Anti Fuzzy filters of pseudo-BL-algebras

Magdalena Wojciechowska-Rysiawa ^∗

Anti Fuzzy filters of pseudo-BL-algebras

Abstract. Characterizations of anti fuzzy filters in a pseudo-BL-algebra are esta- blished. Conditions for a fuzzy set to be an anti fuzzy filter are given. Finally, a study anti of fuzzy prime filters and characterizations of linearly ordered pseudo-BL- algebras via anti fuzzy filters are given.

2000 Mathematics Subject Classification: 03G25,06F35.

Key words and phrases: Pseudo-BL-algebra; Anti fuzzy filter; (anti fuzzy) prime filter.

1. Introduction. H´ajek [9] introduced BL-algebras in 1998. MV-algebras, which are contained in the class of BL-algebras were introduced by Chang [1].

A noncommutative extention of MV-algebras, called pseudo MV-algebras, were in- troduced by Georgescu and Iorgulescu [6]. In 2000 Georgescu and Iorgulescu gave a concept of pseudo-BL-algebras as noncommutative generalization of BL-algebras and pseudo MV-algebras. The pseudo-BL-algebras correspond to a pseudo-basic fuzzy logic (see [9] and [10]). In [2] and [3], basic properties of pseudo-BL-algebras were introduced.

The concept of fuzzy sets was introduced by Zadeh in [18]. The fuzzy ideals in MV-algebras were studied by Hoo in [11] and [12], in pseudo MV-algebras by Jun and Walendziak [15]. Rach˚ unek and ˇSalounov´a introduced fuzzy filters of bounded R`-monoids. Hong, Jun and Jeong introduced a concept of anti fuzzy ideals in BCK-algebras in [13] and [14].

In this paper, we introduce the notion of anti fuzzy filters of pseudo-BL-algebras.

We describe a connection between anti fuzzy filters and filters of a pseudo-BL- algebra. What is more, we study anti fuzzy prime filters and using them, we cha- racterize linearly ordered pseudo-BL algebras.

2. Preliminaries. The notion of pseudo-BL-algebras was defined by Georgescu and Iorgulescu [8] as follows:

∗

Thanks for prof. A. Walendziak

Definition 2.1 A pseudo-BL-algebra is an algebra (A, ∨, ∧, , →, ;, 0, 1) of type (2, 2, 2, 2, 2, 0, 0) satisfies the following axioms for any x, y, z ∈ A :

(C1) (A, ∨, ∧, 0, 1) is a bounded lattice;

(C2) (A, , 1) is a monoid;

(C3) x  y ¬ z ⇔ x ¬ y  z ⇔ y ¬ x ; z;

(C4) x ∧ y = (x → y)  x = x  (x ; y) ;

(C5) (x → y) ∨ (y → x) = (x ; y) ∨ (y ; x) = 1.

Note that every pseudo-BL-algebra satisfying x → y = x ; y for all x, y ∈ A is a BL-algebra.

Lemma 2.2 ( [2]) Let (A, ∨, ∧, , →, ;, 0, 1) be a pseudo-BL-algebra. Then for all x, y, z ∈ A :

(a) y ¬ x → y and y ¬ x ; y;

(b) x  y ¬ x ∧ y;

(c) x  y ¬ x and x  y ¬ y;

(d) x → 1 = x ; 1 = 1;

(e) x ¬ y ⇔ x → y = 1 ⇔ x ; y = 1;

(f) x → x = x ; x = 1;

(g) x → (y → z) = (x  y) → z and x ; (y ; z) = (y  x) ; z;

(h) x ¬ y → (x  y) and x ¬ y ; (y  x) ; (i) x  (y ∨ z) = (x  y) ∨ (x  z) .

Definition 2.3 A subset F of pseudo-BL-algebra A is called a filter if it satisfies the following two conditions:

(F1) if x, y ∈ F , then x  y ∈ F ;

(F2) if x ∈ F , y ∈ A and x ¬ y, then y ∈ F .

Proposition 2.4 ([2]) Let F be a subset of pseudo-BL-algebra A. The following are equivalent:

(i) F is a filter;

(ii) 1 ∈ F and if x, x → y ∈ F , then y ∈ F ;

(iii) 1 ∈ F and if x, x ; y ∈ F , then y ∈ F.

A filter F of pseudo-BL-algebra A is called proper if F 6= A.

The proper filter F is prime if for all x, y ∈ A,

x ∨ y ∈ F implies x ∈ F or y ∈ F.

Proposition 2.5 ([8]) Any proper filter of pseudo-BL-algebra A can be extended to a prime filter.

Now, recall some facts about fuzzy sets and fuzzy filters in a pseudo-BL-algebra.

A fuzzy set of a pseudo-BL-algebra A is a function µ : A → [0; 1] . We need to define operations ∧ and ∨ for all α, β ∈ [0, 1] :

α ∧ β = min {α, β} and α ∨ β = max {α, β} . For any fuzzy sets µ and ν of pseudo-BL-algebra A, we define

ν ¬ µ ⇔ ν (x) ¬ µ (x) for all x ∈ A.

A relation ¬ is an order relation in the set of fuzzy sets in A.

For a fuzzy set µ of pseudo-BL-algebra A we define an upper α-level by U (µ, α) = {x ∈ A : µ (x) ­ α} .

Definition 2.6 ([17]) A fuzzy set µ is called a fuzzy filter of a pseudo-BL-algebra A if for all x, y ∈ A :

(ff1) µ (x  y) ­ µ (x) ∧ µ (y) ; (ff2) x ¬ y ⇒ µ (x) ¬ µ (y) .

It is easily seen that (ff2) implies (ff3) µ (1) ­ µ (x) for all x ∈ A.

Proposition 2.7 ([17]) A fuzzy set µ is a fuzzy filter of a pseudo-BL-algebra A iff it satisfies (ff1) and

(ff4) µ (x ∨ y) ­ µ (x) for all x, y ∈ A.

Proposition 2.8 ([17]) Let µ be a fuzzy set of a pseudo-BL-algebra A. The follo- wing are equivalent:

(1) µ is a fuzzy filter of A;

(2) µ satisfies (ff3) and for all x, y ∈ A µ (y) ­ µ (x) ∧ µ (x → y) ; (3) µ satisfies (ff3) and for all x, y ∈ A µ (y) ­ µ (x) ∧ µ (x ; y) .

In [17], there were proved that if F is a filter of A and α, β ∈ [0, 1] where α > β, then a fuzzy set

µ

(α, β) (x) =

 α if x ∈ F β otherwise

is a fuzzy filter. We will denote χ

= µ

(1, 0), which is called the characteristic function of F .

Definition 2.9 If µ is a fuzzy set of pseudo-BL-algebra A, then the complement of µ denote by µ

, is a fuzzy set of A defined by µ

(x) = 1 − µ (x) for any x ∈ A.

3. Anti fuzzy filters. We give the definition of anti fuzzy filters of a pseudo- BL-algebra.

Definition 3.1 A fuzzy set µ is called an anti fuzzy filter of a pseudo-BL-algebra A if for all x, y ∈ A :

(af1) µ (x  y) ¬ µ (x) ∨ µ (y) ; (af2) x ¬ y ⇒ µ (y) ¬ µ (x) .

Proposition 3.2 Let µ be an anti fuzzy filter of a pseudo-BL-algebra A. Then, for any x ∈ A,

(1) µ (1) ¬ µ (x) .

Proof Since x ¬ 1, we have µ (1) ¬ µ (x) by (af2).

Proposition 3.3 A fuzzy set µ in a pseudo-BL-algebra A is an anti fuzzy filter of A if and only if it satisfies (1) and

(af3) µ (y) ¬ µ (x → y) ∨ µ (x) for all x, y ∈ A.

Proof Let µ be an anti fuzzy filter of A. Since x ∧ y ¬ y, by (C4) and (af1), we have

µ (y) ¬ µ (x ∧ y) = µ ((x → y)  x) ¬ µ (x → y) ∨ µ (x) .

Conversely, let µ satisfy (1) and (af3). Let x, y ∈ A such that x ¬ y. By Lemma 2.2 (e), x → y = 1. Hence

µ (y) ¬ µ (x → y) ∨ µ (x) = µ (1) ∨ µ (x) = µ (x) . Thus (af2) holds.

Now let x, y ∈ A. By Lemma 2.2 (h) and (af3),

µ (x  y) ¬ µ (y) ∨ µ (y → (x  y)) ¬ µ (y) ∨ µ (x) .

The proof is completed.

Corollary 3.4 A fuzzy set µ of a pseudo-Bl-algebra A is an anti fuzzy filter of A if and only if it satisfies (1) and

(af4) µ (y) ¬ µ (x ; y) ∨ µ (x) for all x, y ∈ A.

Proposition 3.5 A fuzzy set µ of a pseudo-Bl-algebra A is a fuzzy filter of A if and only if µ

is an anti fuzzy filter of A.

Proof Let µ be a fuzzy filter of A and x, y ∈ A. Then by (ff1) and definition of µ

we have

µ

(x  y) = 1 − µ (x  y) ¬ 1 − (µ (x) ∧ µ (y)) =

= 1 − 

1 − µ

(x)

∧ 1 − µ

(y)
 = µ

(x) ∨ µ

(y) . Now, let x ¬ y. Then µ (x) ¬ µ (y) and hence 1 − µ (y) ¬ 1 − µ (x). We obtain µ

(y) ¬ µ

(x) . Hence µ

is an anti fuzzy filter of A.

Converse can be proved similarly.

Remark 3.6 Let F be a filter of a pseudo-BL-algebra A and let α, β ∈ [0, 1] such that α > β. The complement µ

(α, β) of µ

(α, β) is given by

µ

(α, β) (x) =

 1 − α if x ∈ F 1 − β otherwise . By Proposition 3.5 it is an anti fuzzy filter of A.

Theorem 3.7 Let µ be an anti fuzzy filter of a pseudo-BL-algebra A. Then the set defined as follows

A

= {x ∈ A : µ (x) = µ (1)}

is a filter of A.

Proof Let x, y ∈ A

. Then µ (x) = µ (y) = µ (1). Hence, by Definition 3.1 µ (x  y) ¬ µ (x) ∨ µ (y) = µ (1) . Applying Proposition 3.2 we get µ (x  y) = µ (1).

Therefore (F1) holds. Now, take x, y ∈ A such that x ∈ A

and x ¬ y. By Definition 3.1 and Proposition 3.2, µ (1) ¬ µ (y) ¬ µ (x) = µ (1). We have y ∈ A

. Concluding

A

is a filter.

Remark 3.8 The conversely theorem does not hold.

Example 3.9 Let A be a pseudo-BL-algebra. Let µ be a fuzzy set of A defined as follows

µ (x) =

 0, 7 if x = 1 0, 5 if x 6= 1 .

Then A

= {1} and it is a filter of A but µ (1) ­ µ (x). Therefore, by Proposition

3.2 µ is not an anti fuzzy filter of A.

Let A be a pseudo-BL-algebra and µ be a fuzzy set of A. Let us define a lower α −level of µ as follows

L (µ; α) := {x ∈ A : µ (x) ¬ α}

where α ∈ [0, 1] .

It is easily seen that L (µ; 1) = A and A = U (µ; α) ∪ L (µ; α) for α ∈ [0, 1] . If α ¬ β, then L (µ; α) ⊆ L (µ; β) .

Theorem 3.10 Let µ be a fuzzy set of a pseudo-BL-algebra A. Then µ is an anti fuzzy filter of A if and only if for each α ∈ [0, 1], L (µ; α) is a filter of A or L (µ; α) =

∅.

Proof Assume that µ is an anti fuzzy filter and let α ∈ [0, 1] such that L (µ; α) 6= ∅ and letx, y ∈ L (µ; α). Then µ (x) ¬ α and µ (y) ¬ α. It follows from (af1) that

µ (x  y) ¬ µ (x) ∨ µ (y) ¬ α.

Thus x  y ∈ L (µ; α). Now suppose that x ¬ y and x ∈ L (µ; α). Then by (af2) µ (y) ¬ µ (x) ¬ α. Hence y ∈ L (µ; α) . Therefore L (µ; α) is a filter of A.

Conversely, let L (µ; α) be a filter of A or L (µ; α) = ∅ for any α ∈ [0, 1]. Assume that (af1) does not valied.

Then there are a, b ∈ A such that µ (a  b) > µ (a) ∨ µ (b) . Let us define β = 1

2 (µ (a  b) + µ (a) ∨ µ (b)) ,

we get µ (a  b) > β > µ (a) ∨ µ (b), β > µ (a) and β > µ (b) . Hence a, b ∈ L (µ, β) and a  b /∈ L (µ, β). This is a contradiction with (F1) of the definition of a filter.

Hence (af1) holds.

Now suppose that (af2) does not hold. Then there exist a, b ∈ A such that a ¬ b and µ (a) ¬ µ (b). Taking

γ = 1

2 (µ (a) + µ (b))

we have µ (a) < γ < µ (b) . Hence a ∈ L (µ, γ) and b /∈ L (µ, γ). It is contradiction

with (F2). Thus µ is an anti fuzzy filter of A.

Corollary 3.11 If µ is an anti fuzzy filter of pseudo-BL-algebra A, then the set A

= {x ∈ A : µ (x) ¬ µ (b)}

is a filter of A for every b ∈ A.

Let µ

be anti fuzzy filters of pseudo-BL-algebra A for every t ∈ T . We define a fuzzy set _

t∈T

µ

as follows:

_

t∈T

µ

!

(x) = _

{µ

(x) : t ∈ T } .

Theorem 3.12 Let µ

be anti fuzzy filters of pseudo BL-algebra A for all t ∈ T . Then _

t∈T

µ

is also an anti fuzzy filter of A.

Proof Let µ

be anti fuzzy filters of A for t ∈ T and µ = _

t∈T

µ

. Let x, y ∈ A.

Then, by (af1), we have µ

(x  y) ¬ µ

(x) ∨ µ

(y). Hence µ (x  y) = _

{µ

(x  y) : t ∈ T } ¬ _

{µ

(x) ∨ µ

(y) : t ∈ T }

= _

{µ

(x) : t ∈ T } ∨ _

{µ

(y) : t ∈ T }

= µ (x) ∨ µ (y) .

Now, let x, y ∈ A such that x ¬ y. Then for every t ∈ T we have µ

(x) ¬ µ

(y) . Thus

µ (x) = _

{µ

(x) : t ∈ T } ¬ _

{µ

(y) : t ∈ T } = µ (y) .

Consequently, µ is an anti fuzzy filter of A.

4. Anti fuzzy prime filters.

Definition 4.1 A non-constant anti fuzzy filter µ of pseudo-BL-algebra A is called an anti fuzzy prime if

µ (x ∨ y) = µ (x) ∧ µ (y) for all x, y ∈ A.

Example 4.2 Let P be a prime filter of pseudo-BL-algebra A and α ∈ (0, 1]. Let us define a fuzzy set µ by

µ (x) =

 0 if x ∈ P, α otherwise .

We show that µ is an anti fuzzy prime filter of A. By Remark 3.6, µ is an anti fuzzy filter. We need to prove that µ is prime.

Let x, y ∈ A such that x ∨ y ∈ P . Then by the definition of prime filters, x ∈ P or y ∈ P . Then

µ (x ∨ y) = 0 = µ (x) ∧ µ (y) .

Now suppose that x ∨ y /∈ P. By (F2), since x, y ¬ x ∨ y, then x, y /∈ P . Thus µ (x ∨ y) = α = µ (x) ∧ µ (y) .

Theorem 4.3 Let µ be a non-constant anti fuzzy filter of a pseudo-BL-algebra A.

Then the following are equivalent:

(i) µ is an anti fuzzy prime filter of A;

(ii) for all x, y ∈ A, if µ (x ∨ y) = µ(1), then µ (x) = µ(1) or µ (y) = µ(1);

(iii) for all x, y ∈ A, µ (x → y) = µ(1) or µ (y → x) = µ(1);

(iv) for all x, y ∈ A, µ (x ; y) = µ(1) or µ (y ; x) = µ(1).

Proof (i) ⇒ (ii) : Letµ be an anti fuzzy prime filter of A and x, y ∈ A such that µ (x ∨ y) = µ(1). Then µ (x) ∧ µ (y) = µ (x ∨ y) = µ(1) and hence µ (x) = µ(1) or µ (y) = µ(1).

(ii) ⇒ (iii) : By (C5) µ ((x → y) ∨ (y → x)) = µ (1) and by (ii), we have µ (x → y) = µ(1) or µ (y → y) = µ(1).

(iii) ⇒ (i) : Let x, y ∈ A. Suppose that for instance, µ (x → y) = 1. By Lemma 2.2 (b), (a) and (i) we have

y ­ (x ∧ y) ∨ ((x → y)  y) = ((x → y)  x) ∨ ((x → y)  y) =

= (x → y)  (x ∨ y) . Hence

µ (y) ¬ µ ((x → y)  (x ∨ y)) ¬ µ (x → y) ∨ µ (x ∨ y) =

= µ (1) ∨ µ (x ∨ y) = µ (x ∨ y) ¬ µ (y) .

Therefore µ (x ∨ y) = µ (y) . Since x ¬ x ∨ y, then µ (x) ­ µ (x ∨ y) = µ (y). Hence µ (x ∨ y) = µ (x) ∧ µ (y). Analogously, from µ (y → x) = µ (1) we have µ (x ∨ y) = µ (x) ∧ µ (y) .

So µ is an anti fuzzy prime filter of A.

Analogously there are proved the implications (ii) ⇒ (iv) ⇒ (i).

Theorem 4.4 Let µ be an anti fuzzy filter of a pseudo-BL-algebra A. Then µ is an anti fuzzy prime filter of A if and only if A

is a prime filter.

Proof Let µ be an anti fuzzy prime filter of a pseudo-BL-algebra A. From Theorem 3.7 follows A

is a filter of A. Now we show that this filter is prime. Let x ∨ y ∈ A

. Then, by assumption and Definition 4.1, µ (1) = µ (x ∨ y) = µ (x)∧µ (y). Therefore µ (x) = µ (1) or µ (y) = µ (1). It is proved that either x ∈ A

or y ∈ A

. Hence A

is a prime filter.

Conversely, let A

be a prime filter of A. Let x, y ∈ A. Then, by (C5) (x → y) ∨ (y → x) = 1 ∈ A

and by definition of prime filters, x → y ∈ A

or y → x ∈ A

. Hence, µ (x → y) = µ (1) or µ (y → x) = µ (1). By Theorem 4.3, µ is an anti fuzzy

prime filter of A.

Corollary 4.5 Let µ be an anti fuzzy prime filter of a pseudo-BL-algebra A.

Then P = {x ∈ A : µ (x) = 0} is either empty set or a prime filter of A.

Theorem 4.6 Let A be a pseudo-BL-algebra, P be a filter of A and α, β ∈ [0, 1]

with α > β. Then P is a prime filter of A if and only if µ

(α, β) defined as in

Remark 3.6 is an anti fuzzy prime filter of A.

Proof Assume that P is a prime filter of A. Since P is proper, µ

(α, β) is non- constant. Let x, y ∈ A. Then, by (C5), (x → y) ∨ (y → x) = 1 ∈ P . Then by the definition of a prime filter, x → y ∈ P or y → x ∈ P , i.e., µ

(α, β) (x → y) = 1 − α = µ

(1) or µ

(α, β) (y → x) = 1 − α = µ

(1) . Hence, by Theorem 4.3, µ

(α, β) is an anti fuzzy prime filter of A.

Conversely, if µ

(α, β) is an anti fuzzy prime filter of A, then by Theorem 4.4, P = A

P(α,β)

is a prime filter of A.

Theorem 4.7 Let µ be a non-constant anti fuzzy filter of a pseudo-BL-algebra A.

Then the following are equivalent:

(i) µ is an anti fuzzy prime filter of A;

(ii) for every α ∈ [0, 1], if L (µ, α) 6= ∅ and L (µ, α) 6= A, then L (µ, α) is a prime filter of A.

Proof (i) ⇒ (ii): Suppose that µ is an anti fuzzy prime filter of a pseudo-BL- algebra A and let L (µ, α) / ∈ {∅, A} . From Theorem 3.10, L (µ, α) is a filter. We need only to show that L (µ, α) is prime. Since L (µ, α) 6= A, it is proper. Let x, y ∈ A such that x ∨ y ∈ L (µ, α). Then µ (x ∨ y) = µ (x) ∧ µ (y) ¬ α. Hence µ (x) ¬ α or µ (y) ¬ α. We have x ∈ L (µ, α) or y ∈ L (µ, α) . Therefore, L (µ, α) is prime.

(ii) ⇒ (i) : Now, let (ii) holds. Suppose that µ is not an anti fuzzy prime filter.

Then µ (x ∨ y) < µ (x) ∧ µ (y) for some x, y ∈ A. Let us define β = 1

2 (µ (x ∨ y) + (µ (x) ∧ µ (y))) . Then we have

µ (x ∨ y) < β < µ (x) ∧ µ (y) .

We obtain that x ∨ y ∈ L (µ, β) and x /∈ L (µ, β) and y /∈ L (µ, β) . Hence L (µ, β) 6= ∅ but L (µ, β) is not prime. It contradicts an assumption. Then µ is an

anti fuzzy prime filter of A.

Theorem 4.8 Let µ and ν be anti fuzzy filters of pseudo-BL-algebra A. If µ is an anti fuzzy prime, µ ­ ν and µ (1) = ν (1), then ν is an anti fuzzy prime filter of A.

Proof Let µ be an anti fuzzy prime filter of A, ν be an anti fuzzy filter such that µ ­ ν and µ (1) = ν (1). By Theorem 4.3, µ (x → y) = µ (1) or µ (y → x) = µ (1).

By assumption, ν (x → y) = ν (1) or ν (y → x) = ν (1). Therefore ν is an anti fuzzy

prime filter.

Theorem 4.9 Let A be a non-trivial pseudo-BL-algebra. The following are equiva- lent:

(i) A is linearly ordered (pseudo-BL-chain);

(ii) Every non-constant anti fuzzy filter of A is an anti fuzzy prime filter of A;

(iii) Every non-constant anti fuzzy filter µ of A such that µ (1) = 0 is an anti fuzzy prime filter of A;

(iv) The anti fuzzy filter χ

is an anti fuzzy prime filter of A.

Proof (i) ⇒ (ii): Suppose that A is a linearly ordered pseudo-BL-algebra and µ is a non-constant anti fuzzy filter of A. Let x, y ∈ A. Then by assumption, x ¬ y or y ¬ x. By Lemma 2.2 (e), we have x → y = 1 or y → x = 1. Hence µ (x → y) = µ (1) or µ (y → x) = µ (1) and by Theorem 4.3, µ is an anti fuzzy prime filter.

(ii) ⇒ (iii) and (iii) ⇒ (iv) Obvious

(iv) ⇒ (i): If χ

is an anti fuzzy prime filter of A, then by Theorem 4.3, χ

(x → y) = χ

(1) or χ

(y → x) = χ

(1), then x → y ∈ {1} or y → x ∈ {1}. By Lemma 2.2 (e), we have x ¬ y or y ¬ x. Hence A is linearly ordered.

Theorem 4.10 Let µ be an anti fuzzy prime filter of pseudo-BL-algebra A and α ∈ (µ (1) , 1]. Then µ ∧ α is an anti fuzzy prime filter of A, where

(µ ∧ α) (x) = µ (x) ∧ α.

Proof Firstly, we prove that µ ∧ α is an anti fuzzy filter of A. Let x, y ∈ A such that x ¬ y. Then µ (y) ¬ µ (x) and hence µ (y) ∧ α ¬ µ (x) ∧ α i.e., (µ ∧ α) (y) ¬ (µ ∧ α) (y). Moreover, since µ (x  y) ¬ µ (x) ∨ µ (y), we conclude that µ (x  y) ∧ α ¬ (µ (x) ∧ α) ∨ µ (y ∧ α) and hence (µ ∧ α) (x  y) ¬ (µ ∧ α) (x) ∨ (µ ∧ α) (y).

Thus µ ∧ α is an anti fuzzy filter.

Since µ is non-constant, µ (x

) > µ (1) for some x

∈ A. Then we get (µ ∧ α) (x

) = µ (x

)∧α > µ (1) ­ µ (1)∧α = (µ ∧ α) (1). Hence µ∧α is a non-constant anti fuzzy filter. We have (µ ∧ α) (1) = µ (1) and µ ∧ α ¬ µ. Then by Theorem 4.8, µ ∧ α is

an anti fuzzy prime filter of A.

Theorem 4.11 Let µ be a non-constant anti fuzzy filter of pseudo-BL-algebra A and µ (1) 6= 0. Then there exists an anti fuzzy prime filter ν such that µ ­ ν.

Proof Let µ be a non-constant anti fuzzy filter of A. Then A

is a proper filter of A.

Then by Proposition 2.5, there is a prime filter P such that A

⊆ P . By Theorem 4.6 χ

is an anti fuzzy prime filter of A. Let us define ν = χ

∧ α, where α =

^ {µ (x) : x ∈ A − P } . Hence α ­ µ (1) > 0. Consequently, α ∈ (0, 1] = χ

, 1  . From Theorem 4.10 we get that ν is an anti fuzzy prime filter of A. Obviously,

µ ­ ν.

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Magdalena Wojciechowska-Rysiawa

Institute of Mathematics and Physics, University of Natural Sciences and Humanities Maja 54, 08–110 Siedlce, Poland

E-mail: magdawojciechowska6@wp.pl

(Received: 27.02.11)