for all
, then for every
,
a lattice with respect to BCK-order
, when
Proof. Since
fuzzy BCK-filter of a bounded commutative BCK-algebra X. and
for all
, then for every
,
is lower semilattice with respect to BCK-order
, (from Theorem 3.10). also
is upper semilattice with respect to BCK-order
, (from Theorem 3.9) combining with Definition 3.6, thus
is a lattice with respect to BCK-order
.
Definition 3.12. [14] A non-empty subset F of a BCK-algebra X is said to be a lattice filter if it satisfies that (D1)
and
implies
; (D2)
g.L.b.
.
Theorem 3.13. Let X be a bounded commutative BCK-algebra,
be a fuzzy BCK-filter of X, then for every
,
is a lattice filter of X, when
and
for all
in X.
Proof. Assume
is a fuzzy BCK-filter of X, then for all
, let
be such that
and
. Then
and so
. Hence
So that
. This shows that
satisfies (D1). The proof of (D’2) is similar to the proof of Theorem 3.10 and omitted. Hence
is a lattice filter of X. This completes the proof.
Theorem 3.14. Let X be a BCK-algebra and
be an arbitrary fuzzy subalgebra of X. Then for every a,x and y in X and for every
, the following hold when
:
(a)
(b) if X it is positive implicative, then
where
is recursively defined as follows
for any
(where
is the set of all the natural numbers).
(c)
(d) if X is bounded commutative, then
(e) if X is positive implicative, then
(f) if X is implicative, then
(g) if X is commutative, them
Proof. For any
in X denote
So
. Hence (a) holds. In any positive implicative BCK-algebra, the following identity hold
for any
thus
so
. Hence (b) holds.
so
and (c) holds.
thus
Which implies
, proving (d).
(e) Since X is a positive implicative BCK-algebra, it follows that
thus
. Hence proving (e).
(f) Since X is an implicative BCK-algebra, it follows that
thus
, so
Hence (f) holds.
(g) Form commutative of X and it follows that
Thus
so
Hence (g) holds. This finishes the proof.
4. Conclusions
1) A level subset of a fuzzy sub-algebra of BCK-algebras X is an ideal X.
2) A level subset of the fuzzy BCK-filter of bounded commutative BCK-algebra X is an upper semilattice with respect to BCK-order ≤.
3) A level subset of a fuzzy BCK-filter of abounds commutative BCK algebras X is a lower semilattice with respect to BCK-order ≤ and If
for all
.
4) A level subset of a fuzzy BCK-filter of abounds commutative BCK algebras X is a lattice with respect to BCK-order ≤.
5) A level subset of a fuzzy BCK-filter of a bounded commutative BCK-algebra X is a lattice filter with respect to BCK-order ≤ and if
for all
.