CHAPTER 2

FUZZY (a, p, q)-IDEALS IN BCI-ALGEBRAS

11

CHAPTER 2

FUZZY (a, p, q)-IDEALS IN BCI-ALGEBRAS

The paper based on the text of this Chapter has been published in the following: (1) International Journal of Mathematical Science, Vol.6, No.2 (2007), 15-25. (2) International Journal of Algebra, Number Theory and Applications, Vol.1, No.1, (2008), 13-24.

12

CHAPTER 2 Fuzzy (a, p, q)-ideals in BCI-algebras 2.1. Introduction In this chapter we define the notion of fuzzy a-ideal in BCI-algebras and we give several characterizations and results about fuzzy a-ideals.

2.2. Fuzzy (a, p, q)-ideals In this section we introduce the concept of fuzzy a-ideal and give several characterizations. We show that a non-empty subset of a BCI-algebra is fuzzy a-ideal if and only if it both fuzzy q-ideal and fuzzy p-ideal. Definition 2.2.1([24]): A non-empty subset I of X is called an a-ideal of X if it satisfies: (A1)

0 ∈ I,

(A2) (x ∗ z) ∗ (0 ∗ y) ∈ I and z ∈ I imply y ∗ x ∈ I, for all x, y, z ∈ X.

Fuzzy (a, p, q)-ideals

13

Definition 2.2.2: A fuzzy set µ in X is called fuzzy a-ideal if it satisfies: (I1)

µ(0) ≥ µ(x), ∀x ∈ X.

(I2)

µ(y ∗ x) ≥ min{µ((x ∗ z) ∗ (0 ∗ y)), µ(z)},

∀x, y, z ∈ X.

Definition 2.2.3 ([24]): A non-empty subset I of X is said to be q-ideal if it satisfies: (A1) and, (I3)

x ∗ (y ∗ z) ∈ I and y ∈ I imply x ∗ z ∈ I,

∀x, y, z ∈ X.

Definition 2.2.4 ([26]): Let µ be a fuzzy set in BCI-algebra X. µ is called a fuzzy q-ideal if it satisfies: (I1) and, (I4)

µ(x ∗ z) ≥ min{µ(x ∗ (y ∗ z)), µ(y)}, ∀x, y, z ∈ X.

Definition2.2.5 ([23]): A fuzzy set µ in BCI-algebra X is said to be fuzzy p-ideal if it satisfies: (I1) and, (I5)

µ(x) ≥ min{µ((x ∗ z) ∗ (y ∗ z)), µ(y)}, ∀x, y, z ∈ X.

Theorem 2.2.6: Any fuzzy a-ideal is a fuzzy ideal in BCI-algebra X but the converse is not true. Proof. Assume µ is a fuzzy a-ideal. We first show that µ is a fuzzy ideal of X. Since µ is a fuzzy a-ideal then we have µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z) ≤ µ(y ∗ x).

14

Chapter 2

Putting y = 0 in (I2) then µ((x ∗ z) ∗ (0 ∗ 0)) ∧ µ(z) ≤ µ(0 ∗ x). Hence µ(x ∗ z) ∧ µ(z) ≤ µ(0 ∗ x).

(1)

Putting z = y = 0 in (I2), it follows that µ((x ∗ 0) ∗ (0 ∗ 0)) ∧ µ(0) ≤ µ(0 ∗ x), hence, µ(x) ≤ µ(0 ∗ x).

(2)

Putting x = z = 0 in (I2), it follows that, µ(y) ≥ µ(0 ∗ (0 ∗ y)) ≥ µ(0 ∗ y) ≥ µ(y ∗ z) ∧ µ(z). Then µ(y) ≥ µ(y ∗ z) ∧ µ(z). Thus, µ is an ideal of X. To show the last part of the Theorem, we use the following example. Example 2.2.7: Let X = {0, a, b, c} be a BCI-algebra with Cayley table as follows. ∗ 0 a b c

0 0 a b c

a 0 0 b c

b 0 0 0 c

c c c c 0

Define µ : X → [0, 1] by µ(0) = t0 , µ(b) = µ(c) = t2 , µ(a) = t1 where t0 , t1 , t2 ∈ [0, 1] such that t0 > t1 > t2 . By routine calculations we get µ is a fuzzy ideal. Now we prove that µ is not a fuzzy a-ideal. Putting in (I2), x = a, y = b, z = 0, we have, t2 = µ(b ∗ a) 6≥ t1 = µ((a ∗ 0) ∗ (0 ∗ b)) ∧ µ(0).

Fuzzy (a, p, q)-ideals

15

Theorem 2.2.8: Let µ be a fuzzy ideal of X. Then the following are equivalent. (i)

µ is a fuzzy a-ideal of X.

(ii)

µ(y ∗ (x ∗ z)) ≥ µ((x ∗ z) ∗ (0 ∗ y)), ∀x, y, z ∈ X.

(iii) µ(y ∗ x) ≥ µ(x ∗ (0 ∗ y)), ∀x, y ∈ X. Proof: (i) ⇒ (ii) Suppose that µ be a fuzzy a-ideal of X, for all x, y, z in X, we have µ(y ∗ (x ∗ z)) ≥ µ((x ∗ z) ∗ s) ∗ (0 ∗ y)) ∧ µ(s). We write s = (x ∗ z) ∗ (0 ∗ y). Then ((x ∗ z) ∗ s) ∗ (0 ∗ y) = ((x ∗ z) ∗ (0 ∗ y)) ∗ s = 0. By (3), we have µ(y ∗ (x ∗ z)) ≥ µ(0) ∧ µ(s) = µ((x ∗ z) ∗ (0 ∗ y)). Hence

µ(y ∗ (x ∗ z)) ≥ µ((x ∗ z) ∗ (0 ∗ y)).

(ii) ⇒ (iii). Letting z = 0 in (ii) we obtain (iii). (iii) ⇒ (i). Since (x ∗ (0 ∗ y)) ∗ ((x ∗ z) ∗ (0 ∗ y)) ≤ x ∗ (x ∗ z) ≤ z. We have µ(x ∗ (0 ∗ y)) ≥ µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z), by (iii) we have µ(y ∗ x) ≥ µ(x ∗ (0 ∗ y)) ≥ µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z). It follows that µ(y ∗ x) ≥ µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z). Hence (I2) holds. Combining (I1), µ is a fuzzy a-ideal of X.

(3)

16

Chapter 2

This completes the proof. Theorem 2.2.9: A fuzzy set µ in X is a fuzzy a-ideal of X if and only if for any t ∈ [0, 1], U (µ, t) is an a-ideal of X whenever U (µ, t) 6= φ. Proof: Assume that µ is a fuzzy a-ideal of X. Let t ∈ [0, 1] such that U (µ, t) 6= φ. As U (µ, t) 6= φ ∃ x ∈ X such that x ∈ U (µ, t), then µ(x) ≥ t. But µ is a fuzzy a-ideal, µ(0) ≥ µ(x) ≥ t. Hence µ(0) ≥ t, implies 0 ∈ U (µ, t). Next let (x ∗ z) ∗ (0 ∗ y) ∈ U (µ, t) and z ∈ U (µ, t). Then µ((x ∗ z) ∗ (0 ∗ y)) ≥ t and µ(z) ≥ t. Since µ is a fuzzy a-ideal of X, therefore µ(y ∗ x) ≥ µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z) ≥ t or µ(y ∗ x) ≥ t. Hence y ∗ x ∈ U (µ, t). This proves that U (µ, t) is an a-ideal of X. Conversely, suppose µ(x0 ) > µ(0) for some x0 ∈ X. Put t0 = 12 (µ(0) + µ(x0 )). Then µ(0) < t0 and 0 ≤ t0 ≤ µ(x0 ) ≤ 1. This implies x0 ∈ U (µ, t0 ) or U (µ, t0 ) 6= φ. As U (µ, t0 ) is an a-ideal of X, we have 0 ∈ U (µ, t0 ) and hence µ(0) ≥ t0 . This contradiction proves that µ(0) ≥ µ(x), for all x ∈ X. Suppose for some x0 , y0 , z0 ∈ X, µ(y0 ∗ x0 ) < µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) ∧ µ(z0 ). Put t1 = 12 [µ(y0 ∗ x0 ) + µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) ∧ µ(z0 )]. Then µ(y0 ∗ x0 ) < t1 and 0 ≤ t1 < µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) ∧ µ(z0 ) ≤ 1, which gives µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) > t1 and µ(z0 ) > t1 .

Fuzzy (a, p, q)-ideals

17

Therefore (x0 ∗ z0 ) ∗ (0 ∗ y0 ) ∈ U (µ, t1 ), z0 ∈ U (µ, t1 ). Since U (µ, t1 ) is a-ideal, we have y0 ∗ x0 ∈ U (µ, t1 ) and hence µ(y0 ∗ x0 ) ≥ t1 , a contradiction. Hence, the supposition is wrong. This completes the proof. Theorem 2.2.10: Let µ be a fuzzy ideal of a BCI-algebra X. Then the following are equivalent: (i)

µ is a fuzzy q-ideal of X.

(ii)

µ((x ∗ y) ∗ z) ≥ µ(x ∗ (y ∗ z)), ∀x, y, z ∈ X.

(iii)

µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)), ∀x, y ∈ X.

Proof:(i) ⇒ (ii). Since µ is a fuzzy q-ideal of X, we have µ((x ∗ y) ∗ z) ≥ µ((x ∗ y) ∗ (0 ∗ z)) ∧ µ(0) = µ((x ∗ y) ∗ (0 ∗ z)). On the other hand (x ∗ y) ∗ (0 ∗ z) = (x ∗ y) ∗ ((y ∗ z) ∗ y) ≤ x ∗ (y ∗ z), therefore µ(x ∗ (y ∗ z)) ≤ µ((x ∗ y) ∗ (0 ∗ z)). Then µ((x ∗ y) ∗ z) ≥ µ(x ∗ (y ∗ z)). (ii) ⇒ (iii). Letting y = 0 and z = y in (I4). (iii) ⇒ (i). Since (x ∗ (0 ∗ y)) ∗ (x ∗ (z ∗ y)) ≤ (z ∗ y) ∗ (0 ∗ y) ≤ z.

18

Chapter 2

Then we have µ(x ∗ (0 ∗ y)) ≥ µ(x ∗ (z ∗ y)) ∧ µ(z). Therefore by hypothesis µ(x ∗ y) ≥ µ(x ∗ (z ∗ y)) ∧ µ(z). Hence µ is a fuzzy q-ideal of X. The proof is complete. Theorem 2.2.11: A fuzzy q-ideal µ of a BCI-algebra X is a fuzzy ideal and a fuzzy subalgebra of X (i.e. closed ideal). Proof: Let µ be a fuzzy q-ideal. Letting z = 0 in (I4). We obtain µ(x) ≥ µ(x ∗ y) ∧ µ(y). Hence µ is a fuzzy ideal of X. If µ be a fuzzy q-ideal, then by (I4), we have µ(x ∗ z) ≥ µ(x ∗ (y ∗ z)) ∧ µ(y). Putting z = y, then µ(x ∗ y) ≥ µ(x) ∧ µ(y). This shows that µ is a fuzzy subalgebra of X and completes the proof. Theorem 2.2.12: Let µ and ν be fuzzy ideals of a BCI-algebra X, such that µ ⊆ ν and µ(0) = ν(0). If µ is a fuzzy q-ideal of X, then so ν. Proof: For any x, y ∈ X by Theorem 2.2.10(iii), we want to show that ν(x ∗ y) ≥ ν(x ∗ (0 ∗ y)).

Fuzzy (a, p, q)-ideals

19

Putting s = x ∗ (0 ∗ y), then (x ∗ s) ∗ (0 ∗ y) = 0. Hence µ((x ∗ s) ∗ (0 ∗ y)) = µ(0) = ν(0). Since µ is a fuzzy q-ideal of X, and using Theorem 2.2.10(iii), we get, µ((x ∗ s) ∗ y) ≥ µ((x ∗ s) ∗ (0 ∗ y)) = ν(0). Thus µ((x ∗ y) ∗ s) ≥ ν(0) ≥ ν(s). Since ν is a fuzzy ideal, we have ν(x ∗ y) ≥ ν((x ∗ y) ∗ s) ∧ ν(s) = ν(s) = ν(x ∗ (0 ∗ y)). It means that ν is a fuzzy q-ideal of X and completing the proof. Theorem 2.2.13: A fuzzy ideal µ of a BCI-algebra X is a fuzzy p-ideal if and only if

µ(x) ≥ µ(0 ∗ (0 ∗ x)).

Proof: Let µ be a fuzzy p-ideal. For all x, y, z ∈ X, we have µ(x) ≥ µ((x ∗ z) ∗ (y ∗ z)) ∧ µ(y). Letting x = z and y = 0, then µ(x) ≥ µ(0 ∗ (0 ∗ x)). Conversely we have (0 ∗ (0 ∗ x)) ∗ ((x ∗ z) ∗ (y ∗ z)) = (((y ∗ z) ∗ (y ∗ z)) ∗ (0 ∗ x)) ∗ ((x ∗ z) ∗ (y ∗ z))

20

Chapter 2

= (((y ∗ z) ∗ (0 ∗ x)) ∗ (y ∗ z)) ∗ ((x ∗ z) ∗ (y ∗ z)) ≤ ((y ∗ z) ∗ (0 ∗ x)) ∗ (x ∗ z) = ((y ∗ z) ∗ (x ∗ z)) ∗ (0 ∗ x) ≤ (y ∗ x) ∗ (0 ∗ x) ≤ y. Then µ(0 ∗ (0 ∗ x)) ≥ µ((x ∗ z) ∗ (y ∗ z)) ∧ µ(y). Therefore µ(x) ≥ µ((x ∗ z) ∗ (y ∗ z)) ∧ µ(y). Thus µ is a fuzzy p-ideal of X. Theorem 2.2.14: Any fuzzy a-ideal is a fuzzy p-ideal. Proof: Let µ be a fuzzy a-ideal of X. Then µ is a fuzzy ideal by Theorem 2.2.6. Setting x = z = 0 in Theorem 2.2.8(ii) we have µ(y) ≥ µ(0 ∗ (0 ∗ y)). From Theorem 2.2.13, µ is a fuzzy p-ideal. Theorem 2.2.15: Any fuzzy a-ideal is a fuzzy q-ideal. Proof: Let µ be a fuzzy a-ideal of X. Then µ is a fuzzy ideal by Theorem 2.2.6. In order to prove that µ is a fuzzy q-ideal, from Theorem 2.2.10(iii) it suffices to show that, µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)).

Fuzzy (a, p, q)-ideals

21

As (0 ∗ (0 ∗ (y ∗ (0 ∗ x)))) ∗ (x ∗ (0 ∗ y)) = ((0 ∗ (0 ∗ y)) ∗ (0 ∗ (0 ∗ (0 ∗ x)))) ∗ (x ∗ (0 ∗ y)) = ((0 ∗ (0 ∗ y)) ∗ (0 ∗ x)) ∗ (x ∗ (0 ∗ y)) ≤ (x ∗ (0 ∗ y)) ∗ (x ∗ (0 ∗ y)) = 0. Hence µ(0 ∗ (0 ∗ (y ∗ (0 ∗ x)))) ≥ µ(x ∗ (0 ∗ y)).

(4)

By Theorem 2.2.14, µ is a fuzzy p-ideal and by Theorem 2.2.13, µ(y ∗ (0 ∗ x)) ≥ µ(0 ∗ (0 ∗ (y ∗ (0 ∗ x)))). By Theorem 2.2.8 (iii) we have µ(x ∗ y) ≥ µ(y ∗ (0 ∗ x)).

(5)

From (4) and (5) we have µ(x ∗ y) ≥ µ(y ∗ (0 ∗ x)) ≥ µ(0 ∗ (0 ∗ (y ∗ (0 ∗ x)))) ≥ µ(x ∗ (0 ∗ y)). Hence µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)). Therefore µ is a fuzzy q-ideal of X. The proof of the theorem is complete.

22

Chapter 2

Theorem 2.2.16: Let µ be a fuzzy ideal of X, µ is a fuzzy a-ideal if and only if it is both a fuzzy p-ideal and a fuzzy q-ideal. Proof: If µ is a fuzzy a-ideal, then µ is a fuzzy p-ideal and a fuzzy q-ideal by Theorem 2.2.14 and Theorem 2.2.15. Conversely, if µ is both a fuzzy p-ideal and a fuzzy q-ideal, then µ is a closed ideal as Theorem 2.2.8, we want to show µ(y ∗ x) ≥ µ(x ∗ (0 ∗ y)). By Theorem 2.2.10(iii), we obtain µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)).

(6)

Because (0 ∗ (y ∗ x)) ∗ (x ∗ y)

= ((0 ∗ y) ∗ (0 ∗ x)) ∗ (x ∗ y) = ((0 ∗ (x ∗ y)) ∗ y) ∗ (0 ∗ x) = (0 ∗ (0 ∗ y)) ∗ y = 0,

then 0 ∗ (y ∗ x) ≤ x ∗ y. Hence µ(0 ∗ (y ∗ x)) ≥ µ(x ∗ y). By (6), we have µ(0 ∗ (y ∗ x)) ≥ µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)).

(7)

Since µ is a fuzzy p-ideal by Theorem 2.2.13, µ(y ∗ x) ≥ µ(0 ∗ (0 ∗ (y ∗ x))).

(8)

Fuzzy (a, p, q)-ideals

23

Since µ is fuzzy closed ideal µ(0 ∗ (y ∗ x)) ≤ µ(0 ∗ (0 ∗ (y ∗ x))). We have µ(y ∗ x) ≥ µ(0 ∗ (0 ∗ (y ∗ x))) ≥ µ(0 ∗ (y ∗ x)) ≥ µ(x ∗ y) ≥ µ(x ∗ (0 ∗ y)). Hence µ(y ∗ x) ≥ µ(x ∗ (0 ∗ y)). Thus, µ is a fuzzy a-ideal by Theorem 2.2.8(iii) and completing the proof. Theorem 2.2.17: An onto homomorphic preimage of a fuzzy a-ideal is a fuzzy a-ideal. Proof: Let f : X → Y be an onto homomorphism. Let ν be a fuzzy a-ideal on Y and let µ be the preimage of ν under f . Let y ∈ Y . Since f is onto, there exists x ∈ X, y = f (x). Since ν is a fuzzy a-ideal of Y , it follows that µ(f (0)) ≥ µ(f (x)) then ν(0) ≥ ν(y) for all y ∈ Y . Next, ν is a fuzzy a-ideal, therefore, for any y1 , y2 , y3 ∈ Y there exists x1 , x2 , x3 in X such that yi = f (xi ), i = 1, 2, 3. Then we have ν(y2 ∗ y1 ) ≥ ν((y1 ∗ y3 ) ∗ (0 ∗ y2 )) ∧ ν(y3 ), i.e. ν(f (x2 ) ∗ f (x1 )) ≥ ν((f (x1 ) ∗ f (x3 )) ∗ (0 ∗ f (x2 ))) ∧ ν(f (x3 )), i.e. ν(f (x2 ∗ x1 )) ≥ ν(f ((x1 ∗ x3 ) ∗ (0 ∗ x2 )) ∧ ν(f (x3 )).

24

Chapter 2

Hence µ(x2 ∗ x1 ) ≥ µ((x1 ∗ x3 ) ∗ (0 ∗ x2 )) ∧ µ(x3 ). Proving that µ is a fuzzy a-ideal of X. Theorem 2.2.18: Let ν be a fuzzy set of a BCI-algebra X and let µν be the strongest fuzzy relation on X. Then ν is a fuzzy a-ideal of X if and only if µν is a fuzzy a-ideal with respect to ν. Proof: Assume that ν is a fuzzy a-ideal of X. Then µν (0, 0) = ν(0) ∧ ν(0) ≥ ν(x) ∧ ν(y) = µν (x, y) for all (x, y) ∈ X × X. Next µν (y1 ∗ x1 , y2 ∗ x2 ) = ν(y1 ∗ x1 ) ∧ ν(y2 ∗ x2 ) ≥ (ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν(z1 )) ∧ (ν((x2 ∗ z2 ) ∗ (0 ∗ y2 )) ∧ ν(z2 )) = (ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν((x2 ∗ z2 ) ∗ (0 ∗ y2 ))) ∧ (ν(z1 ) ∧ ν(z2 )) = µν ((x1 ∗ z1 ) ∗ (0 ∗ y1 ), (x2 ∗ z2 ) ∗ (0 ∗ y2 )) ∧ µν (z1 , z2 )) for any (x1 , x2 ), (y1 , y2 ), (z1 , z2 ) ∈ X × X. Hence µν is a fuzzy a-ideal of X × X. Conversely, suppose that µν is a fuzzy a-ideal of X × X. Then for all x ∈ X, ν(0) ∧ ν(0) = µν (0, 0) ≥ µν (x, x) = ν(x) ∧ ν(x). It follows that ν(0) ≥ ν(x) for all x ∈ X.

Fuzzy (a, p, q)-ideals

25

Now for all (x1 , x2 ), (y1 , y2 ), (z1 , z2 ) ∈ X × X we have ν(y1 ∗ x1 ) ∧ ν(y2 ∗ x2 ) = µν (y1 ∗ x1 , y2 ∗ x2 ) ≥ µν (((x1 , x2 ) ∗ (z1 , z2 )) ∗ (0 ∗ (y1 , y2 ))) ∧ µν (z1 , z2 ) = (µν ((x1 ∗ z1 ) ∗ (0 ∗ y1 ), (x2 ∗ z2 ) ∗ (0 ∗ y2 ))) ∧ µν (z1 , z2 ) = (ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν((x2 ∗ z2 ) ∗ (0 ∗ y2 ))) ∧ (ν(z1 ) ∧ ν(z2 )) = (ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν(z1 )) ∧ (ν((x2 ∗ z2 ) ∗ (0 ∗ y2 )) ∧ ν(z2 )). We take x2 = y2 = z2 = 0, we get ν(y1 ∗ x1 ) ∧ ν(0) ≥ ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν(z1 ) ∧ ν(0) or ν(y1 ∗ x1 ) ≥ ν((x1 ∗ z1 ) ∗ (0 ∗ y1 )) ∧ ν(z1 ). Which proves that ν is a fuzzy a-ideal of X. This completes the proof.

26

Chapter 2

2.3. n-fold fuzzy a-ideals In this section, we discuss the notion of n-fold fuzzy a-ideals of BCIalgebras. We establish the extension property for an n-fold fuzzy a-ideal. We define the notion of An -Noetherian BCI-algebras and give its characterizations. For any elements x, y of a BCI-algebra X, x ∗ y n denotes (· · · ((x ∗ y) ∗ y) ∗ · · ·) ∗ y in which y occurs n times. Definition 2.3.1: Let n be a positive integer. A nonempty subset I of a BCI-algebra X is called an n-fold a-ideal of X if (A1)

and,

(I6)

y n ∗ x ∈ I whenever (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ I and z ∈ I,

for all x, y, z ∈ X. We try to fuzzify the concept of n-fold a-ideal. Definition 2.3.2: Let n be a positive integer. A fuzzy set µ in X is called an n-fold fuzzy a-ideal of X if (F1)

µ(0) ≥ µ(x) for all x ∈ X,

(F4)

µ(y n ∗ x) ≥ min{µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z)} for all x, y, z ∈ X .

n-fold fuzzy a-ideals

27

Theorem 2.3.3: Let I be an ideal of X. Then the following are equivalent: (1)

I is an n-fold a-ideal of X.

(2)

(x ∗ z) ∗ (0 ∗ y n+1 ) ∈ I

(3)

x ∗ (0 ∗ y n+1 ) ∈ I

implies

implies

y n ∗ (x ∗ z) ∈ I.

y n ∗ x ∈ I.

Proof: (1) ⇒ (2). Suppose that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ I. For brevity, we write t = (x ∗ z) ∗ (0 ∗ y n+1 ). Then ((x ∗ z) ∗ t) ∗ (0 ∗ y n+1 ) = ((x ∗ z) ∗ (0 ∗ y n+1 )) ∗ t = 0 ∈ I. By (I6), y n ∗ (x ∗ z) ∈ I. (2) ⇒ (3). Letting z = 0 in (2) we obtain (3). (3) ⇒ (1). Let (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ I and z ∈ I. Since (x ∗ (0 ∗ y n+1 )) ∗ ((x ∗ z) ∗ (0 ∗ y n+1 )) ≤ x ∗ (x ∗ z) ≤ z ∈ I, we have x ∗ (0 ∗ y n+1 ) ∈ I as I is an ideal. By (3), y n ∗ x ∈ I. Hence, (I6) holds. Combining (A1 ), I is an n-fold a-ideal of X. This completes the proof. Theorem 2.3.4: Let µ be a fuzzy ideal of X. Then µ is an n-fold fuzzy a-ideal of X if and only if it satisfies the inequality, µ(y n ∗ x) ≥ µ(x ∗ (0 ∗ y n+1 )) for all x, y ∈ X.

28

Chapter 2

Proof: Suppose that µ is an n-fold fuzzy a-ideal of X and let x, y ∈ X. Then µ(y n ∗ x)

≥ µ((x ∗ 0) ∗ (0 ∗ y n+1 )) ∧ µ(0) = µ(x ∗ (0 ∗ y n+1 )) ∧ µ(0) = µ(x ∗ (0 ∗ y n+1 )).

Conversely, assume that µ be a fuzzy a−ideal of X satisfying the equality µ(y n ∗ x) ≥ µ(x ∗ (0 ∗ y n+1 )) for all x, y ∈ X. Then µ(y n ∗ x) ≥ µ(x ∗ (0 ∗ y n+1 )) ≥ µ((x ∗ z) ∗ (0 ∗ y n+1 )) ∧ µ(z), for all x, y, z ∈ X. Hence µ is an n-fold fuzzy a-ideal of X. Lemma 2.3.5: Let I be a nonempty subset of a BCI-algebra X, and µ be a fuzzy set in X defined by a0 if x ∈ I, µ(x) = a1 otherwise, where a0 > a1 in [0, 1]. Then µ is a fuzzy a-ideal of X if and only if I is an a-ideal of X. Proof: Assume that µ is a fuzzy a-ideal of X. Since µ(0) ≥ µ(x) for all x ∈ X, we have µ(0) = a0 and so 0 ∈ I. Let x, y, z ∈ X be such that (x ∗ z) ∗ (0 ∗ y) ∈ I and z ∈ I. Using (F2), we get µ(y ∗ x) ≥ µ((x ∗ z) ∗ (0 ∗ y)) ∧ µ(z) = a0

n-fold fuzzy a-ideals

29

and so µ(y ∗ x) = a0 , that is y ∗ x ∈ I, consequently, I is an a−ideal of X. Conversely, suppose that I be an a-ideal of X. Since 0 ∈ I, µ(0) = a0 ≥ µ(x) for all x ∈ X. Suppose that (I2) does not hold then there exist x0 , y0 , z0 ∈ X such that µ(y0 ∗ x0 ) = a1 and µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) ∧ µ(z0 ) = a0 . Thus µ((x0 ∗ z0 ) ∗ (0 ∗ y0 )) = a0 = µ(z0 ) and so (x0 ∗ y0 ) ∗ (0 ∗ y0 ) ∈ I and z0 ∈ I. It follows that y0 ∗x0 ∈ I so that µ(y0 ∗z0 ) = a0 . This is a contradiction. Therefore µ is a fuzzy a-ideal of X. Lemma 2.3.6: Let I be an ideal of a BCI-algebra X, n a positive integer and µ be a fuzzy set defined in Lemma 2.3.5. Then µ is an n-fold fuzzy a-ideal of X if and only if I is an n-fold a-ideal of X. Proof: Assume that µ is an n-fold fuzzy a-ideal of X. Let x, y ∈ X be such that x ∗ (0 ∗ y n+1 ) ∈ I. Using Theorem 2.3.4, we get µ(y n ∗ x) ≥ µ(x ∗ (0 ∗ y n+1 )) = a0 , and so µ(y n ∗ x) = a0 , that is y n ∗ x ∈ I. Hence by Theorem 2.3.3, we conclude that I is an n-fold a-ideal of X. Conversely, suppose that I be an n-fold a-ideal of X. For any x, y ∈ X,

30

Chapter 2

either y n ∗ x ∈ I or y n ∗ x ∈ / I. Hence µ(y n ∗ x) = a0 ≥ µ(x ∗ (0 ∗ y n+1 )). In the latter, we know that x ∗ (0 ∗ y n+1 ) ∈ / I. Then µ(y n ∗ x) = a1 = µ(x ∗ (0 ∗ y n+1 )). From Theorem 2.3.4, it follows that µ is an n-fold fuzzy a-ideal of X. Theorem 2.3.7: Let µ be a fuzzy set in X and let n be a positive integer. Then µ is an n-fold fuzzy a-ideal of X if and only if every nonempty level set U (µ, t) of µ is an n-fold a-ideal of X. Proof: Assume that µ is an n-fold fuzzy a-ideal of X and t ∈ [0, 1] such that U (µ, t) is nonempty. Then there exists x ∈ U (µ, t). It follows that µ(0) ≥ µ(x) then 0 ∈ U (µ, t). Let x, y, z ∈ X be such that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ U (µ, t) and z ∈ U (µ, t). We have µ(y n ∗ x) ≥ µ((x ∗ z) ∗ (0 ∗ y n+1 )) ∧ µ(z) ≥ t. Hence y n ∗ x ∈ U (µ, t). Therefore U (µ, t) is an n-fold a-ideal of X. Conversely, suppose that every nonempty level set U (µ, t) is an n-fold a-ideal of X. Let x ∈ X. Put µ(x) = t . Then x ∈ U (µ, t). Since by assume 0 ∈ U (µ, t) we get µ(0) ≥ µ(x) = t and so µ(0) ≥ µ(x) for all x ∈ X. Now assume that there exists x0 , y0 , z0 ∈ X such that µ(y0n ∗ x0 ) < µ((x0 ∗ z0 ) ∗ (0 ∗ y0n+1 )) ∧ µ(z0 )

n-fold fuzzy a-ideals

31

selecting 1 t0 = (µ(y0n ∗ x0 ) + µ((x0 ∗ z0 ) ∗ (0 ∗ y0n+1 )) ∧ µ(z)). 2 Then µ(y0n ∗ x0 ) < t0 < µ((x0 ∗ z0 ) ∗ (0 ∗ y0n+1 )) ∧ µ(z0 ). It follows that / U (µ, t0 ). (x0 ∗ z0 ) ∗ (0 ∗ y0n+1 ) ∈ U (µ, t0 ) and z0 ∈ U (µ, t0 ) and (y0n ∗ x0 ) ∈ This is a contradiction. Hence µ is an n-fold fuzzy a-ideal of X. Theorem 2.3.8: If µ is an n-fold fuzzy a-ideal of X, then the set, Xµ = {x ∈ X | µ(x) = µ(0)} is an n-fold a-ideal of X. Proof: Let µ be an n-fold fuzzy a-ideal of X. Clearly 0 ∈ Xµ . Let x, y, z ∈ X be such that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Xµ and z ∈ Xµ . Then µ(y n ∗ x) ≥ min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) = µ(0). It follows from (F 1) µ(y n ∗ x) = µ(0). So that y n ∗ x ∈ Xµ . Hence Xµ is an n-fold a-ideal of X. Definition 2.3.9: A BCI-algebra X is said to satisfy the An -ascending (resp, An -descending) chain condition (briefly, An -ACC( resp, An -DCC)) if for every ascending (resp, descending) sequence A1 ⊆ A2 ⊆ · · · (resp, A1 ⊇ A2 ⊇ · · ·)

32

Chapter 2

of n-fold a-ideals of X there exists a natural number k such that Ar = Ak for all r ≥ k. If X satisfies the An -ACC, we say that X is a An -Noetherian BCI-algebra. Theorem 2.3.10: Let {Ak | k ∈ N } be a family of n-fold a-ideals of X which is nested, that is, A1 ⊃ A2 ⊃ A3 ⊃ · · · . Let µ be a fuzzy set in X defined by k k+1 if x ∈ Ak − Ak+1 , k = 0, 1, 2, · · · ∞ T µ(x) = Ak , 1 if x ∈ k=0

for all x ∈ X, where A0 stands for X. Then µ is an n-fold fuzzy a-ideal of X. Proof: Clearly µ(0) ≥ µ(x), for all x ∈ X . Let x, y, z ∈ X. Suppose that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak − Ak+1 , z ∈ Ar − Ar+1 for k = 0, 1, 2, · · ·,

r = 0, 1, 2, · · ·. Without loss of generality we may

assume that k ≤ r. Then obviously z ∈ Ak . Since Ak is an n−fold a-ideal, it follows that y n ∗ x ∈ Ak so that k = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . µ(y n ∗ x) ≥ k+1 ∞ ∞ ∞ T T T If (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak and z ∈ Ak , then y n ∗ x ∈ Ak . k=0

k=0

Hence µ(y n ∗ x) = 1 = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) .

k=0

n-fold fuzzy a-ideals

If (x ∗ z) ∗ (0 ∗ y

33

n+1

)∈ /

∞ T

Ak and z ∈

k=0

∞ T

Ak , then there exists i ∈ N

k=0

such that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ai − Ai+1 . It follows that y n ∗ x ∈ Ai so that, i = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . µ(y n ∗ x) ≥ i+1 ∞ ∞ T T Finally, assume that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak and z ∈ / Ak . Then n

k=0

k=0

z ∈ Aj − Aj+1 for some j ∈ N . Hence y ∗ x ∈ Aj , and thus j µ(y n ∗ x) ≥ = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . j+1 Consequently, µ is an n-fold fuzzy a-ideal of X. Corollary 2.3.11: If every n-fold fuzzy a-ideal has finite number of values, then X satisfies the An -DCC. Proof: Suppose, every fuzzy a-ideal of X has finite number of values. Suppose X does not satisfy An -DCC. Then there exists a sequence A1 ⊃ A2 ⊃ A3 ⊃ · · · of n-fold a-ideals such that it does not terminate, that is Ak −Ak+1 6= φ, for all k ∈ N . But, then by Theorem 2.3.10 the function µ defined by k k+1 if x ∈ Ak − Ak+1 , k = 0, 1, 2, · · · ∞ T µ(x) = 1 if x ∈ Ak , k=0

is an n-fold fuzzy a-ideal having infinitely many values. This is a contradiction.

34

Chapter 2

Theorem 2.3.10 tells that if every n-fold fuzzy a-ideal of X has a finite number of values, then X satisfies the An -DCC. Now we consider the converse of Theorem 2.3.10. Theorem 2.3.12: Let X be a BCI-algebra satisfying An -DCC and let µ be an n-fold fuzzy a-ideal of X. If a sequence of elements of Im(µ) is strictly increasing, then µ has a finite number of values. Proof: Let {λk } be a strictly increasing of elements of Im(µ). Hence 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ 1. Then U (µ, α) is an n-fold a-ideal of X for all α = 2, 3, · · ·. Let x ∈ U (µ, α) where U (µ, α) = {x ∈ X | µ(x) ≥ λα }. Then µ(x) ≥ λα ≥ λα−1 , and so x ∈ U (µ, α − 1). Hence U (µ, α) ⊆ U (µ, α − 1). Since λα−1 ∈ Im(µ) there exists xα−1 ∈ X such that µ(xα−1 ) = λα−1 . It follows that xα−1 ∈ U (µ, α − 1) but xα−1 ∈ / U (µ, α). Thus U (µ, α) ⊂ U (µ, α − 1), and so we obtain a strictly descending sequence, U (µ, 1) ⊃ U (µ, 2) ⊃ U (µ, 3) ⊃ · · · of n-fold a-ideals of X which is not terminating. This contradicts the assumption that X satisfies the An -DCC. Then µ has a finite number of value.

n-fold fuzzy a-ideals

35

We note that a set is well ordered if and only if it does not contain any infinite descending sequence. Theorem 2.3.13: The following are equivalent: (i) X is a An -Noetherian BCI-algebra. (ii) The set of values of any n-fold fuzzy a-ideal of X is a well ordered subset of [0, 1]. Proof: (i)⇒ (ii). Let µ be an n-fold fuzzy a-ideal of X. Assume that the set of values of µ is not a well-ordered subset of [0, 1]. Then there exists a strictly decreasing sequence {λk } such that µ(xk ) = λk . It follows that U (µ, 1) ⊂ U (µ, 2) ⊂ U (µ, 3) ⊂ · · ·

(1)

is a strictly ascending chain of n-fold a-ideal of X where U (µ, λ) = {x ∈ X | µ(x) ≥ tλ } for every λ = 1, 2, · · ·. This contradicts the assumption that X is a An -Noetherian. (ii) ⇒ (i). Assume that condition (ii) is satisfied and X is not An -Noetherian. Then there exists a strictly ascending chain, A1 ⊂ A2 ⊂ A3 ⊂ · · · of n-fold a-ideals of X. Let A =

S

k∈N

Ak . Then A is an n-fold a-ideal.

Define a fuzzy set ν in X by 0 if x∈ / Ak , ν(x) = 1 r if r = min {k ∈ N | x ∈ Ak } .

36

Chapter 2

We claim that ν is an n-fold fuzzy a-ideal of X. Since 0 ∈ Ak for all k = 1, 2, · · · , we have ν(0) = 1 ≥ ν(x) for all x ∈ X . Let x, y, z ∈ X. If (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak − Ak−1 and z ∈ Ak − Ak−1 for k = 2, 3, · · · then y n ∗ x ∈ Ak . It follows that ν(y n ∗ x) ≥

1 = min ν((x ∗ z) ∗ (0 ∗ y n+1 )), ν(z) . k

Suppose that z ∈ Ak − Ar and (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak for all r ≤ k. Since Ak is an n-fold a-ideal, it follows that y n ∗ x ∈ Ak . Hence 1 ν(y n ∗x) ≥ k1 ≥ r+1 ≥ ν(z) , ν(y n ∗x) ≥ min ν((x ∗ z) ∗ (0 ∗ y n+1 )), ν(z) . Similarly for the case (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak − Ar and z ∈ Ak , we have ν(y n ∗ x) ≥ min ν((x ∗ z) ∗ (0 ∗ y n+1 )), ν(z) . Thus ν is an n-fold fuzzy a-ideal of X. Since the chain (1) is not terminating, ν has a strictly descending sequence of values. This contradicts the assumption that the value set of any n-fold fuzzy a-ideal is well ordered. Therefore X is An -Noetherian. This completes the proof. Theorem 2.3.14: Let S = {λk | k = 1, 2, · · ·} ∪ {0} where {λk } is a strictly descending sequence in (0, 1). Then a BCI-algebra X is An -Noetherian if and only if for each n-fold fuzzy a-ideal µ of X, Im(µ) ⊆ S implies that there exists a natural number k such that Im(µ) ⊆ {λ1 , λ2 · · · λk } ∪ {0} . Proof: Assume that X is a An -Noetherian BCI-algebra and let µ be an n-fold fuzzy a-ideal of X . Then by Theorem 2.3.13, we know that Im(µ) is a well-ordered subset of [0, 1], and so the condition is neces-

n-fold fuzzy a-ideals

37

sary. Conversely, suppose that the condition is satisfied. Assume that X is not An -Noetherian. Then there exists a strictly ascending chain of n-fold a-ideals of X such that A1 ⊂ A2 ⊂ A3 ⊂ · · · . Define a fuzzy set µ in X λ 1 λk µ(x) = 0

by if x ∈ A1 , if x ∈ Ak − Ak−1 , k = 2, 3, · · · ∞ S if x ∈ X − Ak . k=0

Since 0 ∈ A1 , we have µ(0) = λ1 ≥ µ(x) for all x ∈ X. If either ∞ S n+1 (x ∗ z) ∗ (0 ∗ y ) or z belongs to X − Ak , then either k=0

µ((x ∗ z) ∗ (0 ∗ y n+1 )) or µ(z) is equal to 0 and hence µ(y n ∗ x) ≥ 0 = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . If (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ A1 and z ∈ A1 , then y n ∗ x ∈ A1 and thus µ(y n ∗ x) = λ1 = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . If (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak − Ak−1 and z ∈ Ak − Ak−1 , then y n ∗ x ∈ Ak . Hence µ(y n ∗ x) ≥ λk = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . Assume that (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ A1 and z ∈ Ak − Ak−1 for k = 2, 3, · · ·. Then y n ∗ x ∈ Ak and therefore µ(y n ∗ x) ≥ λk = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) .

38

Chapter 2

Similarly for (x ∗ z) ∗ (0 ∗ y n+1 ) ∈ Ak − Ak−1 and z ∈ A1 , k = 2, 3, · · ·, we obtain µ(y n ∗ x) ≥ λk = min µ((x ∗ z) ∗ (0 ∗ y n+1 )), µ(z) . Consequently, µ is an n-fold fuzzy a-ideal of X. This contradicts our assumption.