((¯ ∈, ∈ ∨ ¯ q)-FUZZY IDEALS, FUZZY IDEALS WITH THRESHOLDS) ¯ OF BCK-ALGEBRAS
MUHAMMAD ZULFIQAR and MUHAMMAD SHABIR Communicated by the former editorial board
In this paper, we introduce the concepts of positive implicative (∈,∈ ∨q)-fuzzy ideal and positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of BCK-algebra and investi-¯ gate some of their related properties.
AMS 2010 Subject Classification: 03G10, 03B05, 03B52, 06F35.
Key words: BCK-algebra, positive implicative (∈,∈ ∨q)-fuzzy ideal, positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal, fuzzy ideals with threshold.¯
1. INTRODUCTION
The concept of BCK-algebras was initiated by Imai and Iseki in [9]. The notion of a fuzzy set, which was published by Zadeh in his classical paper [27] of 1965, was applied by many researchers to generalize some of the basic concepts of algebra. The fuzzy algebraic structures play a vital role in Mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, real analysis, measure theory etc. In 1991, Xi applied fuzzy subsets in BCK-algebras [26] and studied fuzzy BCK-algebras.
He defined the concepts of fuzzy ideal and fuzzy positive implicative ideal.
For the general development of BCK-algebras, the fuzzy ideal theory plays an important role [21–22]. In the ideal theory of BCK-algebras, generation of an ideal by a subset in BCK-algebras is an important problem [11, 18, 20].
In 1971, Rosenfeld laid the foundations of fuzzy groups in [25]. Murali, defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset in [23]. The idea of quasi-coincidence of a fuzzy point with a fuzzy set given in [24], plays a vital role to generate some different types of fuzzy subgroups, called (α, β)-fuzzy subgroups, introduced by Bhakat and Das [5]. In particular, (∈,∈ ∨q)-fuzzy subgroup is an impor- tant and useful generalization of the Rosenfeld’s fuzzy subgroups [25]. In [6], Biswas defined Rosenfeld’s fuzzy subgroups with interval valued membership
MATH. REPORTS16(66),2(2014), 219–241
functions. In [1, 2], Bhakat and Bhakat and Das [3, 4], studied the concept of (∈ ∨q)-level subsets, (∈,∈ ∨q)-fuzzy normal, quasi-normal and maximal subgroups. Zhan et al. [28], discussed (∈,∈ ∨q)-fuzzy ideals in BCI-algebras.
In [7], Davvaz discussed (∈,∈ ∨q)-fuzzy subnearrings and ideals. Jun [13, 14] introduced the concept of (α, β)-fuzzy subalgebras (ideals) of BCK/BCI- algebras. Davvaz and Corsini redefined fuzzy Hv-submodule and many-valued implications in [8]. Jun [15] defined (∈,∈ ∨q)-fuzzy subalgebras in BCK/BCI- algebras. Zulfiqar initiated the notion of (α, β)-fuzzy positive implicative ideals in BCK-algebras [29]. In [17], Maet al. studied (¯∈,∈ ∨¯ q)-fuzzy filters of BL-¯ algebras.
In this paper, we show that every positive implicative (∈,∈ ∨q)-fuzzy ideal of a BCK-algebra X is an (∈,∈ ∨q)-fuzzy ideal of X. We prove that a fuzzy set µof a BCK-algebra X is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X if and only if [µ]t(6=φ) is a positive implicative ideal of X for all t ∈ (0, 1]. We show that a fuzzy set µ of a BCK-algebra X is a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X if and only if it satisfies the conditions (C) and (L),¯ where
(C) µ(0)∨0.5≥µ(x),
(L) µ(x∗z)∨0.5≥µ((x∗y)∗z)∧µ(y∗z), for all x, y, z ∈X.
We show that a fuzzy set µ of a BCK-algebra X is a positive implicative fuzzy ideal with thresholds ε and δ of X, with ε < δ if and only if µt = {x ∈ X | µ(x) ≥ t} is a positive implicative ideal of X for all ε < t ≤ δ.
In Section 2, we recall the notions of ideal and positive implicative ideal of BCK-algebra; in Section 3, we review some fuzzy logic concepts; in Section 4, we define the concept of positive implicative (∈,∈ ∨q)-fuzzy ideal of a BCK- algebra and investigate some of their properties; in Section 5, we introduce the concept of (¯∈,∈ ∨¯ q)-fuzzy ideal and positive implicative (¯¯ ∈,∈ ∨¯ q)-fuzzy ideal¯ of BCK-algebras and discuss some of their properties; in Section 6, we define the concepts of fuzzy ideal with thresholds and positive implicative fuzzy ideal with thresholds of BCK-algebras and investigate some of their properties.
2. SECTION 2 (CRISP SETS - LEVEL 0)
Throughout this paper, X always denotes a BCK-algebra unless otherwise specified.
Definition 2.1 ([12]).By a BCK-algebra, we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the axioms:
(BCK-I) ((x ∗y) ∗ (x∗ z))∗ (z∗y) = 0 (BCK-II) (x ∗(x ∗ y))∗ y = 0
(BCK-III) x ∗x = 0 (BCK-IV) 0 ∗ x = 0
(BCK-V) x∗ y = 0 and y ∗ x = 0 imply x = y for all x, y, z ∈X.
We can define a partial order ≤on X by x ≤y if and only if x ∗y = 0.
Proposition 2.2 ([12, 20, 21]). In any BCK-algebra X, the following are true:
(1) (x∗ y)∗ z = (x∗ z)∗ y (2) (x∗ z)∗ (y∗ z)≤x∗ y (3) (x∗ y)∗ (x∗ z)≤z ∗ y (4) x ∗0 = x
(5) x ∗(x ∗ (x∗ y)) = x∗y for all x, y, z ∈X.
A BCK-algebra X is called positive implicative [20] if it satisfies (x∗ z)∗(y ∗z) = (x ∗ y)∗ z,
for all x, y, z ∈X.
Note that positive implicative BCK-algebras coincide with Hilbert alge- bras [10].
Definition 2.3 ([20]).A nonempty subset S of a BCK-algebra X is called a subalgebra of X if it satisfies
x∗y ∈S, for all x, y ∈S.
Definition 2.4 ([11]). A nonempty subset I of a BCK-algebra X is called an ideal of X if it satisfies the conditions (I1) and (I2), where
(I1) 0∈ I,
(I2) x ∗y∈ I and y∈I imply x ∈I, for all x, y ∈X.
Definition 2.5 ([16]). A nonempty subset I of a BCK-algebra X is called a positive implicative ideal of X if it satisfies the conditions (I1) and (I3), where (I1) 0 ∈I,
(I3) (x ∗ y)∗ z∈ I and y∗ z∈ I imply x∗ z∈ I, for all x, y, z ∈X.
Theorem 2.6 ([19]). A BCK-algebra X is a positive implicative if and only if every ideal of X is a positive implicative ideal.
Proposition 2.7 ([20]). Any positive implicative ideal of a BCK-algebra X is an ideal of X, but the converse is not true in general.
Theorem 2.8 ([20]). Let X be a BCK-algebra. Then an ideal I of X is positive implicative ideal of X if and only if the condition
(for all x, y∈ X) ((x ∗ y)∗ y ∈ I ⇒ x ∗ y ∈I) is satisfied.
3. SECTION 3 (LEVEL 1 OF FUZZIFICATION)
We now review some fuzzy logic concepts. A fuzzy set µ of a universe X is a function from X into the unit closed interval [0, 1], that is µ: X→ [0, 1].
Definition 3.1 ([26]). For a fuzzy set µ of a BCK-algebra X and t ∈(0, 1], the crisp set
µt={x∈X|µ(x)≥t}
is called the level subset ofµ.
Recall that ([0, 1],∧= min,∨= max, 0, 1) is a complete lattice (chain).
Definition 3.2 ([26]).Let X be a BCK-algebra. A fuzzy setµin X is said to be a fuzzy subalgebra of X if it satisfies
µ(x∗y)≥µ(x)∧µ(y), for all x, y ∈X.
Definition 3.3 ([21]).A fuzzy set µof a BCK-algebra X is called a fuzzy ideal of X if it satisfies the conditions (F1) and (F2), where
(F1) µ(0)≥µ(x),
(F2) µ(x)≥µ(x∗y)∧µ(y), for all x, y ∈X.
Definition 3.4 ([16]). A fuzzy setµof a BCK-algebra X is called a positive implicative fuzzy ideal of X if it satisfies the conditions (F1) and (F3), where (F1) µ(0)≥µ(x),
(F3) µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z), for all x, y, z ∈X.
Theorem 3.5 ([16]). Every positive implicative fuzzy ideal of a BCK- algebra X is a fuzzy ideal of X.
Remark 3.6.A fuzzy ideal may not be a positive implicative fuzzy ideal, as shown in the following example.
Example 3.7.Let X = {0,1,2,3} in which ∗ is given by the table:
* 0 1 2 3
0 0 0 0 0
1 1 0 0 1
2 2 1 0 2
3 3 3 3 0
Then X is a BCK-algebra [20]. Define a fuzzy set µin X by µ(0) = 0.9, µ(1) = µ(2) = 0.7 andµ(3) = 0.4. Simple calculations show that µ is a fuzzy ideal of X, but it is not a positive implicative fuzzy ideal of X.
Because for x = 2, y = 1, z = 1, (F3) becomes µ(2∗1)≥µ((2∗1)∗1)∧µ(1∗1)
µ(1)≥µ(1∗1)∧µ(0) 0.7≥µ(0)∧µ(0)
≥0.9∧0.9
≥0.9 but
0.70.9.
Theorem 3.8 ([16]). A fuzzy ideal µ of a BCK-algebra X is a positive implicative fuzzy ideal of X if and only if it satisfies the condition
µ(x∗y)≥µ((x∗y)∗y)
for all x, y ∈ X.
Theorem 3.9 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative fuzzy ideal of X if and only if, for every t ∈ [0, 1], µt(6= φ) is a positive implicative ideal of X.
Theorem 3.10 ([29]). Let µ be a fuzzy set of a BCK-algebra X. If µis a positive implicative fuzzy ideal of X, then the set
I ={x∈X|µ(x) =µ(0)}
is a positive implicative ideal of X.
4. SECTION 4 (LEVEL 2.1 OF FUZZIFICATION)
In this section, we introduce the concept of positive implicative (∈,∈ ∨q)- fuzzy ideal of a BCK-algebra and investigate some of their properties.
Definition 4.1 ([13]).A fuzzy setµ of a BCK-algebra X having the form µ(y) =
t∈(0,1], ify =x 0, ify 6=x.
is said to be a fuzzy point with support x and value t and is denoted byxt. A fuzzy pointxtis said to belong to (resp., quasi-coincident with) a fuzzy set µ, written as xt∈ µ(resp., xtqµ) if µ(x) ≥ t (resp., µ(x) + t >1). If xt ∈µ or xtqµ, then we writext∈ ∨qµ.
Definition 4.2 ([13]). A fuzzy set µ of a BCK-algebra X is called an (∈,∈ ∨q)-fuzzy ideal of X if it satisfies the conditions (A) and (B), where (A) xt∈µ ⇒ 0t∈ ∨qµ,
(B) (x∗y)t∈µ, yr∈µ⇒ xt∧r∈ ∨qµ, for all t, r∈ (0, 1] and x, y∈X.
Theorem 4.3 ([13]). Every fuzzy ideal of a BCK-algebra X is an (∈,∈
∨q)-fuzzy ideal of X.
Lemma 4.4 ([13]). Let µ be a fuzzy set of a BCK-algebra X. Then µt is an ideal of X for all 0.5 < t ≤ 1 if and only if it satisfies the conditions (C) and (D), where
(C) µ(0)∨0.5≥µ(x),
(D) µ(x)∨0.5≥µ(x∗y)∧µ(y), for all x, y, z ∈X.
Definition 4.5.A fuzzy set µ of a BCK-algebra X is called a positive implicative (∈,∈ ∨q)-fuzzy ideal of X if it satisfies the conditions (A) and (E), where
(A) xt∈µ ⇒ 0t∈ ∨qµ,
(E) ((x∗y)∗z)t∈µ,(y∗z)r∈µ⇒ (x∗z)t∧r∈ ∨qµ, for all t, r∈ (0, 1] and x, y∈X.
Example 4.6.Let X ={0,1,2,3}be a BCK-algebra with the Cayley table as follow [20]:
* 0 1 2 3
0 0 0 0 0
1 1 0 0 1
2 2 1 0 2
3 3 3 3 0
Let µ be a fuzzy set in X defined by µ(0) = 0.7, µ(1) = µ(2) = µ(3) = 0.6. Simple calculations show that µ is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X.
Theorem4.7. Every positive implicative(∈,∈ ∨q)-fuzzy ideal of a BCK- algebra X is an (∈,∈ ∨q)-fuzzy ideal of X.
Proof. Let µ be a positive implicative (∈,∈ ∨q)-fuzzy ideal of X. Then for all t, r∈ (0, 1] and for all x, y, z∈ X, we have
((x∗y)∗z)t∈µ,(y∗z)r ∈µ⇒ (x∗z)t∧r ∈ ∨qµ.
Put z = 0 in above, we get
((x∗y)∗0)t∈µ,(y∗0)r ∈µ⇒ (x∗0)t∧r∈ ∨qµ.
This implies
(x∗y)t∈µ, yr ∈µ⇒ xt∧r∈ ∨qµ(by PROPOSITION 2.2(4)).
This means thatµsatisfies the condition (B). Combining with (A) implies that µis an (∈,∈ ∨q)-fuzzy ideal of X.
The converse of the above theorem does not hold (see Example 4.10).
Theorem4.8. A(∈,∈ ∨q)-fuzzy ideal µof a BCK-algebra X is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X if and only if it satisfies the condition
µ(x∗y)≥µ((x∗y)∗y)∧0.5 for all x, y ∈ X.
Proof. The proof is similar to the proof of THEOREM 3.8.
Theorem 4.9 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X if and only if it satisfies the conditions (F) and (G), where
(F) µ(0)≥µ(x)∧0.5,
(G) µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5, for all x, y, z ∈X.
Example 4.10. Let X ={0, a, b, c}be a BCK-algebra in which∗is defined as follows [16]:
* 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0
Define a map µ: X→ [0, 1] by µ(0) = 0.7, µ(a) =µ(b) = 0.4 andµ(c)
= 0.3. Simple calculations show thatµis an (∈,∈ ∨q)-fuzzy ideal of X, but it is not a positive implicative (∈,∈ ∨q)-fuzzy ideal of X.
Because for x = b, y = a, z = a, (G) becomes
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5 µ(b∗a)≥µ((b∗a)∗a)∧µ(a∗a)∧0.5
µ(a)≥µ(a∗a)∧µ(0)∧0.5 µ(a)≥µ(0)∧µ(0)∧0.5
0.4≥0.7∧0.7∧0.5
≥0.5 but
0.40.5
Theorem 4.11. Let µ be a positive implicative (∈,∈ ∨q)-fuzzy ideal of a BCK-algebra X. Then
(1) If there exists x ∈ X, such that µ(x)≥0.5, then µ(0)≥0.5.
(2) If µ(0)< 0.5, then µ is a positive implicative (∈,∈)-fuzzy ideal of X.
Proof. (1) It follows from THEOREM 4.9(F).
(2) The proof is similar to [29].
Theorem 4.12 ([29]). A fuzzy set µ of a BCK-algebra X is a positive implicative(∈,∈ ∨q)-fuzzy ideal of X if and only if the setµt(6=φ) is a positive implicative ideal of X, for all 0< t≤0.5.
In the next theorem we can show a similar result for the case when µt is a positive implicative ideal of a BCK-algebra X for 0.5< t≤1.
Theorem 4.13. Let µ be a fuzzy set of a BCK-algebra X. Then µt(6=φ) is a positive implicative ideal of X for all 0.5< t ≤1 if and only if it satisfies the conditions (C) and (H), where
(C) µ(0)∨0.5≥µ(x),
(H) µ(x∗z)∨0.5≥µ((x∗y)∗z)∧µ(y∗z), for all x, y, z ∈X.
Proof. Supposeµt(6=φ) is a positive implicative ideal of X. From LEMMA 4.4, it follows that (C) hold. Assume that there exist x, y, z ∈X such that
µ(x∗z)∨0.5< µ((x∗y)∗z)∧µ(y∗z) =t.
Then
µ(x∗z)< t and (x∗y)∗z∈µt, y∗z∈µt.
Since µt is a positive implicative ideal of X, we have x∗z∈µt.
Thus,
µ(x∗z) =t, a contradiction. Hence, (H) holds.
Conversely, assume that the conditions (C) and (H) hold. By LEMMA 4.4, we know thatµt is a positive implicative ideal of X. Suppose that
0.5< t≤1, (x∗y)∗z∈µt, y∗z∈µt
Then
0.5< t≤µ((x∗y)∗z)∧µ(y∗z)
≤µ(x∗z)∨0.5 (by using condition (H))
≤µ(x∗z).
Hence, µt is a positive implicative ideal of X.
Theorem 4.14. Every positive implicative fuzzy ideal of a BCK-algebra X is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X.
Proof. Supposeµis a positive implicative fuzzy ideal of X. Then it is also a fuzzy ideal of X by THEOREM 3.5. By using THEOREM 4.3, it follow that µis an (∈,∈ ∨q)-fuzzy ideal of X. Then by Definition 3.4 (F3), for any x, y, z
∈ X, we have
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z).
(i) If µ((x∗y)∗z)∧µ(y∗z)≥0.5, then
µ((x∗y)∗z)≥0.5 and µ(y∗z)≥0.5.
This implies that
µ(x∗z)≥0.5.
Thus,
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5.
(ii) If µ((x∗y)∗z)∧µ(y∗z)<0.5, then
µ((x∗y)∗z)∧µ(y∗z) =µ((x∗y)∗z)∧µ(y∗z)∧0.5.
Thus,
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5.
This satisfies the condition (G). Hence, µ is a positive implicative (∈,∈
∨q)-fuzzy ideal of X.
For any fuzzy set µof a BCK-algebra X and t ∈(0, 1], we denote Q(µ;t) ={x∈X |xtqµ}
and
[µ]t={x∈X |xt∈ ∨qµ}.
It is clear that
[µ]t=µt∪Q(µ;t).
Theorem4.15. A fuzzy setµof a BCK-algebra X is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X if and only if [µ]t(6=φ) is a positive implicative ideal of X for all t ∈ (0, 1].
Proof. Suppose µ is a positive implicative (∈,∈ ∨q)-fuzzy ideal of X.
Then
µ(0)≥µ(x)∧0.5, for all x∈[µ]t. Sincex∈[µ]t. Then
xt∈ ∨qµ, i.e.,
µ(x)≥t or µ(x) +t >1.
Case 1: µ(x)≥t.
(1) If t >0.5, then
µ(0)≥µ(x)∧0.5
≥t∧0.5
= 0.5.
So that
µ(0) +t >1, i.e.,
0tqµ.
(2) If t ≤0.5, then
µ(0)≥t, i.e.,
0t∈µ.
Case 2: µ(x) +t >1.
(1) If t >0.5, then
µ(0)≥µ(x)∧0.5
>(1−t)∧0.5
= 1−t.
Hence,
µ(0) +t >1.
Thus, we have
0tqµ.
(2) If t ≤0.5, then
µ(0)>(1−t)∧0.5
= 0.5.
So,
0t∈µ.
Thus, in any case, we have 0∈[µ]t.
Suppose (x∗y)∗z,y∗z∈[µ]t for t ∈(0, 1]. Then ((x∗y)∗z)t∈ ∨qµ or (y∗z)t∈ ∨qµ.
Thus,
µ((x∗y)∗z)≥t or µ((x∗y)∗z) +t >1 and
µ(y∗z)≥t or µ(y∗z) +t >1.
Since µis a positive implicative (∈,∈ ∨q)-fuzzy ideal of X, we have µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5.
Case 1: µ((x∗y)∗z)≥tand µ(y∗z)≥t.
(1) If t >0.5, then
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5
≥t∧t∧0.5
≥t∧0.5
= 0.5.
So, we have
(x∗z)tqµ.
(2) If t ≤0.5, thenµ(x∗z)≥t. Thus, we have (x∗z)t∈µ.
Case 2: µ((x∗y)∗z)≥tand µ(y∗z) +t >1.
(1) If t >0.5, then
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5
≥t∧(1−t)∧0.5
= 1−t.
i.e.,
µ(x∗z) +t >1.
Thus, we have
(x∗z)tqµ.
(2) If t <0.5, then
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5
≥t∧(1−t)∧0.5
=t.
So,
(x∗z)t∈µ.
Case 3: µ((x∗y)∗z) +t >1 andµ(y∗z)≥t.
The proof is similar to Case 2.
Case 4: µ((x∗y)∗z) +t >1 andµ(y∗z) +t >1.
(1) If t >0.5, then
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5
>(1−t)∧(1−t)∧0.5
>(1−t)∧0.5
= 1−t.
i.e.,
µ(x∗z) +t >1 and thus, we have
(x∗z)tqµ.
(2) If t ≤0.5, then
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)∧0.5
≥(1−t)∧(1−t)∧0.5
≥(1−t)∧0.5
= 0.5
=t.
Thus,
(x∗z)t∈µ.
Therefore, in any case, we have
(x∗z)t∈ ∨qµ and so that
x∗z∈[µ]t.
Hence, [µ]tis a positive implicative ideal of X.
Conversely, assume that µis a fuzzy set of X and t ∈ (0, 1] be such that [µ]t is a positive implicative ideal of X. If
µ(0)< t≤µ(x)∧0.5
for some t ∈(0, 0.5), thenµ(0)< t <0.5. Since 0∈[µ]t, then µ(0)≥t or µ(0) +t >1,
a contradiction. If
µ(x∗z)< t≤µ((x∗y)∗z)∧µ(y∗z)∧0.5 for some t ∈(0, 0.5). Then
µ((x∗y)∗z)≥t and µ(y∗z)≥t.
i.e.,
(x∗y)∗z, y∗z∈µt⊆[µ]t. Thus,
x∗z∈[µ]t. Hence, we have
µ(x∗z)≥t or µ(x∗z) +t >1,
a contradiction. Hence,µis a positive implicative (∈,∈ ∨q)-fuzzy ideal of X.
5. SECTION 5 (LEVEL 2.2 OF FUZZIFICATION)
In this section, we introduce the concepts of (¯∈,∈ ∨¯ q)-fuzzy ideal and¯ positive implicative (¯∈,∈∨¯ q)-fuzzy ideal of BCK-algebras and investigate some¯ of their properties.
Definition 5.1. Letµbe a fuzzy set of a BCK-algebra X. Thenµis called a (¯∈,∈ ∨¯ q)-fuzzy ideal of X if it satisfies the conditions (I) and (J), where¯ (I) 0t∈µ¯ ⇒ xt∈ ∨¯ qµ,¯
(J) xt∧r∈µ¯ ⇒ (x∗y)t∈ ∨¯ qµ¯ oryr∈ ∨¯ qµ,¯ for all x, y ∈X and t, r ∈[0, 1].
Definition 5.2. Letµbe a fuzzy set of a BCK-algebra X. Thenµis called a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X if it satisfies the conditions (I)¯ and (K), where
(I) 0t∈µ¯ ⇒ xt∈ ∨¯ qµ,¯
(K) (x∗z)t∧r∈µ¯ ⇒ ((x∗y)∗z)t∈ ∨¯ qµ¯ or (y∗z)r∈ ∨¯ qµ,¯ for all x, y, z ∈X and t, r ∈[0, 1].
* 0 1 2 3
0 0 0 0 0
1 1 0 0 1
2 2 1 0 2
3 3 3 3 0
Example 5.3.Let X ={0,1,2,3}be a BCK-algebra with Cayley table as follows [20]:
Let µ be a fuzzy set in X defined by µ(0) = 0.5, µ(3) = 0.2 and µ(1) = µ(2) = 0.4. Simple calculations show thatµis an (¯∈,∈ ∨¯ q)-fuzzy ideal as well¯ as a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X.¯
Theorem5.4. Every positive implicative(¯∈,∈ ∨¯ q)-fuzzy ideal of a BCK-¯ algebra X is an (¯∈,∈ ∨¯ q)-fuzzy ideal of X.¯
Proof. Let µ be a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X. Then¯ for all t, r (0, 1] and for all x, y, z∈X, we have
(x∗z)t∧r∈µ¯ ⇒ ((x∗y)∗z)t∈ ∨¯ qµ¯ or (y∗z)r∈ ∨¯ qµ.¯ Put z = 0 in above, we get
(x∗0)t∧r∈µ¯ ⇒ ((x∗y)∗0)t∈ ∨¯ qµ¯ or (y∗0)r∈ ∨¯ qµ.¯ This implies
xt∧r∈µ¯ ⇒ (x∗y)t∈ ∨¯ qµ¯ or yr∈ ∨¯ qµ¯ (by Proposition 2.2(4)).
This means that µsatisfies the condition (J). Combining with (I) implies that µis an (¯∈,∈ ∨¯ q)-fuzzy ideal of X.¯
The converse of the above theorem does not hold (see Example 5.7).
Theorem5.5. A fuzzy set µof a BCK-algebra X is a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X if and only if it satisfies the conditions (C) and (L),¯ where
(C) µ(0)∨0.5≥µ(x),
(L) µ(x∗z)∨0.5≥µ((x∗y)∗z)∧µ(y∗z), for all x, y, z ∈X.
Proof. (I)⇒ (C) Let x ∈X be such that
µ(x)> µ(0)∨0.5.
Select t such that
µ(x)≥t > µ(0)∨0.5.
Then 0t∈µ. But¯ µ(x) ≥ t and µ(x) +t > 1, that is xt ∈ µ and xtqµ, which is a contradiction. Hence,
µ(0)∨0.5≥µ(x).
(C)⇒ (I)
Let 0t∈µ. Then¯ µ(0)< t.
If µ(0)≥0.5, then by condition (C) µ(0)≥µ(x)
and so µ(x)< t, that is xt∈µ. If¯ µ(0)<0.5, then by condition (C) 0.5≥µ(x).
Suppose xt∈µ. Then µ(x)≥t. Thus, 0.5 ≥t. Hence, µ(x) +t≤0.5 + 0.5 = 1,
that is xtqµ. This implies that¯
xt∈ ∨¯ qµ.¯ (K) ⇒ (L)
Suppose there exist x, y, z ∈X such that
µ((x∗y)∗z)∧µ(y∗z)> µ(x∗z)∨0.5.
Select t such that
µ((x∗y)∗z)∧µ(y∗z)≥t > µ(x∗z)∨0.5.
Then (x ∗z)t∈µ¯ but ((x ∗ y) ∗z)t ∈ µ and (y∗ z)t ∈ µ, which is a contradiction. Hence,
µ(x∗z)∨0.5≥µ((x∗y)∗z)∧µ(y∗z).
(L) ⇒ (K)
Let x, z ∈X and t, r ∈ (0, 1] be such that (x∗z)t∧r∈µ.¯ Then
µ(x∗z)< t∧r.
(a) If µ(x∗z)≥0.5, then by condition (L)
µ(x∗z)≥µ((x∗y)∗z)∧µ(y∗z)
and soµ((x∗y)∗z)< t orµ(y∗z)< r, that is ((x∗y)∗z)t∈µ¯ or (y∗z)r∈µ.¯ Hence,
((x∗y)∗z)t∈ ∨¯ qµ¯ or (y∗z)r∈ ∨¯ qµ.¯ (b) If µ(x∗z)<0.5, then by condition (L)
0.5≥µ((x∗y)∗z)∧µ(y∗z).
Suppose ((x∗y)∗z)t∈µ, (y∗z)r∈µ. Then
µ((x∗y)∗z)≥t and µ(y∗z)≥r.
Thus, 0.5≥t∧r. Hence,
µ((x∗y)∗z)∧µ(y∗z) +t∧r≤0.5 + 0.5 = 1, that is ((x∗y)∗z)tqµ¯ or (y∗z)rqµ. This implies that¯
((x∗y)∗z)t∈ ∨¯ qµ¯ or(y∗z)r∈ ∨¯ qµ.¯
Theorem5.6. A(¯∈,∈ ∨¯ q)-fuzzy ideal¯ µof a BCK-algebra X is a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X if and only if it satisfies the condition¯
µ(x∗y)∨0.5≥µ((x∗y)∗y) for all x, y ∈ X.
Proof. Letµbe a positive implicative (¯∈,∈ ∨¯ q) - fuzzy ideal of X. If z is¯ replaced by y in (L), we have
µ(x∗y)∨0.5≥µ((x∗y)∗y)∧µ(y∗y)
≥µ((x∗y)∗y)∧µ(0) (BCK-III)
≥µ((x∗y)∗y) (by condition (C))
Conversely, assume that µis a positive implicative (¯∈,∈ ∨¯ q) - fuzzy ideal¯ of X and
µ(x∗y)∨0.5≥µ((x∗y)∗y).
Since µis an (¯∈,∈ ∨¯ q)-fuzzy ideal of X, we have¯ µ(0)∨0.5≥µ(x), for all x ∈X. Now,
µ(x∗z)∨0.5≥µ((x∗z)∗z) µ(x∗z)∨0.5∨0.5≥µ((x∗z)∗z)∨0.5.
Since µis an (¯∈,∈ ∨¯ q)-fuzzy ideal of X, we have¯ µ(x∗z)∨0.5≥µ(((x∗z)∗z)∗(y∗z))∧µ(y∗z)
≥µ((x∗z)∗y)∧µ(y∗z) (by PROPOSITION 2.2(2))
≥µ((x∗y)∗z)∧µ(y∗z) (by PROPOSITION 2.2(1)).
Hence, µis a positive implicative (¯∈,∈ ∨¯ q) - fuzzy ideal of X.¯
Example 5.7.Let X = {0, a, b, c}be a BCK-algebra with Cayley table as follows [29]:
* 0 a b c
0 0 0 0 0
a a 0 0 a
b b a 0 b
c c c c 0
Let µ be a fuzzy set in X defined byµ(0) = 0.6, µ(a) =µ(b) = 0.4 and µ(c) = 0.3. Simple calculations show thatµis an (¯∈,∈ ∨¯ q) - fuzzy ideal of X,¯ butµis not a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X.¯
Because for x = b, y = a, z = a, (L) becomes
µ(b∗a)∨0.5≥µ((b∗a)∗a)∧µ(a∗a) µ(a)∨0.5≥µ(a∗a)∧µ(0)
0.4∨0.5≥µ(0)∧µ(0) 0.5≥0.6∧0.6 0.5≥0.6 but
0.50.6.
Theorem5.8. A fuzzy set µof a BCK-algebra X is a positive implicative (¯∈,∈ ∨¯ q)-fuzzy ideal of X if and only if for any t¯ ∈ (0.5, 1], µt ={x ∈ X | µ(x)≥t} is a positive implicative ideal of X.
Proof. The proof is similar to the proof of THEOREM 4.12.
Remark 5.9.Let µbe a fuzzy set of a BCK - algebra X and It={t|t∈(0,1] andµtis a positive implicative ideal of X}.
In particular,
(1) IfIt= (0, 1], thenµis a positive implicative fuzzy ideal of a BCK - algebra X (THEOREM 3.9).
(2) If It = (0, 0.5], then µis a positive implicative (∈,∈ ∨q) - fuzzy ideal of a BCK - algebra X (THEOREM 4.12).
(3) If It = (0.5, 1], thenµis a positive implicative (¯∈,∈ ∨¯ q) - fuzzy ideal of a¯ BCK - algebra X (THEOREM 5.8).
Corollary 5.10. Every positive implicative fuzzy ideal of a BCK - alge- bra X is a positive implicative (¯∈,∈ ∨¯ q)¯ - fuzzy ideal of X.
6. SECTION 6 (LEVEL 3 OF FUZZIFICATION)
In this section, we introduce the concepts of fuzzy ideal with thresholds and positive implicative fuzzy ideal with thresholds of BCK - algebras and investigate some of their properties.
Definition 6.1. A fuzzy setµof a BCK - algebra X is called a fuzzy ideal with thresholdsεandδof X,ε, δ∈(0,1] withε < δ, if it satisfies the conditions (M) and (N), where
(M) µ(0)∨ε≥µ(x)∧δ,
(N) µ(x)∨ε≥µ(x∗y)∧µ(y)∧δ, for all x, y ∈X.
Example 6.2. Let X ={0,1,2,3,4}in which the operation∗ is defined as follows:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 1 0 0
2 2 2 0 0 0
3 3 3 3 0 0
4 4 3 4 3 0
Then (X, ∗ , 0) is a BCK - algebra [20]. Let s0, s1, s2 ∈ [0, 1] be such that s0 > s1 > s2. We define a map µ : X → [0, 1] by µ(0) = s0, µ(1) =s1
and µ(2) =µ(3) =µ(4) =s2. Simple calculations show thatµis a fuzzy ideal of X with thresholdsε=s2 and δ=s0.
Definition 6.3.A fuzzy set µ of a BCK-algebra X is called a positive implicative fuzzy ideal with thresholdsεand δ of X, ε, δ ∈(0,1] withε < δ, if it satisfies the conditions (M) and (O), where
(M) µ(0)∨ε≥µ(x)∧δ,
(O) µ(x∗z)∨ε≥µ((x∗y)∗z)∧µ(y∗z)∧δ, for all x, y, z ∈X.
Example 6.4.Let X = {0,1,2,3,4} be a BCK - algebra in which ∗ is defined as follows [20]:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 1 0 0
2 2 2 0 0 0
3 3 3 3 0 0
4 4 3 4 0 0
Let µ be a fuzzy set of a BCK - algebra X defined by µ(0) = µ(1) = 0.6, µ(2) = 1, µ(3) = 0 and µ(4) = 0.2. Simple calculations show thatµ is a positive implicative fuzzy ideal of X with thresholdsε= 0.2 and δ = 0.6.
Theorem 6.5. Every positive implicative fuzzy ideal with thresholds of a BCK-algebra X is a fuzzy ideal with thresholds of X.
Proof. Let µ be a positive implicative fuzzy ideal with thresholds of X.
Then for all ε, δ∈(0,1] and for all x, y, z ∈X, we have µ(x∗z)∨ε≥µ((x∗y)∗z)∧µ(y∗z)∧δ.
Put z = 0 in above, we get
µ(x∗0)∨ε≥µ((x∗y)∗0)∧µ(y∗0)∧δ.
This implies
µ(x)∨ε≥µ(x∗y)∧µ(y)∧δ (by PROPOSITION 2.2(4)).
This means thatµsatisfies the condition (O). Combining with (M) implies that µis fuzzy ideal with thresholds of X.
Example 6.6. Let X ={0,1,2,3,4}in which the operation∗ is defined as follows:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 1 0 0
2 2 2 0 0 0
3 3 3 3 0 0
4 4 3 4 3 0
Then (X, ∗ , 0) is a BCK-algebra [20]. Lets0, s1, s2 ∈[0,1] be such that s0 > s1 > s2. We define a map µ : X → [0, 1] by µ(0) = s0, µ(1) = s1 and µ(2) = µ(3) = µ(4) = s2. Routine calculations give that µ is a fuzzy ideal of X with thresholdsε=s2 and δ=s0. Butµ is not a positive implicative fuzzy ideal of X with thresholdsε=s2 and δ=s0, because
Put x = 2, y = 3, z = 4 in (O) we get
µ(2∗4)∨ε≥µ((2∗3)∗4)∧µ(3∗4)∧δ µ(0)∨ε≥µ(0∗4)∧(0)∧δ
µ(0)∨ε≥µ(0)∧µ(0)∧δ s0∨s2 ≥s0∧s0∧s0
s2∨s2 ≥s0
s2 ≥s0
but
s2 s0.
Theorem6.7. A fuzzy idealµwith thresholdsεandδof a BCK - algebra X is a positive implicative fuzzy ideal with thresholds of X if and only if it satisfies the condition
µ(x∗y)∨ε≥µ((x∗y)∗y)∧δ for all x, y ∈ X.
Proof. The proof is similar to the proof of THEOREM 3.8 and THEO-
REM 4.8.
Theorem6.8. A fuzzy setµof a BCK - algebra X is a positive implicative fuzzy ideal with thresholds ε and δ of X, with ε < δ if and only if µt = {x ∈ X |µ(x)≥t} is a positive implicative ideal of X for all ε < t≤δ.
Proof. Supposeµis a positive implicative fuzzy ideal of X and ε < t≤δ.
Letx∈µt. Then µ(x)≥t. So
µ(0)∨ε≥µ(x)∧δ
≥t∧δ
≥t
> ε.
Thus,
µ(0)≥t.
We get
0∈µt. Let (x∗y)∗z∈µtand y∗z∈µt. Then
µ((x∗y)∗z)≥t and µ(y∗z)≥t.
By using condition (O), we have
µ(x∗z)∨ε≥µ((x∗y)∗z)∧µ(y∗z)∧δ
≥t∧t∧δ
≥t∧δ
≥t
> ε.
It follows that
µ(x∗z)≥t, and so that
x∗z∈µt.
Hence, µt is a positive implicative ideal of X.
Conversely, assume that µt is a positive implicative ideal of X for all ε < t≤δ. If there exist x, y, z ∈X such that
µ(x∗z)∨ε < t0 =µ((x∗y)∗z)∧µ(y∗z)∧δ.
Then
ε < t0≤δ and (x∗y)∗z∈µt0, y∗z∈µt0, µ(x∗z)< t0. Since µt0 is a positive implicative ideal of X, so
x∗z∈µt0 and µ(x∗z)≥t0. This is a contradiction with
µ(x∗z)< t0. Therefore,
µ(x∗z)∨ε≥µ((x∗y)∗z)∧µ(y∗z)∧δ.
Similarly, we can prove that
µ(0)∨ε≥µ(x)∧δ.
From Definition 6.3, the following result holds:
Theorem 6.9. Let µ be a positive implicative fuzzy ideal with thresholds of ε and δ of a BCK - algebra X, with ε < δ. Then
(i) µis a positive implicative fuzzy ideal when ε= 0, δ = 1.
(ii) µ is a positive implicative (∈,∈ ∨q) - fuzzy ideal when ε = 0,δ = 0.5.
(iii) µis a positive implicative (¯∈,∈ ∨¯ q)¯ - fuzzy ideal when ε = 0.5, δ = 1.
Proof. Straightforward.
7. CONCLUSION
In the study of fuzzy algebraic system, we see that the positive implicative fuzzy ideals with special properties always play a central role.
In this paper, we define the concepts of positive implicative (∈,∈ ∨q) - fuzzy ideal and positive implicative (¯∈,∈ ∨¯ q) - fuzzy ideal of BCK - algebra¯ and give several characterizations of a positive implicative fuzzy ideal in BCK - algebras in terms of these notions.
We believe that the research along this direction can be continued, and in fact, some results in this paper have already constituted a foundation for further investigation concerning the further development of fuzzy BCK - alge- bras and their applications in other branches of algebra. In the future study of
fuzzy BCK - algebras, perhaps the following topics are worth to be considered:
(1) To characterize other classes of BCK - algebras by using this notion;
(2) To apply this notion to some other algebraic structures;
(3) To consider these results to some possible applications in computer sci- ences and information systems in the future.
Acknowledgments.The authors are very grateful to referees for their valuable com- ments and suggestions for improving this paper.
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Received 12 January 2012 GC University Lahore,
Department of Mathematics, Pakistan
mzulfiqarshafi@hotmail.com Quaid-i-Azam University, Department of Mathematics,
Islamabad, Pakistan
mshabirbhatti@yahoo.co.uk
