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 .
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