Fuzzy filters in pseudo hoops

1 Introduction

Non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information. The algebraic counterparts of some non-classical logics satisfy residuation and those logics can be considered in a frame of residuated lattices. For example, Hájek’s BL (basic logic [16]), Lukasiewiczs MV (many-valued logic [8]) and MTL (monoidal t-norm based logic [11]) are determined by the class of BL-algebras, MV-algebras and MTL-algebras, respectively. Pseudo-MV algebras were introduced as a non-commutative generalization of MV-algebras (see [13]). Equivalent structures were defined and studied in [7], under the name of pseudo-Wajsberg algebras. Pseudo-Wajsberg algebras are a non-commutative version of Wajsberg algebras. A. Dvurecenskij proved in [10] that the category of pseudo-MV algebras is equivalent to the category of ℓ-groups with strong unit. In [9], pseudo-BL algebras were defined as a common extension of BL-algebras and pseudo-MV algebras. The main source of examples of pseudo-BL algebras is ℓ-group theory. Hoops are naturally ordered commutative residuated integral monoids, introduced by B. Bosbach in [5, 6], then studied by J.R. Büchi and T.M. Owens in a paper never published. In the last years, hoops theory was enriched with deep structure theorems. Many of these results have a strong impact with fuzzy logic. Particularly, from the structure theorem of finite basic hoops ([1], Corollary 2.10) one obtains an elegant short proof of the completeness theorem for propositional basic logic ([1], Theorem 3.8). In [16], G. Georgescu, L. Leustean and V. Preoteasa presented pseudo hoops which were originally introduced by Bosbach in [5, 6] under the name residuated integral monoids. The prefix “pseudo” stands for non-commutative or not necessarily commutative type of algebra. Pseudo-hoops are monoids endowed with orders. Moreover, the orders are canonical (actually inverse canonical) they are given by divisibility relations w.r.t. the monoid operation and the orders have residuals. In [16], Georgescu studied about their basic properties and defined filters and normal filters and proved that the lattice of normal filters and the lattice of congruences of a pseudo-hoop are isomorphic. They investigated some classes of pseudo-hoops, namely, cancellative pseudo-hoops,Wajsberg pseudo-hoops, basic pseudo-hoops, product pseudo-hoops, and (strongly) simple pseudo-hoops. The most important of these classes are Wajsberg pseudo-hoops and basic pseudo-hoops. Pseudo BL-algebras will appear as particular cases of pseudo-hoops. Also, all we know that the concept of fuzzy set was introduced by Zadeh [18] and this idea has been applied to other algebraic structures such as groups, semigroups, rings and topologies. With the development of fuzzy set, it is widely used in many fields. Now, in this paper, we introduce the notion of some types of fuzzy filters (implicative, positive implicative and fantastic filters) in pseudo hoops and study some of their properties. Also, we defined the notion of fuzzy congruence and fuzzy quotient algebras on pseudo hoops and we investigated fuzzy quotient that is made by different kinds of fuzzy filters.

2 Preliminaries

In this section, we recollect some definitions and results which will be used in this paper. A pseudo hoop is an algebra (A, ⊙ , → , ⇝ , 1) such that, for all x, y, z ∈ A,

(PH-1) (A, ⊙ , 1) is a monoid with unit 1,

(PH-2) x ⊙ y ≤ z if and only if x ≤ y → z if and only if y ≤ x ⇝ z,

(PH-3) (x → y) ⊙ x = x ⊙ (x ⇝ y). (Divisibility condition)

A pseudo hoop can be thought as an algebra (A, ⊙ , → , ⇝ , 1), where for all x, y ∈ A, x ∧ y = x ⊙ (x ⇝ y). Then, it is easy to see that (A, ∧ , 1) is a ∧-semilattice.

On pseudo hoop A, we define x ≤ y if and only if x → y = 1 (this is equivalent to x ⇝ y = 1) and it is easy to see that ≤ is a partial order relation on A. If ⊙ is commutative (equivalent → =⇝), A is called hoop. We say that a pseudo hoop A is bounded if it has an element 0 ∈ A such that 0 ≤ x, for all x ∈ A. We let x⁰ = 1, x ⁿ  = x^n-1 ⊙ x, for any n ∈ ℕ. We define two unary operations, x^- = x → 0 and x ~ = x ⇝ 0, for all x ∈ A. If (x^-) ^∼ = (x^∼) ^- = x, for all x ∈ A, then the bounded pseudo hoop A is said to have the pseudo double negation property, (PDN) for short (see [16]).

A lattice-ordered group or ℓ-group is a group which is also a lattice that satisfies identities x (y ∧ z) t = xyt ∧ xzt and x (y ∨ z) t = xyt ∨ xzt. Throughout we write x ≤ y instead of x ∨ y = y or x ∧ y = x, and ℓ-group as an observation for lattice-ordered group.

Example 2.1. [16] Let (G, + , - , 0, ∧ , ∨) be an arbitrary ℓ-group. For an arbitrary element 0 ≤ u ∈ G define the following operations, on the set [0, u], x ⊙ y = ( x - u + y ) ∨ 0 , x → y = ( y - x + u ) ∧ u and x ⇝ y = ( u - x + y ) ∧ u for any x, y ∈ [0, u]. By routine calculation, we can see that ([0, u] , ⊙ , → , ⇝ , u) is a bounded pseudo hoop.

The following proposition provide some properties of pseudo hoops.

Proposition 2.2. [5, 6] Let (A, ⊙ , → , ⇝ , 1) be a pseudo hoop. Then the following conditions hold, for all x, y, z, a ∈ A,

x → x = x ⇝ x = 1,

Let A be a pseudo hoop. A non-empty subset F of A is a filter if it satisfies,

x, y ∈ F implies x ⊙ y ∈ F,

A filter F of A is proper if and only if F ≠ A. Denote, F ( A ) the set of all fuzzy filter of A.

Proposition 2.3. [16] Let A be a pseudo hoop and F be a non-empty subset of A such that 1 ∈ F. Then the following statements are equivalent, for any x, y, z ∈ A,

F is a filter,

Let A be a pseudo hoop and F ∈ F ( A ). Then F is called normal if x → y ∈ F if and only if x ⇝ y ∈ F, for all x, y ∈ A.

Let A be a pseudo hoop. We define the binary operations ∨₁ and ∨₂ on A as follows: x ∨ 1 y = ( ( x → y ) ⇝ y ) ∧ ( ( y → x ) ⇝ x ) and x ∨ 2 y = ( ( x ⇝ y ) → y ) ∧ ( ( y ⇝ x ) → x )

Now, if ∨₁ (respectively ∨₂) is associative, then by [Proposition 2.12, [4]], ∨₁ (respectively ∨₂) is a join operation on A and so by [Lemma 2.9(iii), [4]], that (A, ∧ , ∨ ₁) is a distributive lattice.

Let A be a pseudo hoop and x, y, z ∈ A. If x ∨ y exists, then (x ∨ y) → z = (x → z) ∧ (y → z) and (x ∨ y) ⇝ z = (x ⇝ z) ∧ (y ⇝ z).

A fuzzy set on a non-empty set A, is a map μ : A → [0, 1]. Let μ be a fuzzy set in A. Then, for all α ∈ [0, 1], the set μ _α  = {x ∈ A ∣ μ (x) ≥ α} is called a level subset of μ.

Note: From now one, we let A = (A, ⊙ , → , ⇝ , 1) be a pseudo hoop and μ be a fuzzy set on A, unless otherwise state.

3 Fuzzy filters in pseudo hoops

In this section, we introduce the notion of fuzzy filter in a pseudo hoop and investigate some of it’s properties. Also, we define a fuzzy congruence relation on a pseudo hoop.

Definition 3.1. Let μ be a fuzzy set on A. Then μ is called a fuzzy filter of A, if for all x, y ∈ A,

μ (x ⊙ y) ≥ μ (x) ∧ μ (y),

Example 3.2. In Example 2.1, A = ([0, u] , ⊙ , ⇝ , →) is a bounded pseudo hoop. We define the fuzzy set μ on A by μ (a) = m and μ (u) = n, for all a ∈ A, where a ≠ u, such that m, n ∈ [0, 1] and m ≤ n. Hence μ is a fuzzy filter of A.

Theorem 3.3. μ is a fuzzy filter of A if and only if, for all α ∈ [0, 1], μ _α ≠ ∅ is a filter of A.

Proof. The proof is straight forward. □

Proposition 3.4. μ is a fuzzy filter of A if and only if, for all x, y ∈ A, the following statements hold:

μ (x) ≤ μ (1),

Proof. (⇒) Let μ be a fuzzy filter of A. Since x ≤ 1, by (FF2), μ (x) ≤ μ (1). Moreover, by Proposition 2.2 (vii), (x → y) ⊙ x ≤ y. Then by (FF2), μ ((x → y) ⊙ x) ≤ μ (y). Also, by (FF1), μ (x) ∧ μ (x → y) ≤ μ ((x → y) ⊙ x)   ≤  μ (y). Hence, μ (x) ∧ μ (x → y) ≤ μ (y). The proof of other case is similar.

(⇐) Let x ≤ y. Then x ∧ y = x and so (x ∧ y) → y = 1. By (ii), μ ((x ∧ y) → y) ∧ μ (x ∧ y) ≤ μ (y). Then μ (1) ∧ μ (x) ≤ μ (y) and so μ (x) ≤ μ (y). Since x ⊙ y ≤ x ⊙ y, by (PH-2), x ≤ y → (x ⊙ y). Hence by (FF2), μ (x) ≤ μ (y → (x ⊙ y)) and so, μ (y) ∧ μ (x) ≤ μ (y) ∧ μ (y → (x ⊙ y)). Then by (ii), μ (y) ∧ μ (y → (x ⊙ y)) ≤ μ (x ⊙ y). Hence, μ (x) ∧ μ (y) ≤ μ (x ⊙ y). Therefore, μ is a fuzzy filter of A. □

Proposition 3.5. Let μ be a fuzzy filter of A. Then the following statements are equivalent, for all x, y ∈ A,

μ (x² → y) ≤ μ (x → y) and μ (x² ⇝ y) ≤ μ (x ⇝ y),

Proof. (i ⇒ ii) Let μ be a fuzzy filter of A. Then by Proposition 2.2(i), 1 = x² → x². Thus by Proposition 2.2(v), 1 = x² → (1 ⇝ x²), and so μ (1) = μ (x² → (1 ⇝ x²)). By assumption, μ (1) = μ (x² → (1 ⇝ x²)) ≤ μ (x → (1 ⇝ x²)) = μ (x → x²). Therefore, by Proposition 3.4(i), μ (x → x²) = μ (1). The proof of other case is similar.

(ii ⇒ i) Let μ be a fuzzy filter of A. Then by Proposition 2.2(ix), x → x² ≤ (x² → y) ⇝ (x → y). By assumption, μ (1) = μ (x → x²) ≤ μ ((x² → y) ⇝ (x → y)). Hence, by Proposition 3.4(i), μ ((x² → y) ⇝ (x → y)) = μ (1). By Proposition 3.4(iii), μ ((x² → y) ⇝ (x → y)) ∧ μ (x² → y) ≤ μ (x → y). Then μ (1) ∧ μ (x² → y) ≤ μ (x → y), and so μ (x² → y) ≤ μ (x → y). The proof of other case is similar. □

Definition 3.6. Fuzzy filter μ on A is called a fuzzy normal filter if, for all x, y ∈ A, μ (x → y) = μ (1) if and only if μ (x ⇝ y) = μ (1).

Theorem 3.7. μ is a fuzzy normal filter of A if and only if for all α ∈ [0, 1], μ _α ≠ ∅ is a normal filter of A.

Proof. The proof is straightforward. □

Example 3.8. In Example 2.1, A = ([0, u] , ⊙ , ⇝ , →) is a bounded pseudo hoop. By [16], let K be a normal convex ℓ-group of G and μ _α  = {a ∈ [0, u] ∣ u - a ∈ K} is a normal filter of [0, u]. Now, we define fuzzy set μ on A by for all x ∈ A, where x ∈ μ _α , μ (x) = μ (1) and x ∉ μ _α , μ (x) = n such that n ∈ [0, 1]. Hence by Theorem 3.7, μ is a fuzzy normal filter of A.

Lemma 3.9. Let μ be a fuzzy filter. We define a relation ∼ _{μ μ(1)} on A as follows, x ∼ μ μ ( 1 ) y if and only if x → y ∈ μ μ ( 1 ) , y → x ∈ μ μ ( 1 ) ( or x ⇝ y ∈ μ μ ( 1 ) , y ⇝ x ∈ μ μ ( 1 ) )

Then, for all x, y ∈ A,

x ∼  _{μ μ(1)} y if and only if α ⊙ x = β ⊙ y, for some α, β ∈ μ_μ(1),

Proof. (i) Suppose x ∼  _{μ μ(1)} y, for x, y ∈ A. Then, by Lemma 3.9, x → y, y → x ∈ μ_μ(1). Let α = x → y and β = y → x. Since (x → y) ⊙ x = x ∧ y = y ∧ x = (y → x) ⊙ y, we get α ⊙ x = β ⊙ y. Conversely, let α ⊙ x = β ⊙ y and α, β ∈ μ_μ(1). By Proposition 2.2 (xv), β → (y → x) = (β ⊙ y) → x = (α ⊙ x) → x = 1. So, β ≤ y → x. Since β ∈ μ_μ(1), by (F2), y → x ∈ μ_μ(1). Similarly, α → (x → y) =1, and so x → y ∈ μ_μ(1). Hence, x ∼  _{μ μ(1)} y.

(ii) The proof is similar to the proof of (i). □

Theorem 3.10. Let μ be a fuzzy filter of A. Then ∼ _{μ μ(1)} is a congruence relation on A.

Proof. It is clear that μ_μ(1)≠ ∅ and ∼ _{μ μ(1)} is reflexive and symmetric. Let x ∼  _{μ μ(1)} y and y ∼  _{μ μ(1)} z. Then x ⇝ y, y ⇝ x ∈ μ_μ(1) and y ⇝ z, z ⇝ y ∈ μ_μ(1). By Proposition 2.2(x), (x ⇝ y) ⊙ (y ⇝ z) ≤ x ⇝ z. Then by (FF1) and (FF2), μ (x ⇝ y) ∧ μ (y ⇝ z) ≤ μ ((x ⇝ y) ⊙ (y ⇝ z)) ≤ μ (x ⇝ z). Since μ (x ⇝ y) = μ (y ⇝ z) = μ (1), we get μ (x ⇝ z) = μ (1). By the similar way, μ (z ⇝ x) = μ (1). Hence, x ∼  _{μ μ(1)} z. Therefore, ∼ _{μ μ(1)} is an equivalence relation on A. Suppose x ∼  _{μ μ(1)} y. Then by Lemma 3.9, there exist α, β ∈ μ_μ(1) such that α ⊙ x = β ⊙ y. Thus, for all t ∈ A, (α ⊙ x) ⊙ t = (β ⊙ y) ⊙ t, and so, α ⊙ (x ⊙ t) = β ⊙ (y ⊙ t). Hence, by Lemma 3.9(i), x ⊙ t ∼  _{μ μ(1)} y ⊙ t. Similarly, t ⊙ x ∼  _{μ μ(1)} t ⊙ y. By Proposition 2.2(x), (x → y) ⊙ (t → x) ≤ t → y and (y → x) ⊙ (t → y) ≤ t → x. Then by (PH-2), x → y ≤ (t → x) → (t → y) and y → x ≤ (t → y) → (t → x). Since μ_μ(1) is a filter of A, by (F2), (t → x) → (t → y) ∈ μ_μ(1) and (t → y) → (t → x) ∈ μ_μ(1). Hence, (t → x) ∼  _{μ μ(1)}  (t → y). Similarly, (x → t) ∼  _{μ μ(1)}  (y → t). The compatibility of ∼ _{μ μ(1)} with ⇝ is proved in a similar way. Therefore, ∼ _{μ μ(1)} is a congruence relation on A. □

Definition 3.11. Let μ be a fuzzy filter of A. Then the fuzzy set μ ^x  : A → [0, 1] which is defined by μ ^x  (y) = μ (x → y) ∧ μ (y → x) and μ _x  (y) = μ (x ⇝ y) ∧ μ (y ⇝ x) is called a coset of the fuzzy filter μ.

Example 3.12. Let A = {0, a, b, c, 1}. We define ⊙, → and ⇝ on A as follows:

Routine calculations show that A with these operations is a hoop. Define a fuzzy set μ on A as follows: μ (0) = μ (a) = μ (b) = μ (c) = m, μ (1) = n such that m, n ∈ [0, 1] and m ≤ n. It is easy to see that μ is a fuzzy filter of A. Also, we can see that the coset μ ^x is defined as follows,

Proposition 3.13. Let μ be a fuzzy filter of A. Then μ ^x  = μ ^y if and only if x ∼  _{μ μ(1)} y, for any x, y ∈ A.

Proof. (⇒) Let x, y ∈ A. If μ ^x  = μ ^y , then μ ^x  (x) = μ ^y  (x), and so μ (x → x) = μ (1) = μ (x → y) ∧ μ (y → x). It is clear that, μ (x → y) ∧ μ (y → x) ≤ μ (x → y). Since μ (x → y) ∧ μ (y → x) = μ (1), we get μ (x → y) = μ (1). Hence, x → y ∈ μ_μ(1). By the similar way, y → x ∈ μ_μ(1). Therefore, x ∼  _{μ μ(1)} y.

(⇐) By Proposition 2.2(x) and (FF2), (y → x) ⊙ (z → y) ≤ z → x and μ ((y → x) ⊙ (z → y)) ≤ μ (z → x). Since, (z → y) ≤ (y → x) ⇝ ((y → x) ⊙ (z → y)), we get μ (z → y) ≤ μ ((y → x) ⇝ ((y → x) ⊙ (z → y))). Then μ (y → x) ∧ μ (z → y) ≤ μ (y → x) ∧  μ ((y → x)   ⇝ ((y → x) ⊙    (z → y))). By Proposition 3.4(iii), μ (y → x) ∧ μ (z → y) ≤ μ ((y → x)   ⊙   (z → y)) ≤  μ (z → x) .   By the similar way, μ (x → y) ∧ μ (z → x) ≤ μ (z → y). Since x ∼  _{μ μ(1)} y, x → y, y → x ∈ μ_μ(1), then μ (x → y) = μ (y → x) = μ (1). Thus, μ (z → y) ≤ μ (z → x) and μ (z → x) ≤ μ (z → y). Hence, μ (z → x) = μ (z → y). By the similar way, μ (x → z) = μ (y → z). Therefore, μ ^x  = μ ^y . □

Let μ be a fuzzy filter of A and A/μ = {μ ^x  ∣ x ∈ A}. For all μ ^x , μ ^y  ∈ A/μ, define μ ^x  ∧ μ ^y  = μ^x∧y, μ ^x  ⊙ μ ^y  = μ^x⊙y, μ ^x  → μ ^y  = μ^x→y, μ ^x  ⇝ μ ^y  = μ^x⇝y. If μ ^x  = μ ^y , μ ^t  = μ ^z , for x, y, t, z ∈ A, then by Proposition 3.13, x ∼  _{μ μ(1)} y, t ∼  _{μ μ(1)} z. Since ∼ _{μ μ(1)} is a congruence relation, (x ∧ t) ∼  _{μ μ(1)}  (y ∧ z). Thus, μ^x∧t = μ^y∧z and the operation ∧ is well-defined. By the similar way, we have μ^x⊙t = μ^y⊙z, μ^x→t = μ^y→z, μ^x⇝t = μ^y⇝z. Hence, A/μ = (A/μ, ∧ , ⊙ , → , ⇝ , μ¹) is an algebraic structure.

Proposition 3.14. Let μ be a fuzzy filter of A. Then, for all x, y ∈ A, μ ^x  ≤ μ ^y if and only if μ (x → y) = μ (1).

Proof. (⇒) Suppose μ ^x  ≤ μ ^y , for x, y ∈ A. Then μ ^x  (x) ≤ μ ^y  (x). By Definition 3.11, μ (x → x) ≤ μ (y → x) ∧ μ (x → y). Since μ (x → x) = μ (1), by Proposition 3.4(i), μ (y → x) ∧ μ (x → y) = μ (1). It is clear that, μ (y → x) ∧ μ (x → y) ≤ μ (x → y). Since μ (y → x) ∧ μ (x → y) = μ (1), we have μ (x → y) = μ (1).

(⇐) Let μ (x → y) = μ (1) and μ ^x  ≰ μ ^y . Then, for all z ∈ A, μ ^x  (z) ≰ μ ^y  (z). Since μ ^x , μ ^y  ∈ [0, 1] and μ ^x  ≰ μ ^y , we get μ ^y  < μ ^x . Then μ (y → x) = μ (1), y → x ∈ μ_μ(1). Moreover, by assumption, μ (x → y) = μ (1), then x → y ∈ μ_μ(1). Hence, x ∼  _{μ μ(1)} y and so by Proposition 3.13, μ ^x  = μ ^y , which is a contradiction. Therefore, μ ^x  ≤ μ ^y . □

Theorem 3.15. Let μ be a fuzzy filter of A. Then A/μ = (A/μ, ∧ , ⊙ , → , ⇝ , μ¹) is a pseudo hoop.

Proof. Let μ be a fuzzy filter of A. By Proposition 3.14, it is clear that ≤ is a partial order. Let μ ^x , μ ^y , μ ^z  ∈ A/μ. μ ^x  ⊙ μ ^y  ≤ μ ^z if and only if μ^x⊙y ≤ μ ^z if and only if μ ((x ⊙ y) → z) = μ (1) if and only if μ (x → (y → z)) = μ (1) if and only if μ ^x  ≤ μ^y→z if and only if μ ^x  ≤ μ ^y  → μ ^z . By the similar way, μ ^x  ≤ μ ^y  ⇝ μ ^z . Thus (PH-2) holds. Clearly A/μ satisfies (PH-1) and (PH-3). Hence, A/μ is a pseudo hoop.□

4 Fuzzy implicative filters

In this section, we introduce the notion of fuzzy implicative filter on a pseudo hoops and investigate their properties of it.

Definition 4.1. Let F be a non-empty subset of A. Then F is called an implicative filter of A if, for any x, y, z ∈ A,

1 ∈ F,

Definition 4.2. The fuzzy set μ on A is called a fuzzy implicative filter of A if, for all x, y, z ∈ A,

μ (x) ≤ μ (1).

Example 4.3. Let A = {0, a, b, c, d, 1}. We define ⊙, → and ⇝ on A as follows:

Routine calculations show that A is a pseudo hoop. Define a fuzzy set μ on A as follows: μ (0) = μ (a) = μ (b) = μ (c) = μ (d) = m, μ (1) = n such that m, n ∈ [0, 1] and m ≤ n. One can check that μ is a fuzzy implicative filter of A.

Theorem 4.4. Every fuzzy implicative filter of A is a fuzzy filter.

Proof. Let μ be a fuzzy implicative filter of A. By (FI1), μ (x) ≤ μ (1). Let z = 1 in (FI2). Then μ (x → ((y → 1) ⇝ y) ∧ μ (x) ≤ μ (y) and by Proposition 2.2(v) and (vi), μ (x → y) ∧ μ (x) ≤ μ (y). By the similar way, μ (x ⇝ y) ∧ μ (x) ≤ μ (y). So, by Proposition 3.4, μ is a fuzzy filter. □

The following example shows that the converse of Theorem 4.4 may not be true, in general.

Example 4.5. Let (A, ⊙ , → , ⇝) be pseudo hoop as Example 4.3. We define a fuzzy set μ on A as follows: μ (b) = μ (c) = m, μ (0) = μ (d) = μ (a) = q, μ (1) = n such that q < m < n, q, m, n ∈ [0, 1]. Then μ is a fuzzy filter but it is not a fuzzy implicative filter, because, μ (1 → ((b → c) ⇝ b)) ∧ μ (1) = n ≰ m = μ (b).

Theorem 4.6. μ is a fuzzy implicative filter of A if and only if μ _α ≠ ∅ is an implicative filter of A, for all α ∈ [0, 1].

Proof. The proof is straightforward. □

Theorem 4.7. Let A be bounded and μ be a fuzzy filter of A. Then, for all x, y, z ∈ A, the following conditions are equivalent,

μ is a fuzzy implicative filter of A,

Proof. (i ⇒ ii) Let μ be a fuzzy implicative filter of A. Then, by Proposition 3.4(i), μ (x) ≤ μ (1), for all x ∈ A. By Proposition 2.2(v), (x → y) ⇝ x = 1 → ((x → y) ⇝ x). Then by (FF1) and (FI2), μ (1 → ((x → y) ⇝ x)) ∧ μ (1) ≤ μ (x).

(ii ⇒ i) Let μ be a fuzzy filter of A. Then, by Proposition 3.4(i), μ (x) ≤ μ (1), for all x ∈ A. Then (FI1) holds. By Proposition 3.4(ii), μ ((x → y) ⇝ x) ≥ μ (z → ((x → y) ⇝ x)) ∧ μ (z). By Proposition 2.2(iv), x ≤ (x → y) ⇝ x. Then by (FF2), μ (x) ≤ μ ((x → y) ⇝ x). By assumption, we have μ (x) = μ ((x → y) ⇝ x). Consequently, μ (x) ≥ μ (z) ∧ μ (z → ((x → y) ⇝ x)). Therefore, (FI2) holds. The proof of other case is similar.

(iii ⇒ i) By Proposition 3.4(ii), μ (x) ∧ μ (x → ((y → z) ⇝ y)) ≤ μ ((y → z) ⇝ y). Since μ ((y → z) ⇝ y) ∧ μ ((y → z) ⇝ y) ⇝ y) ≤ μ (y) and by assumption μ (1) = μ (((y → z) ⇝ y) ⇝ y), we have μ ((y → z) ⇝ y) ∧ μ (1) ≤ μ (y). Then μ (x) ∧ μ (x → ((y → z) ⇝ y)) ≤ μ (y). So (FI2) holds. The proof of (FI3) is similar.

(iii ⇒ iv) Since A is bounded, it is enough to take y = 0 in (iii). Then μ (((x ⇝ 0) → x) → x) = μ ((x^∼ → x) → x) = μ (1), for all x ∈ A. The proof of other case is similar.

(iv ⇒ iii) Since A is bounded, for any y ∈ A, 0 ≤ y. By Proposition 2.2(xiv), x^- ≤ x → y. Now, by Proposition 2.2(xii), (x → y) ⇝ x ≤ x^- ⇝ x, and so (x^- ⇝ x) ⇝ x ≤ ((x → y) ⇝ x) ⇝ x. By (FF2), μ ((x^- ⇝ x) ⇝ x) ≤ μ (((x → y) ⇝ x) ⇝ x). Since, μ ((x^- ⇝ x) ⇝ x) = μ (1), by Proposition 3.4(i), μ (((x → y) ⇝ x) ⇝ x) = μ (1).

(v ⇒ vi) Since μ ((y^- ⊙ x) ⇝ y) ∧ μ (y ⇝ y) ≤ μ (x ⇝ y), for x, y ∈ A, by Proposition 2.2(i) and (FF2), μ (y ⇝ y) = μ (1). Then, μ ((y^- ⊙ x) ⇝ y) ≤ μ (x ⇝ y). The proof of other case is similar.

(vi ⇒ v) By Propositions 2.2(x) and 3.4(i), μ ((z^- ⊙ x) ⇝ y) ∧ μ (y ⇝ z) ≤ μ (((z^- ⊙ x) ⇝ y) ⊙ (y ⇝ z)) ≤ μ ((z^- ⊙ x) ⇝ z). By assumption, μ ((z^- ⊙ x) ⇝ z) ≤ μ (x ⇝ z). Then, μ ((z^- ⊙ x) ⇝ y) ∧ μ (y ⇝ z) ≤ μ (x ⇝ z). The proof of other case is similar.

(vi ⇒ iv) By Proposition 2.2 (i), (x^- ⊙ (x^- ⇝ x)) ⇝ x = 1. Then by (FF2), μ ((x^- ⊙ (x^- ⇝ x)) ⇝ x) = μ (1). By (vi), μ (1) = μ ((x^- ⊙ (x^- ⇝ x)) ⇝ x) ≤ μ ((x^- ⇝ x) ⇝ x). By Proposition 3.4(i), μ ((x^- ⇝ x) ⇝ x) = μ (1). Therefore, (iv) holds.

(i ⇒ vi) Let μ be a fuzzy implicative filter of A. Suppose μ is a fuzzy normal filter. Then by Theorem 3.7, μ _α ≠ ∅ is a normal filter, for all α ∈ [0, 1]. By Proposition 2.2(iv), y ≤ x → y. Then by Proposition 2.2(xii), (x → y) →0 ≤ y → 0, Thus, by Proposition 2.2(xii), (y^- ⇝ y) ≤ (x → y) ^- ⇝ y. Now, by Proposition 2.2(xiii), x ⇝ (y^- ⇝ y) ≤ x ⇝ ((x → y) ^- ⇝ y). Hence, by Proposition 2.2(xv), (y^- ⊙ x) ⇝ y ≤ ((x → y) ^- ⊙ x) ⇝ y, and so μ ((y^- ⊙ x) ⇝ y) ≤ μ (((x → y) ^- ⊙ x) ⇝ y). Let μ ((y^- ⊙ x) ⇝ y) = α and α ≤ μ (((x → y) ^- ⊙ x) ⇝ y) = β such that α, β ∈ [0, 1]. Thus, (y^- ⊙ x) ⇝ y ∈ μ _α and ((x → y) ^- ⊙ x) ⇝ y ∈ μ _β . Since α ≤ β, μ _β  ⊆ μ _α , then ((x → y) ^- ⊙ x) ⇝ y ∈ μ _α , and so ((x → y) ^- ⇝ (x → y)) ∈ μ _α . Since μ is a fuzzy implicative filter, by Theorem 4.6, μ _α is an implicative filter, for all α ∈ [0, 1]. Then x → y ∈ μ _α . Moreover, since μ _α is a normal filter, x ⇝ y ∈ μ _α . Then, μ (x ⇝ y) ≥ α = μ ((y^- ⊙ x) ⇝ y). The proof of other case is similar. □

Theorem 4.8 Let A be bounded and μ and η be two fuzzy filters of A such that μ ⊆ η and μ (1) = η (1). If μ is a fuzzy normal and fuzzy implicative filter of A, then η is a fuzzy implicative filter of A, too.

Proof. Let μ and η be two fuzzy filters of A such that μ ⊆ η and μ be a fuzzy implicative filter of A. By Theorem 4.7, it suffices to prove that (vi) holds. Let t = (y^- ⊙ x) ⇝ y. By Proposition 2.2(i), t → t = 1, then t → ((y^- ⊙ x) ⇝ y) =1. By Proposition 2.2(ii), (y^- ⊙ x) ⇝ (t → y) =1. Then by (FF2), μ ((y^- ⊙ x) ⇝ (t → y)) = μ (1). By Proposition 2.2(iv), y ≤ t → y and by Proposition 2.2(xii), (t → y) →0 ≤ y → 0. Then by Proposition 2.2(xi), (t → y) ^- ⊙ x ≤ y^- ⊙ x. Again, by Proposition 2.2(xii), (y^- ⊙ x) ⇝ (t → y) ≤ ((t → y) ^- ⊙ x) ⇝ (t → y). By (FF2), μ ((y^- ⊙ x) ⇝ (t → y)) ≤ μ (((t → y) ^- ⊙ x) ⇝ (t → y)). Consequently, μ (((t → y) ^- ⊙ x) ⇝ (t → y)) = μ (1). Since μ is a fuzzy implicative filter, by Theorem 4.7(v), μ (1) = μ (((t → y) ^- ⊙ x) ⇝ (t → y)) ≤ μ (x ⇝ (t → y)). Then μ (x ⇝ (t → y)) = μ (1). Also, from μ ⊆ η, μ (1) = μ (x ⇝ (t → y)) ⊆ η (x ⇝ (t → y)). By assumption, μ (1) = η (1), then η (x ⇝ (t → y)) = η (1). Thus, by Proposition 2.2(ii), η (t) ∧ η (t → (x ⇝ y)) ≤ η (x ⇝ y), and so η (t) ≤ η (x ⇝ y). Hence, η ((y^- ⊙ x) ⇝ y) ≤ η (x ⇝ y). Therefore, η is a fuzzy implicative filter. □

Proposition 4.9. Let μ be a fuzzy implicative filter of A. Then μ ((x ⇝ y) → y) = μ ((y ⇝ x) → x) and μ ((x → y) ⇝ y) = μ ((y → x) ⇝ x), for all x, y ∈ A.

Proof. Let μ be a fuzzy implicative filter. Then by Theorem 4.6, μ _α ≠ ∅ is an implicative filter, for all α ∈ [0, 1]. Suppose μ ((x ⇝ y) → y) = α. Then (x ⇝ y) → y ∈ μ _α . By Proposition 2.2(viii), y ≤ (y → x) ⇝ x. Thus, by Proposition 2.2(xiii), (x → y) ⇝ y ≤ (x → y) ⇝ ((y → x) ⇝ x), by (F2), (x → y) ⇝ ((y → x) ⇝ x) ∈ μ _α . By the similar way, we have x ≤ (y → x) ⇝ x, then by Proposition 2.2(xii), ((y → x) ⇝ x) → y ≤ x → y and (x → y) ⇝ ((y → x) ⇝ x) ≤ (((y → x) ⇝ x) → y) ⇝ ((y → x) ⇝ x). Consequently, by (F2), (((y → x) ⇝ x) → y) ⇝ ((y → x) ⇝ x) ∈ μ _α . Since μ _α is an implicative filter, by Proposition 2.2(v), 1 → ((((y → x) ⇝ x) → y) ⇝ ((y → x) ⇝ x)) ∈ μ _α . Also, from 1 ∈ μ _α , by (IF2), (y → x) ⇝ x ∈ μ _α . So, μ ((y → x) ⇝ x) ≥ α. Hence, μ ((y → x) ⇝ x) ≥ μ ((x → y) ⇝ y). By the similar way, we have μ ((y → x) ⇝ x) ≤ μ ((x → y) ⇝ y). The proof of other case is similar. □

Theorem 4.10. Let A be bounded with (PDN) such that the operations ∨₁ and ∨₂ exist. If μ is a fuzzy normal implicative filter of A, then A/μ is a Boolean algebra.

Proof. Since the operations ∨₁ and ∨₂ exist, A is a distributive lattice. Let μ be a fuzzy implicative filter. By Theorem 4.7(iv), μ ((x^∼ → x) → x) = μ (1). Since μ is a fuzzy normal filter, μ ((x^∼ → x) ⇝ x) = μ (1). By Proposition 4.9, μ ((x → x^∼) ⇝ x^∼) = μ (1), μ (x ∨ ₁x^∼) = μ ((x → x^∼) ⇝ x^∼) ∧ μ ((x^∼ → x) ⇝ x) = μ (1). Since μ (x ∨ ₁x^∼) = μ (1), by Definition 3.9 and Proposition 3.13, μ^x∨₁x∼ = μ¹. By the similar way, μ^x∨₂x∼ = μ¹. Hence, μ ^x  ∨ ₁μ ^{x ∼}  = μ¹ and μ ^x  ∨ ₂μ ^{x ∼}  = μ¹. Moreover, since x ∧ x^∼ ≤ x, by Proposition 2.2(xii), x → 0 ≤ (x ∧ x^∼) →0, x^- ≤ (x ∧ x^∼) →0. By the similar way, x ∧ x^∼ ≤ x^∼, x^∼- ≤ (x ∧ x^∼) →0. Since A has (PDN), x ≤ (x ∧ x^∼) →0. Hence, (x ∧ x^∼) →0 is an upper bound of {x, x^-}. Thus, x ∨ ₂x^- ≤ (x ∧ x^∼) →0, and so μ (x ∨ ₂x^-) ≤ μ ((x ∧ x^∼) →0). Since μ (x ∨ ₂x^-) = μ (1), by (FI1), μ ((x ∧ x^∼) →0) = μ (1). By the similar way, μ ((x ∧ x^∼) ⇝0) = μ (1). Then x ∧ x^∼ ∼  _{μ μ(1)} 0. Hence, μ^x∧x- = μ⁰ = μ^x∧x∼. Therefore, A/μ is a Boolean algebra. □

5 Fuzzy positive implicative filters

In this section, we introduce the notion of fuzzy positive implicative filters in pseudo hoops and investigate their properties and we study the relation between them and fuzzy implicative filters.

Definition 5.1. [4] A non-empty subset F of A is called a positive implicative filter of A if,

1 ∈ F,

Definition 5.2. The fuzzy subset μ of A is called a fuzzy positive implicative filter of A if for all x, y, z ∈ A,

μ (x) ≤ μ (1),

Example 5.3. In Example 3.12, we define the fuzzy set μ on A as by μ (0) = μ (a) = μ (b) = μ (c) = m, μ (1) = n and m, n ∈ [0, 1] such that m ≤ n. It is easy to see that μ is a fuzzy positive implicative filter of A.

Theorem 5.4. Let μ be a fuzzy filter of A. Then μ is a fuzzy positive implicative filter of A if and only if μ _α ≠ ∅ is a positive implicative filter of A, for all α ∈ [0, 1].

Proof. The proof is straight forward. □

Theorem 5.5. Every fuzzy positive implicative filter of A is a fuzzy filter.

Proof. Let μ be a fuzzy positive implicative filter of A. By (FPI1), μ (x) ≤ μ (1). Let x = 1 in (FPI2). Then μ (1 → (y → z)) ∧ μ (1 ⇝ y) ≤ μ (1 → z) and by Proposition 2.2(v) and (vi), μ (y → z) ∧ μ (y) ≤ μ (z). By the similar way, μ (y ⇝ z) ∧ μ (y) ≤ μ (z). Therefore, μ is a fuzzy filter of A. □

The following example shows that the converse of Theorem 5.5 may not be true, in general.

Example 5.6. According to Example 4.3, μ is a fuzzy filter of A but it is not a fuzzy positive implicative filter, because μ (d → (b → 0)) ∧ μ (d ⇝ b) = n ≰ m = μ (d → 0).

Theorem 5.7. Let A be bounded, μ and η be two fuzzy filters of A such that μ ⊆ η. If μ is a fuzzy positive implicative filter, then η is a fuzzy positive implicative filter, too.

Proof. Let μ and η be two fuzzy filters of bounded pseudo hoop A such that μ ⊆ η and μ be a fuzzy positive implicative filter. Suppose t = x ⇝ (x ⇝ y) and η (x ⇝ (x ⇝ y)) = α. Then t ∈ η _α . By Proposition 2.2(i) and by Proposition 3.4(i), μ (t → t) = μ (1) ≥ α. Then t → t ∈ μ _α . Thus, by Proposition 2.2(xv), 1 = t → (x ⇝ (x ⇝ y)) = t → ((x ⊙ x) ⇝ y) ∈ μ _α . By Proposition 2.2(ii), (x ⊙ x) ⇝ (t → y) ∈ μ _α . Again, by Proposition 2.2(xv), x ⇝ (x ⇝ (t → y)) ∈ μ _α . Since μ _α is a positive implicative filter and x → x = 1 ∈ μ _α , by (PIF3), x ⇝ (t → y) ∈ μ _α . Also, by Proposition 2.2(ii), t → (x ⇝ y) ∈ μ _α , and so μ (t → (x ⇝ y)) ≥ α. Since μ ⊆ η, α ≤ μ (t → (x ⇝ y)) ≤ η (t → (x ⇝ y)). Then t → (x ⇝ y) ∈ η _α . Moreover, since η _α is a filter and t ∈ η _α , by Theorem 2.3(ii), x ⇝ y ∈ η _α . Hence, η _α is a positive implicative filter. Then by Theorem 5.4, η is a fuzzy positive implicative filter. □

Lemma 5.8. Let μ be a fuzzy positive implicative filter of A. Then μ is a fuzzy normal filter.

Proof. Let μ be a fuzzy positive implicative filter. Suppose μ (x ⇝ y) = μ (1). Then by (FPI2), μ (x → (y → y)) ∧ μ (x ⇝ y) ≤ μ (x → y). Since μ (x ⇝ y) = μ (1) and μ (1) = μ (x → 1) = μ (x → (y → y)), by (FPI1), μ (x → y) = μ (1). Therefore, μ is a fuzzy normal filter. □

Theorem 5.9. Let μ be a fuzzy positive implicative filter of A such that μ ((x → y) ⇝ y) ≤ μ ((y → x) ⇝ x), for all x, y ∈ A. Then μ is a fuzzy implicative filter.

Proof. Let μ be a fuzzy positive implicative filter of A. By Proposition 2.2(viii), x ≤ (x → y) ⇝ y. Then by Proposition 2.2(xiii), (x → y) ⇝ x ≤ (x → y) ⇝ ((x → y) ⇝ y). By (FF2), μ ((x → y) ⇝ x) ≤ μ ((x → y) ⇝ ((x → y) ⇝ y)). Since μ is a fuzzy positive implicative filter, μ ((x → y) ⇝ ((x → y) ⇝ y)) ∧ μ ((x → y) → (x → y)) ≤ μ ((x → y) ⇝ y). By Proposition 2.2(i) and (FF2), μ ((x → y) ⇝ ((x → y) ⇝ y)) ≤ μ ((x → y) ⇝ y). By assumption, μ ((x → y) ⇝ ((x → y) ⇝ y)) ≤ μ ((y → x) ⇝ x). Hence, μ ((x → y) ⇝ x) ≤ μ ((y → x) ⇝ x). Let μ ((x → y) ⇝ x) = α, μ ((y → x) ⇝ x) = β. Then (x → y) ⇝ x ∈ μ _α and (y → x) ⇝ x ∈ μ _β . Since α ≤ β, μ _β  ⊆ μ _α , and so (y → x) ⇝ x ∈ μ _α . By Proposition 2.2(iv) and (xii), y ≤ x → y, (x → y) ⇝ x ≤ y ⇝ x, and so μ ((x → y) ⇝ x) ≤ μ (y ⇝ x). Let μ (y ⇝ x) = γ. Then y ⇝ x ∈ μ _γ . By Lemma 5.8 and Theorem 3.7, μ _γ is a normal filter. Thus, y → x ∈ μ _γ . Since α ≤ γ, then μ _γ  ⊆ μ _α . Hence, y → x ∈ μ _α . Since (y → x) ⇝ x ∈ μ _α , by (F2), x ∈ μ _α , μ (x) ≥ α = μ ((x → y) ⇝ x). Therefore, by Theorem 4.7(i), μ is a fuzzy implicative filter. □

Theorem 5.10. Let A be bounded with (PDN) such that ∨₁ and ∨₂ exist. If μ is a fuzzy positive implicative filter of A and for all x, y ∈ A, μ ((x → y) ⇝ y) ≤ μ ((y → x) ⇝ x), then A/μ is a Boolean algebra.

Proof. By Theorems 5.9 and 4.10, the proof is clear.

6 Fuzzy fantastic filters

In this section, we introduce the notion of fuzzy fantastic filter in pseudo hoops and investigate some properties of it and we get the relation between different kinds of fuzzy filter.

Definition 6.1.[4] Let F be a non-empty subset of A. Then F is called a fantastic filter of A, if it satisfies the following properties,

1 ∈ F,

Definition 6.2. The fuzzy set μ of A is called a fuzzy fantastic filter A if, for any x, y, z ∈ A,

μ (x) ≤ μ (1),

Example 6.3. According to Example 2.1, A = ([0, u] , ⊙ , → , ⇝ , u) is a bounded pseudo hoop. By [16], let K be a normal convex ℓ-group of G and F = {a ∈ [0, u] ∣ u - a ∈ K} is a normal filter of [0, u]. Suppose z → (x → y) ∈ F and z ∈ F. Then ( ( y → x ) ⇝ x ) → y = ( ( ( x - y + u ) ∧ u ) ⇝ x ) → y = ( ( u - ( ( x - y + u ) ∧ u ) + x ) ∧ u ) → y = ( ( ( u - u + y - x + x ) ∨ ( u - u + x ) ∧ u ) ) → y = ( y ∨ x ) → y = ( y - ( y ∨ x ) + u ) ∧ u = u ∈ F .

Thus, we have (FA2). By the similar way (FA3) holds. Therefore, F is a fantastic filter. Now, we define the fuzzy set μ on A by for all a ∈ A, where a ≠ u, μ (a) = m, μ (u) = n such that m, n ∈ [0, 1] and m ≤ n. Hence μ is a fuzzy fantastic filter of A.

Theorem 6.4. Every fuzzy fantastic filter of A is a fuzzy filter.

Proof. Let μ be a fuzzy fantastic filter of A. Then by (FFA1), μ (x) ≤ μ (1). By Proposition 2.2(v), x → y = x → (1 → y). Then μ (x → y) = μ (x → (1 → y)). Hence, μ (x) ∧ μ (x → y) = μ (x) ∧ μ (x → (1 → y)). Since μ is a fuzzy fantastic filter, by (FFA2), μ (x) ∧ μ (x → (1 → y)) ≤ μ (((y → 1) ⇝1) → y). Again, by Proposition 2.2(v) and (vi), μ (x) ∧ μ (x → y) ≤ μ (y). By the similar way, μ (x) ∧ μ (x ⇝ y) ≤ μ (y). Then, by Proposition 3.4, μ is a fuzzy filter. □

Example 6.5. In Example 3.12, μ is a fuzzy filter but it is not a fuzzy fantastic filter of A, because μ (0 → a) = μ (1) = n ≰ m = μ (((a → 0) ⇝0) → a).

Theorem 6.6. μ is a fuzzy fantastic filter of A if and only if μ _α ≠ ∅ is a fantastic filter of A, for all α ∈ [0, 1].

Proof. The proof is straight forward. □

Theorem 6.7. Let μ be a fuzzy filter of A. Then μ is a fuzzy fantastic filter if and only if μ (x → y) ≤ μ (((y → x) ⇝ x) → y).

Proof. (⇒) Let μ be a fuzzy fantastic filter and z = 1. By (FFA2), μ (((x → y) ⇝ y) → x) ≥ μ (1 → (y → x)) ∧ μ (1) = μ (1 → (y → x)). Then, μ (((x → y) ⇝ y) → x) ≥ μ (y → x).

(⇐) Let μ be a fuzzy filter. Then by Proposition 3.4(i), μ (x) ≤ μ (1). By Proposition 3.4(ii), μ (y → x) ≥ μ (z → (y → x)) ∧ μ (z). Then by assumption, μ (((x → y) ⇝ y) → x) ≥ μ (z → (y → x)) ∧ μ (z). Therefore, μ is a fuzzy fantastic filter. □

Theorem 6.8. Let A be bounded and μ be a fuzzy implicative filter of A. Then μ is a fuzzy fantastic filter.

Proof. Let μ be a fuzzy implicative filter of A. Then by Proposition 2.2(xiv), x^- ≤ x → y. Then (x → y) ⇝ y ≤ x^- ⇝ y, and so ((x → y) ⇝ y) ⇝ (x^- ⇝ y) =1. Then by Proposition 2.2(xv), (x^- ⊙ ((x → y) ⇝ y)) ⇝ y = 1, and by Proposition 3.4(i), μ ((x^- ⊙ ((x → y) ⇝ y)) ⇝ y) = μ (1). Since μ is a fuzzy implicative filter, by Proposition 4.7(v), μ ((x^- ⊙ ((x → y) ⇝ y)) ⇝ y) ∧ μ (y ⇝ x) ≤ μ ((((x → y) ⇝ y)) ⇝ x). Then, μ (y ⇝ x) ≤ μ ((((x → y) ⇝ y)) ⇝ x). By Theorem 6.7, μ is a fuzzy fantastic filter of A. □

Theorem 6.9. Let μ and η be two fuzzy filters of A such that μ ⊆ η and μ (1) = η (1). If μ is a fuzzy fantastic filter, then η is a fuzzy fantastic filter, too.

Proof. Let μ and η be two fuzzy filters of A such that μ ⊆ η and μ be a fuzzy fantastic filter. Then by Theorem 6.6, μ _α ≠ ∅ is a fantastic filter for all α ∈ [0, 1]. Since μ ⊆ η, μ (x) ≤ η (x), for all x ∈ A. Suppose x ∈ μ _α . Then μ (x) ≥ α, and so η (x) ≥ α. Thus, x ∈ η _α , and so μ _α  ⊆ η _α . Let η _α ≠ ∅. Since μ (1) = η (1), μ _α ≠ ∅. Let y → x ∈ η _α , for x, y ∈ A and for all α ∈ [0, 1]. By Proposition 2.2(i), (y → x) ⇝ (y → x) =1. Then by Proposition 2.2(ii), y → ((y → x) ⇝ x) =1, and so by (FA1), y → ((y → x) ⇝ x) =1 ∈ μ _α , for all α ∈ [0, 1]. By Proposition 2.2(v), 1 → (y → ((y → x) ⇝ x)) =1 ∈ μ _α . Since μ _α is a fantastic filter by (FA2), we have ((((y → x) ⇝ x) → y) ⇝ y) → ((y → x) ⇝ x) = (y → x) ⇝ (((((y → x) ⇝ x) → y) ⇝ y) → x) ∈ μ _α . Since μ _α  ⊆ η _α , (y → x) ⇝ (((((y → x) ⇝ x) → y) ⇝ y) → x) ∈ η _α . Since y → x ∈ η _α , by Proposition 2.3(ii), (((((y → x) ⇝ x) → y) ⇝ y) → x) ∈ η _α . Let t = (((((y → x) ⇝ x) → y) ⇝ y) → x), we get t ⇝ ( ( ( x → y ) ⇝ y ) → x ) = ( ( ( ( ( y → x ) ⇝ x ) → y ) ⇝ y ) → x ) ⇝ ( ( ( x → y ) ⇝ y ) → x ) ≥ ( ( x → y ) ⇝ y ) → ( ( ( ( y → x ) ⇝ x ) → y ) ⇝ y ) By Proposition 2.2 ( ix ) ≥ ( ( ( y → x ) ⇝ x ) → y ) ⇝ ( x → y ) By Proposition 2.2 ( ii ) ≥ x → ( ( y → x ) ⇝ x ) By Proposition 2.2 ( ix ) = ( y → x ) ⇝ ( x → x ) By Proposition 2.2 ( ii ) = 1 By Proposition 2.2 ( i )

By assumption, η _α is a filter. Then 1 ∈ η _α and by (F2), t ⇝ (((x → y) ⇝ y) → x) ∈ η _α . Since t ∈ η _α , by Proposition 2.3(ii), ((x → y) ⇝ y) → x ∈ η _α . Then, η _α is a fantastic filter, for all α ∈ [0, 1]. Thus, by Theorem 6.6, η is a fuzzy fantastic filter. □

Theorem 6.10. Let μ be a fuzzy fantastic and fuzzy positive implicative filter of A such that x² = x, for all x ∈ A. Then μ is a fuzzy implicative filter.

Proof. Let μ be a fuzzy fantastic and fuzzy positive implicative filter. Since μ is a fuzzy fantastic filter, μ ((x → y) ⇝ x) ≤ μ (((((x → (x → y)) ⇝ (x → y)) → x))). By Proposition 2.2(xv), ((((x → (x → y) ⇝ (x → y)) → x))) = ((x² → y) ⇝ (x → y)) → x. By Proposition 2.2(ix), x → x² ≤ (x² → y) ⇝ (x → y), and so μ (x → x²) ≤ μ ((x² → y) ⇝ (x → y)). Since μ is a fuzzy positive implicative filter and x² = x, for all x ∈ A, by Proposition 3.5(ii), μ (1) = μ (x → x²) ≤ μ ((x² → y) ⇝ (x → y)), and so μ ((x² → y) ⇝ (x → y)) = μ (1). By Theorem 5.5, μ is a fuzzy filter. Then by Proposition 3.4(ii), μ (((((x → (x → y)) ⇝ (x → y)) → x)))) ∧ μ ((((x → (x → y)) ⇝ (x → y)))) ≤ μ (x), and so μ ((((x → (x → y)) ⇝ (x → y)) → x))) ∧ μ (1) ≤ μ (x). Since μ ((x → y) ⇝ x) ≤ μ ((((x → (x → y)) ⇝ (x → y)) → x))), we get, μ ((x → y) ⇝ x) ≤ μ (x). By the similar way, μ ((x ⇝ y) → x) ≤ μ (x). Then, by Proposition 4.7(ii), μ is a fuzzy implicative filter. □

Theorem 6.11. Let A be bounded with (PDN) such that ∨₁ and ∨₂ exist. If μ is a fuzzy positive implicative and fuzzy fantastic filter of A such that x² = x, for all x ∈ A, then A/μ is a Boolean algebra.

Proof. By Lemma 5.8, Theorems 4.10 and 6.10, the proof is clear.

7 Conclusions and future works

The aim of this paper is to introduce the notions of fuzzy (implicative, positive implicative, fantastic) filters in pseudo hoops and investigate some properties of them. Several characterizations of these fuzzy filters are derived but in this paper we try to investigate the relation between them. In the next research, we can devoted to provide theoretical foundation to design intelligent information processing system.