Fuzzy filters of Sheffer stroke Hilbert algebras

1 Introduction

The concept of fuzzy theory was originally introduced by Lotfi Zadeh [14]. Since this theory are closely related to human’s intelligence and behaviors, it has been commonly used by many researchers who want to produce devices imitating human behaviors. Thus, the fuzzy theory plays an important role to improve artificial intelligence, industry, science and technology. Filter theory has an important role in logic, and so there exist some applications in logical algebras. Therefore, this theory arrests many scientist’s attention. Hilbert algebras were introduced by L. Henkin and T. Skolem for studies in intuitionistic and other non-classical logics [6] in the 1950’s. Then A. Diego analyzed Hilbert algebras, their deductive systems and various properties. He proved that Hilbert algebras form a variety [5]. Also, Hilbert algebras, related concepts and deductive systems are examined by Busneag ([1, 2]) and Jun [7]. In the meantime, Sheffer operation was introduced by H.M. Sheffer [13]. And later, McCune et al. proved that any Boolean operation can be expressed by this operation [8]. Sheffer operation has all diods on the chips in a computer, and so it leads to generate a single diod instead of different diods for every Boolean operation. Hence, it has importance in producing computer chips. In addition to this, due to properties of Sheffer operation, many algebraic structures with the operation have quitely useful axiom systems. For that reason, checking these axiom systems is easy and effortless. Therefore, scientists have started to study algebraic structures with Sheffer operation such as Sheffer stroke non-associative MV-algebras [4], (fuzzy) filters of Sheffer stroke BL-algebras [10], filters of strong Sheffer stroke non-associative MV-algebras [11] and Sheffer stroke BG-algebras [12]. Also, Sheffer stroke Hilbert algebras, their properties, deductive system and ideals are presented in [9]. We give a deductive system and a filter of a Sheffer stroke Hilbert algebra, and prove that its filter is equivalent to its deductive system. By describing a fuzzy filter and a homomorphism on a Sheffer stroke Hilbert algebra, it is shown that a new fuzzy filter of a Sheffer stroke Hilbert algebra is defined by the compound function of fuzzy filter and a homomorphism on this algebra. Thereafter a quotient a Sheffer stroke Hilbert algebra is built by its fuzzy filter, and then it is shown that this quotient is also a Sheffer stroke Hilbert algebra. Also, normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra are introduced and some properties are given. It is proved that f⁺ is a normal fuzzy filter of a Sheffer stroke Hilbert algebra H if f is its fuzzy filter, and f is a fuzzy filter of H if f⁺ is its fuzzy filter, where f⁺ (x) = f (x) +1 - f (1), for any x ∈ H. Besides, it is demonstrated that f is a normal fuzzy filter of a Sheffer stroke Hilbert algebra if and only if f⁺ = f. It is stated that the compound function of a fuzzy filter f of a Sheffer stroke Hilbert algebra H and an increasing function g : [0, f (1)] ⟶ [0, 1] is a fuzzy filter of H. Indeed, it is shown that the sets of all normal fuzzy filters of a Sheffer stroke Hilbert algebra is a poset while it is not a lattice. Then the relationships between normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra are expressed. After determining (strongest) fuzzy relation S _g on a Sheffer stroke Hilbert algebra H for a fuzzy subset g of H, it is proved that g is a fuzzy filter of H if and only if S _g is a fuzzy filter of a Sheffer stroke Hilbert algebra H × H. Finally, the Cartesian product of fuzzy filters and related concepts presented.

2 Preliminaries

In this section, we give fundamental definitions and notions about Sheffer stroke Hilbert algebras.

Definition 2.1. [3] Let H = 〈 H , | 〉 be a groupoid. The operation | is said to be Sheffer stroke if it satisfies the conditions:

x|y = y|x,

Definition 2.2. [9] A Sheffer stroke Hilbert algebra is a structure 〈H, |〉 of type (2), in which H is a non-empty set and | is Sheffer stroke on H such that the following identities are satisfied, for all x, y, z ∈ H:

(SHa₁) (x| ((y| (z|z)) | (y| (z|z)))) | (((x| (y|y)) | ((x| (z|

z)) | (x| (z|z)))) | ((x| (y|y)) | ((x| (z|z)) | (x| (z|

z))))) = x| (x|x) ,

(SHa₂) If x| (y|y) = y| (x|x) = x| (x|x) then x = y.

Example 2.1. [9] Consider the Sheffer stroke Hilbert algebra 〈H, |〉, where the set H = {0, x, y, z, t, u, v, 1} and the binary operation | on H has the Cayley table in Table 1:

The Hasse diagram of this algebra follows:

Example 2.2. [9] Consider the Sheffer stroke Hilbert algebra 〈H, |〉 with H : = {0, p, q, 1}, and with the Cayley table in Table 2:

The Hasse diagram of this algebra follows:

Lemma 2.1. [9] Let 〈H, |〉 be a Sheffer Stroke Hilbert algebra. Then the following identities hold for all x ∈ H:

x| (x|x) =1,

Lemma 2.2. [9] Let 〈H, |〉 be a Sheffer stroke Hilbert algebra. Then the relation ≤ defined by x ≤ y ⇔ x| (y|y) =1 is a partial order on H, which is called the natural ordering on H. With respect to this ordering, 1 is the largest element of H.

Lemma 2.3. [9] Let 〈H, |〉 be a Sheffer stroke Hilbert algebra. Then the following hold for all x, y, z ∈ H:

(Shb₁) x ≤ y| (x|x),

(Shb₂) x| ((y| (z|z)) | (y| (z|z))) = (x| (y|y)) | ((x| (z|z)) | (x| (z|z))) ,

(Shb₃) (x| (y|y)) | (y|y) = (y| (x|x)) | (x|x) ,

(Shb₄) x| ((y| (z|z)) | (y| (z|z))) = y| ((x| (z|z)) | (x| (z| z))) ,

(Shb₆) ((x| (y|y)) | (y|y)) | (y|y) = x| (y|y) ,

(Shb₇) x| (y|y) ≤ (y| (z|z)) | ((x| (z|z)) | (x| (z|z))) .

Definition 2.3. [9] A subset S of H is called a deductive system of H if it satisfies

(i) 1 ∈ S,

(ii) x ∈ S and x| (y|y) ∈ S imply y ∈ S.

Lemma 2.4. Let 〈H, |〉 be a Sheffer stroke Hilbert algebra. Then x | ( y | y ) ≤ ( z | ( x | x ) ) | ( ( z | ( y | y ) ) | ( z | ( y | y ) ) ) , for all x, y, z ∈ H.

Proof. Since

(x| (y|y)) | (((z| (x|x)) | ((z| (y|y)) | (z| (y|

y)))) | ((z| (x|x)) | ((z| (y|y)) | (z| (y|y)))))

= (x| (y|y)) | ((z| ((x| (y|y)) | (x| (y|

y)))) | (z| ((x| (y|y)) | (x| (y|y)))))

= z| (((x| (y|y)) | ((x| (y|y)) | (x| (y|y))))

| ((x| (y|y)) | ((x| (y|y)) | (x| (y|y)))))

= z|(1|1)

= 1

from (Shb₂), (Shb₄), Lemma 2.1 (i) and (ii), it follows that x| (y|y) ≤ (z| (x|x)) | ((z| (y|y)) | (z| (y|y))) , for all x, y, z ∈ H.

Theorem 2.1. Let 〈H, | _H 〉 and 〈G, | _G 〉 be two Sheffer stroke Hilbert algebras. Then 〈H × G, |_H×G〉 is a Sheffer stroke Hilbert algebra, where the Sheffer stroke |_H×G on H × G is defined by (x₁, y₁) |_H×G (x₂, y₂) = (x₁| _H x₂, y₁| _G y₂), for all (x₁, y₁) , (x₂, y₂) ∈ H × G.

In this section, we introduce a filter of Sheffer stroke Hilbert algebras. Assume that H states a Sheffer stroke Hilbert algebra unless otherwise specified.

Definition 3.1. A non-empty subset F of H is called a filter if

(f₁) 1 ∈ F,

(f₂) x| (y|y) ∈ F for x ∈ H and y ∈ F, (f₃) (x| (y₁|y₂)) | (y₁|y₂) ∈ F for x ∈ H and y₁, y₂ ∈  F.

Example 3.1. Consider the Sheffer stroke Hilbert algebra H in Example 2.1. Then {u, 1}, {v, 1}, {t, 1}, {x, t, u, 1}, {z, u, v, 1}, {y, v, t, 1}, {1} and H are filters of H.

Theorem 3.1. A subset F of H is a filter of H if and only if it is a deductive system of H.

Proof. (⇒) Let F be a filter of H. Then 1 ∈ F from (f₁). Let x ∈ F and x| (y|y) ∈ F. By substituting [y₁ : = x], [y₂ : = x| (y|y)] and [x : = y] in (f₃), simultaneously, it follows from (S1)-(S3), (Shb₃), (Shb₆) and Lemma 2.1 that (y| (x| (x| (y|y)))) | (x| (x| (y|y)))

= ((((x|x) |y) |y) |y) | (((x|x) |y) |y)

= (y| ((x|x) |y)) | ((x|x) |y)

= ((((x|x) |y) | ((x|x) |y)) | (y|y)) | (y|y)

= ((x|x) | ((y| (y|y)) | (y| (y|y)))) | (y|y)

= ((x|x) |(1|1)) | (y|y)

= y ∈ F .

(⇐) Let F be a deductive system of H. Then

(f₁): 1 ∈ F from Definition 2.3 (i).

(f₂): Let x ∈ H and y ∈ F. Since y ≤ x| (y|y), then y | ( ( x | ( y | y ) ) | ( x | ( y | y ) ) ) = 1 from Lemma 2.2. It follows that x| (y|y) ∈ F.

(f₃): Let x ∈ H and y₁, y₂ ∈ F. Since y₁| (((y₁|y₂) |y₂) | ((y₁|y₂) |y₂)) = (y₁|y₂) | ((y₁|y₂) | (y₁|y₂)) =1 ∈ F from (Shb₄), Lemma 2.1 (i) and Definition 2.3 (i), it is obtained from Definition 2.3 (ii) that (y₁|y₂) |y₂ ∈ F. Also, since y₂| (((y₁|y₂) | (y₁|y₂)) | ((y₁|y₂) | (y₁|y₂))) = (y₁|y₂) |y₂ ∈ F from (S1)-(S2), we have from Definition 2.3 (ii) that (y₁|y₂) | (y₁|y₂) ∈ F. It follows from (f₂) and (S2) that (x| (y₁|y₂)) | (y₁|y₂) = (x| (((y₁|y₂) | (y₁|y₂)) | ((y₁|y₂) | (y₁|y₂)))) | (((y₁|y₂) | (y₁|y₂)) | ((y₁|y₂) | (y₁|y₂))) (x| (y₁|y₂)) | (y₁|y₂) ∈ F.

4 Fuzzy filters of sheffer stroke hilbert algebras

In this section, we introduce fuzzy filters of Sheffer stroke Hilbert algebras.

The inclusion ⊆ between two fuzzy subsets f and g of H is defined by f ⊆ g ⇔ f ( x ) ≤ g ( x ) , for all x, y ∈ H.

Definition 4.1. A fuzzy subset f of H is callis said to be a fuzzy filter of H if the following hold for all x, y, y₁, y₂ ∈ H:

(FH1) f (x) ≤ f (1),

(FH2) f (y) ≤ f (x| (y|y)),

(FH3) min {f (y₁) , f (y₂)} ≤ f ((x| (y₁|y₂)) | (y₁|y₂)).

Example 4.1. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Define a fuzzy subset f of H by f ( x ) = { 0.21 if x=0, p 0.33 if x = q 0.56 if x = 1 . Then f is a fuzzy filter of H.

Lemma 4.1. Let f be a fuzzy filter of H. Then f ( x ) ≤ f ( ( x | ( y | y ) ) | ( y | y ) ) for all x, y ∈ H .

Proof. It is obtained that f ((x| (y|y)) | (y|y)) = f ((y| (x|x)) | (x|x)) for all x, y ∈ H by (Shb₃). By substituting [y₁ : = x], [y₂ : = x] and [x : = y] in (FH3), simultaneously, it follows that f (x) = min {f (x) , f (x)} ≤ f ((y| (x|x)) | (x|x)) = f ((x| (y|y)) | (y|y)).

Lemma 4.2. Every fuzzy filter f of H is order-preserving, i.e., f (x) ≤ f (y) if x ≤ y.

Proof. Let x ≤ y. Then x| (y|y) =1. Thus, f (x) ≤ f ((x| (y|y)) | (y|y)) = f (1| (y|y)) = f (y) from Lemma 4.1 and Lemma 2.1 (iii).

We demonstrate that x ≰ y whenever f (x) ≤ f (y) for all x, y ∈ H.

Example 4.2. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Define a fuzzy filter f of H by f ( x ) = { 0.79 if x=1 0.21 otherwise . Then f (p) ≤ f (q) but p ≰ q.

We give an example to show that, in general, an order-preserving fuzzy subset of H is not fuzzy filter of H.

Example 4.3. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Let a fuzzy subset f of H be defined by f (0) =0.13, f (p) =0.14, f (q) =0.15 and f (1) =0.72. Then f is order-preserving but is not a fuzzy filter of H since 0.14 = f (p) = min {f (p) , f (q)} > f ((0| (p|q)) | (p|q)) = f (0) =0.13.

Lemma 4.3. Let f be a fuzzy subset of H. Then f is a fuzzy filter of H if and only if

(a) f is order-preserving,

(b) min {f (x) , f (y)} ≤ f ((x|y) | (x|y)), for all x, y ∈ H.

Proof. (⇒) Let f be a fuzzy filter of H. Then it is obvious from Lemma 4.2 that f is order-preserving. By substituting [x : =1|1], [y₁ : = x] and [y₂ : = y] in (FH3), it is obtained from (S1), (S2), Lemma 2.1 (ii) and (iii) that min {f (x) , f (y)} ≤ f (((1|1) | (x|y)) | (x|y)) = f ((x|y) | (x|y)), for all x, y ∈ H.

(⇐) Let f be a fuzzy subset of H satisfying (a) and (b).

(FH1): Since x ≤ 1, by using (a) we have f (x) ≤ f (1), for all x ∈ H.

(FH2): Since y ≤ x| (y|y), it is obtained from (a) that f (y) ≤ f (x| (y|y)), for all x, y ∈ H.

(FH3): By substituting [x : = y₁] and [y : = y₂] in (b), it follows that min {f (y₁) , f (y₂)} ≤ f ((y₁|y₂) | (y₁|y₂)) ≤ f ((x| (y₁|y₂)) | (y₁|y₂)), for all x, y₁, y₂ ∈ H, from (Shb₁), (S2) and (a).

Let f be a fuzzy subset of H. We define a subset H f = { x ∈ H : f ( x ) = f ( 1 ) } of H.

Proposition 4.1. Let A be a nonempty subset of H and f _A be a fuzzy subset of H defined by f A ( x ) = { t 1 if x∈ A t 2 otherwise where t₁, t₂ ∈ [0, 1] such that t₁ > t₂. Then f _A is a fuzzy filter of H if and only if A is a filter of H. Also, H _{f A}  = A.

Proof. Let f _A be a fuzzy filter of H. Since f _A  (1) = t₁, then 1 ∈ A. Let x ∈ H and y ∈ A. Then f _A  (y) = t₁. Since t₁ = f _A  (y) ≤ f _A  (x| (y|y)) from (FH2), f _A  (x| (y|y)) = t₁, and so x| (y|y) ∈ A . Also, let x ∈ H and y₁, y₂ ∈ A. Then t 1 = min { f ( y 1 ) , f ( y 2 ) } ≤ f ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) ) . Thus, f ((x| (y₁|y₂)) | (y₁|y₂)) = t₁, which means (x| (y₁|y₂)) | (y₁|y₂) ∈ A.

Conversely, let A be a filter of H. Since 1 ∈ A, f _A  (x) ≤ f _A  (1) = t₁, for all x ∈ H. Let x and y be any elements in H. If y ∈ A, then x| (y|y) ∈ A from f₂. Thus, f _A  (x| (y|y)) = t₁ = f _A  (y). If y ∉ A, then f _A  (y) = t₂. Hence, f _A  (y) ≤ f _A  (x| (y|y)). Moreover, let x, y₁ and y₂ be any elements in H. If y₁, y₂ ∈ A, then (x| (y₁|y₂)) | (y₁|y₂) ∈ A from f₃. Thus, f _A  ((x| (y₁|y₂)) | (y₁|y₂)) = t₁ = min {f _A  (y₁) , f _A  (y₂)}. If y₁, y₂ ∉ A, then t₂ = min {f _A  (y₁) , f _A  (y₂)} ≤ f _A  ((x| (y₁|y₂)) | (y₁|y₂)). Also, if y₁ ∈ A and y₂ ∉ A (or y₁ ∉ A and y₂ ∈ A), then f _A  (y₁) = t₁ and f _A  (y₂) = t₂ (or f _A  (y₁) = t₂ and f _A  (y₂) = t₁). So, t₂ = min {f _A  (y₁) , f _A  (y₂)} ≤ f _A  ((x| (y₁|y₂)) | (y₁|y₂)).

Since A is a filter of H, H f A = { x ∈ H : f A ( x ) = f A ( 1 ) } = { x ∈ H : f A ( x ) = t 1 } = { x ∈ H : x ∈ A } = A .

Lemma 4.4. A fuzzy subset f of H is a fuzzy filter of H if and only if for every t ∈ [0, 1], the set f _t  = {x ∈ H : f (x) ≥ t} is either empty or a filter of H.

Proof. Let f be a fuzzy filter of H and f _t ≠ ∅. Then there exists x ∈ H such that t ≤ f (x). Since t ≤ f (x) ≤ f (1) from (FH1), 1 ∈ f _t . Let y ∈ f _t . Then t ≤ f (y). Since f (y) ≤ f (x| (y|y)) from (FH2), t ≤ f (y) ≤ f (x| (y|y)). Thus, x| (y|y) ∈ f _t . Also, let y₁, y₂ ∈ f _t . Then t ≤ f (y₁) and t ≤ f (y₂). Since t ≤ min {f (y₁) , f (y₂)} ≤ f ((x| (y₁|y₂)) | (y₁|y₂)) from (FH3), (x| (y₁|y₂)) | (y₁|y₂) ∈ f _t .

Conversely, let f _t ≠ ∅ be a filter of H. Let some x ∈ H such that f (1) < f (x). If t = f ( x ) + f ( 1 ) 2 , then f (1) < t < f (x). Thus, 1 ∉ f _t which contradicts with (f₁). Hence, f (x) ≤ f (1) for all x ∈ H. Let x, y ∈ H such that f (x| (y|y)) < f (y). If t = f ( x | ( y | y ) ) + f ( y ) 2 , then f (x| (y|y)) < t < f (y). So, y ∈ f _t but x| (y|y) ∉ f _t which is a contradiction with (f₂). Thus, f (y) ≤ f (x| (y|y)) for all x, y ∈ H. Moreover, let x, y₁, y₂ ∈ H such that f ((x| (y₁|y₂)) | (y₁|y₂)) < min {f (y₁) , f (y₂)}. If t = t 1 + t 2 2 where f ((x| (y₁|y₂)) | (y₁|y₂)) = t₁ and min {f (y₁) , f (y₂)} = t₂, then f ((x| (y₁|y₂)) | (y₁|y₂)) < t < min {f (y₁) , f (y₂)}. Hence, y₁, y₂ ∈ f _t but (x| (y₁|y₂)) | (y₁|y₂) ∉ f _t . This contradicts with (f₃). Thus, min {f (y₁) , f (y₂)} ≤ f ((x| (y₁|y₂)) | (y₁|y₂)) for all x, y₁, y₂ ∈ H.

Lemma 4.5. Let f _s and f _t be two filters of H such that s < t. Then f _s and f _t are equal if and only if there exist no a₀ ∈ H such that s ≤ f (a₀) < t.

Proof. Suppose that f _s  = f _t such that s < t. Then f _s  = {x ∈ H : s ≤ f (x)} = {x ∈ H : t ≤ f (x)} = f _t . If there exists a₀ ∈ H such that s ≤ f (a₀) < t, then a₀ ∉ f _t  = f _s which contradicts with a₀ ∈ f _s . Hence, there is no a₀ ∈ H such that s ≤ f (a₀) < t.

Conversely, assume that there is no a₀ ∈ H such that s ≤ f (a₀) < t. Let f _s  ≠ f _t . Then there exists a₀ ∈ H such that s ≤ f (a₀) < t. This is a contradictionn. Thus, f _s  = f _t

Corollary 4.1. Let f be a fuzzy filter of H. Then f _s  = f _t , for s, t ∈ Im (f), if and only if s = t.

Proof. (⇒) Let f _s  = f _t for s, t ∈ Im (f). Then f (a₀) = s and f (a₁) = t for a₀, a₁ ∈ H. Thus, a₀ ∈ f _s  = f _t and a₁ ∈ f _t  = f _s , and so s = f (a₀) ≥ t and t = f (a₁) ≥ s, respectively. Hence, s = t.

(⇐) It is clear that f _s  = f _t for s, t ∈ Im (f) if s = t.

Lemma 4.6. Let f be a fuzzy filter of H and a₀ ∈ H. Then f (a₀) = t if and only if a₀ ∈ f _t and a₀ ∉ f _s , for all s > t.

Proof. (⇒) Let f (a₀) = t. Since f (a₀) = t < s, it follows that a₀ ∈ f _t and a₀ ∉ f _s .

(⇐) Let a₀ ∈ f _t and a₀ ∉ f _s , for all s > t. Then t ≤ f (a₀) < s. Assume that t ≤ f (a₀) = s₀. So, a₀ ∈ f _{s 0} which is a contradiction. Thus, f (a₀) = t.

Definition 4.2. Let 〈G, | _G 〉 and 〈H, | _H 〉 be two Sheffer stroke Hilbert algebras. A mapping h : G ⟶ H is called a homomorphism if h ( x | G y ) = h ( x ) | H h ( y ) for all x, y ∈ G. Also, h (1 _G ) =1 _H .

Let h be an endomorphism and f be a fuzzy subset of H. Define a new fuzzy subset f ^h of H by f h ( x ) = f ( h ( x ) ) , for all x ∈ H.

Theorem 4.1 expresses that the compound function of an endomorphism h and a fuzzy filter f of a Sheffer stroke Hilbert algebra H is a fuzzy filter of H. Theorem 4.1. Let h be an endomorphism of H. If f is a fuzzy filter of H, then f ^h is a fuzzy filter of H.

Proof. Let h be an endomorphism and f be a fuzzy filter of H. Then f h ( x ) = f ( h ( x ) ) ≤ f ( 1 ) = f ( h ( 1 ) ) = f h ( 1 ) . Likewise, we have f h ( x | ( y | y ) ) = f ( h ( x | ( y | y ) ) ) = f ( h ( x ) | ( h ( y ) | h ( y ) ) ) ≥ f ( h ( y ) ) = f h ( y ) and f h ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) ) = f ( h ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) ) ) = f ( ( h ( x ) | ( h ( y 1 ) | h ( y 2 ) ) ) | ( h ( y 1 ) | h ( y 2 ) ) ) ≥ min { f ( h ( y 1 ) ) , f ( h ( y 2 ) ) } = min { f h ( y 1 ) , f h ( y 2 ) } , for all x, y, y₁, y₂ ∈ H.

Theorem 4.2 states that the inverse of Theorem 4.1 holds when h is a surjective endomorphism of H.

Theorem 4.2. Let h be a surjective endomorphism of H. If f ^h is a fuzzy filter of H, then f is a fuzzy filter of H.

Proof. Let h be a surjective endomorphism of H and f ^h be a fuzzy filter of H. Then f ( x ) = f ( h ( u ) ) = f h ( u ) ≤ f h ( 1 ) = f ( h ( 1 ) ) = f ( 1 ) . Besides, we get f ( x | ( y | y ) ) = f ( h ( u ) | ( h ( v ) | h ( v ) ) ) = f ( h ( u | ( v | v ) ) ) ≥ f h ( v ) = f ( h ( v ) ) = f ( y ) and

f ((x| (y₁|y₂)) | (y₁|y₂))

= f ((h (u) | (h (v₁) |h (v₂))) | (h (v₁) |h (v₂)))

= f (h ((u| (v₁|v₂)) | (v₁|v₂)))

= f ^h  ((u| (v₁|v₂)) | (v₁|v₂))

≥min {f ^h  (v₁) , f ^h  (v₂)}

= min {f (h (v₁)) , f (h (v₂))}

= min {f (y₁) , f (y₂)} ,

where x = h (u), y = h (v), y₁ = h (v₁) and y₂ = h (v₂), for all x, y, u, v, y₁, y₂, v₁, v₂ ∈ H.

Theorem 4.3. Let h be an automorphism of H and f be a fuzzy filter of H. Then f ^h  = f if and only if h (f _t ) = f _t , for all t ∈ Im (f).

Proof. Let f ^h  = f, t ∈ Im (f) and x ∈ f _t . It follows h (x) ∈ h (f _t ). Then f (h (x)) = f ^h  (x) = f (x) ≥ t, and so h (x) ∈ f _t . Thus, h (f _t ) ⊆ f _t . Moreover, let x ∈ f _t and y ∈ H be such that h (y) = x. Then f (y) = f ^h  (y) = f (h (y)) = f (x) ≥ t which means y ∈ f _t . Thus, x = h (y) ∈ h (f _t ). Hence, f _t  ⊆ h (f _t ). Therefore, h (f _t ) = f _t .

Conversely, let h (f _t ) = f _t , for all t ∈ Im (f), and f (x) = t. Then x ∈ f _t and x ∉ f _s , for all s > t by Lemma 4.6. Thus, it follows that h (x) ∈ h (f _t ) = f _t , and so f ^h  (x) = f (h (x)) ≥ t. Assume that f ^h  (x) = s. Since f ^h  (x) = f (h (x)) = s, h (x) ∈ f _s  = h (f _s ). Due to the fact that h is one-to-one, it is obtained that x ∈ f _s , which is a contradiction. Hence, f ^h  (x) = f (h (x)) = t = f (x), for every x ∈ H which means f ^h  = f.

If h is not an automorphism on H, Theorem 4.3 does not hold.

Example 4.4. Consider the fuzzy filter f of H in Example 4.1. Let a homomorphism h on H be defined by h : H ⟶ H 0 , p ⟼ 0 q , 1 ⟼ 1 . Since h is not surjective, it is not an automorphism on H. Then f ^h  ≠ f since f ^h  (q) = f (h (q)) = f (1) ≠ f (q) and h (f_0.21) = h (H) = {0, 1} ≠ H = f_0.21, for 0.21 ∈ Im (f).

Definition 4.3. Let f be a fuzzy filter of H. Define the binary relation ∼ _f on H as follows: x ∼ f y ⇔ f ( x | ( y | y ) ) = f ( 1 ) andf ( y | ( x | x ) ) = f ( 1 ) , for all x, y ∈ H.

Example 4.5. Consider the Sheffer stroke Hilbert algebra H in Example 2.1. For a fuzzy filter f of H by f ( a ) = { 0.11 if a=0, x, y, t 0.87 if a=z, u, v, 1 . Then ∼ _f  = {(0, 0) , (x, x) , (y, y) , (z, z) , (t, t) , (u, u) , (v, v) , (1, 1) , (0, x) , (x, 0) , (0, y) , (y, 0) , (0, t) , (t, 0) , (x, y) , (y, x) , (x, t) , (t, x) , (y, t) , (t, y) , (z, u) , (u, z) , (z, v) , (v, z) , (z, 1) , (1, z) , (1, u) , (u, 1) , (1, v) , (v, 1) , (v, u) , (u, v)} is a binary relation on H. It is also an equivalence relation on H.

Definition 4.4. If xαy implies x|zαy|z for all x, y, z ∈ H, then the equivalence relation α is called a congruence relation on H.

Example 4.6. Consider the equivalence relation ∼ _f on H in Example 4.5. Then it is a congruence relation on H.

Lemma 4.7. An equivalence relation α is a congruence relation on H if and only if xαy and zαt imply x|zαy|t, for all x, y, z, t ∈ H.

Proof. Let α be a congruence relation on H and x, y, z and t be any elements in H such that xαy and zαt. Then x|zαy|z and y|zαy|t from (S1). Thus, it is obtained from the transitivity of α that x|zαy|t.

Conversely, suppose that xαy and zαt imply x|zαy|t for all x, y, z, t ∈ H. Let x, y and z be any elements in H such that xαy. Since zαz, x|zαy|z. Then α is a congruence relation on H.

Lemma 4.8. Let f be a fuzzy filter of H and ∼ _f be defined as in Definition 4.3. Then ∼ _f is a congruence relation on H.

Proof.

•Reflexive: since f (x| (x|x)) = f (1) from Lemma 2.1 (i), it follows x ∼  _f x, for all x ∈ H.

•Symmetric: let x and y be any elements in H such that x ∼  _f y. Then f (x| (y|y)) = f (1) and f (y| (x|x)) = f (1). Since f (y| (x|x)) = f (1) and f (x| (y|y)) = f (1), it follows that xy ∼  _f x.

•Transitive: let x, y and z be any elements in H such that x ∼  _f y and y ∼  _f z. Then f (x| (y|y)) = f (1) = f (y| (x|x)) and f (y| (z|z)) = f (1) = f (z| (y|y)). Since f (1) = f (x| (y|y)) ≤ f ((y| (z|z)) | ((x| (z|z)) | (x| (z|z)))) from (Shb₇) and Lemma 4.2, we have f ((y| (z|z)) | ((x| (z|z)) | (x| (z|z)))) = f (1). So, y| (z|z) ∈ f_f(1) and (y| (z|z)) | ((x| (z|z)) | (x| (z|z))) ∈ f_f(1). Since f_f(1) is a filter of H, x| (z|z) ∈ f_f(1) by Theorem 3.1 and Definition 2.3 (ii). Thus, f (x| (z|z)) ≥ f (1) which implies f (x| (z|z)) = f (1). Also, it follows that f ((z| (y|y)) | ((z| (x|x)) | (z| (x|x)))) = f (1) since f (1) = f (y| (x|x)) ≤ f ((z| (y|y)) | ((z| (x|x)) | (z| (x|x)))) from Lemma 2.4 and Lemma 4.2. Then z| (y|y) ∈ f_f(1) and (z| (y|y)) | ((z| (x|x)) | (z| (x|x))) ∈ f_f(1). Since f_f(1) is a filter of H, it is obtained from Theorem 3.1 and Definition 2.3 (ii) that z| (x|x) ∈ f_f(1). Hence, f (z| (x|x)) ≥ f (1) which means that f (z| (x|x)) = f (1). Therefore, x ∼  _f z.

Now, it is shown that the equivalence relation ∼ _f is a congruence relation on H. Let x, y, z and t be any elements in H such that x ∼  _f y and z ∼  _f t. Then f (x| (y|y)) = f (1) = f (y| (x|x)) and f (z| (t|t)) = f (1) = f (t| (z|z)).

(a) Since f (1) = f (x| (y|y)) = f ((y|y) | ((x|x) | (x|x))) ≤ f ((z| ((y|y) | (y|y))) | ((z| ((x|x) | (x|x))) | (z| ((x|x) | (x|x))))) = f ((y|z) | ((x|z) | (x|z))) from (S1), (S2), Lemma 2.4 and Lemma 4.2, we have f ((y|z) | ((x|z) | (x|z))) = f (1), and similarly, f ((x|z) | ((y|z) | (y|z))) = f (1). Hence, x|z ∼  _f y|z.

(b) by substituting [x : = z], [y : = t] and [z : = y] in (a), simultaneously, it follows from (S1) that y|z ∼  _f y|t.

Therefore, it is obtained from the transitivity of ∼ _f that x|z ∼  _f y|t.

Theorem 4.4. Let f be a fuzzy filter of H and ∼ be a congruence relation on H defined by f. Then 〈H/∼ , |_∼〉 is a Sheffer stroke Hilbert algebra, where the quotient set H/∼ = {[x] _∼ : x ∈ H} and the Sheffer stroke |_∼ is defined by [x] _∼|_∼ [y] _∼ = [x|y] _∼ for all x, y ∈ H.

Proof. Let [x] _∼, [y] _∼ and [z] _∼ be any elements of H. Then ([x] _∼|_∼ (([y] _∼|_∼ ([z] _∼|_∼ [z] _∼)) |_∼ ([y] _∼|_∼ ([z] _∼|_∼ [z] _∼)))) |_∼ ((([x] _∼|_∼ ([y] _∼|_∼ [y] _∼)) |_∼ (([x] _∼|_∼ ([z] _∼|_∼ [z] _∼)) |_∼ ([x] _∼|_∼ ([z] _∼|_∼ [z] _∼)))) |_∼ (([x] _∼|_∼ ([y] _∼|_∼ [y] _∼))

| (([x] _∼|_∼ ([z] _∼|_∼ [z] _∼)) |_∼ ([x] _∼|_∼ ([z] _∼|_∼ [z] _∼)))))

= [(x| ((y| (z|z)) | (y| (z|z)))) | (((x|

(y|y)) | ((x| (z|z)) | (x| (z|z)))) | ((x

| (y|y)) | ((x| (z|z)) | (x| (z|z)))))] _∼

= [x| (x|x)] _∼

= [x] _∼|_∼ ([x] _∼|_∼ [x] _∼) .

Let [x] _∼|_∼ ([y] _∼|_∼ [y] _∼) = [y] _∼|_∼ ([x] _∼|_∼ [x] _∼) = [x] _∼|_∼ ([x] _∼|_∼ [x] _∼). Then [x| (y|y)] _∼ = [y| (x|x)] _∼ = [x| (x|x)] _∼ = [1] _∼ from Lemma 2.1 (i). Thus, x| (y|y) ∼1 and y| (x|x) ∼1, and so, f ((x| (y|y)) |(1|1)) = f (1) = f (x| (y|y)) = f (1| ((x| (y|y)) (x| (y|y)))) and f ((y| (x|x)) |(1|1)) = f (1) = f (y| (x|x)) = f (1| ((y| (x|x)) (y| (x|x)))) from Lemma 2.1 (ii) and (iii). Hence, x ∼ y which implies [x] _∼ = [y] _∼.

Example 4.7. Consider the Sheffer stroke Hilbert algebra H in Example 2.1. Define a fuzzy filter f by f ( a ) = { 0.73 if a=u, 1 0.27 otherwise . Let ∼ f = { ( 0 , 0 ) , ( x , x ) , ( y , y ) , ( z , z ) , ( t , t ) , ( u , u ) , ( v , ) ˇ , ( 1 , 1 ) , ( 1 , u ) , ( 0 , y ) , ( y , 0 ) , ( u , 1 ) , ( x , t ) , ( t , x ) , ( z , ) ˇ , ( v , z ) } be a congruence relation on H defined by f. Then 〈H/∼ , |_∼〉 is a Sheffer stroke Hilbert algebra with the Hasse diagram in Fig. 3, where the quotient set H/∼ = {[0] _∼, [x] _∼, [v] _∼, [1] _∼}. The Sheffer stroke |_∼ on H/∼ has the Cayley table in Table 3.

OPEN IN VIEWER

Fig. 1

The Hasse diagram of the Sheffer stroke Hilbert algebra in Example 2.1.

OPEN IN VIEWER

Fig. 2

The Hasse diagram of the Sheffer stroke Hilbert algebra in Example 2.2.

OPEN IN VIEWER

Fig. 3

The Hasse diagram of the Sheffer stroke Hilbert algebra in Example 4.7.

In this section, we introduce {normal and maximal fuzzy filters of Sheffer Stroke Hilbert algebras.

Definition 5.1. A fuzzy filter f of H is called normal if there exists x ∈ H such that f (x) =1.

Example 5.1. Consider the Sheffer stroke Hilbert algebra H in Example 2.1. Let f (0) = f (y) = f (z) = f (v) =0.01 and f (x) = f (t) = f (u) = f (1) =1. Then f is a normal fuzzy filter of H. However, a fuzzy filter f of H in Example 4.1 is not normal since there is no x ∈ H such that f (x) =1.

Proposition 5.1. A fuzzy filter f of H is normal if and only if f (1) =1.

Proof. (⇒) Let f be a normal fuzzy filter of H. Then there exists x ∈ H such that f (x) =1. Since f (x) ≤ f (1) for all x ∈ H, 1 = f (x) ≤ f (1) which implies f (1) =1.

(⇐) Let f be a fuzzy filter of H such that f (1) =1. Since there exists x = 1 ∈ H such that f (x) =1, it follows that f is normal.

Proposition 5.2. For a fuzzy filter f of H, let f⁺ be a fuzzy subset of H defined by f⁺ (x) = f (x) +1 - f (1), for all x ∈ H. Then f⁺ is a normal fuzzy filter of H, and f ⊆ f⁺.

Proof. We first show that f⁺ is a fuzzy filter of H:

(FH1): f⁺ (1) = f (1) +1 - f (1) ≥ f (x) +1 - f (1) = f⁺ (x),

(FH2): f⁺ (x| (y|y)) = f (x| (y|y)) +1 - f (1) ≥ f (y) +1 - f (1) = f⁺ (y) and

(FH3):

f⁺ ((x| (y₁|y₂)) | (y₁|y₂))

= f ((x| (y₁|y₂)) | (y₁|y₂)) +1 - f (1)

≥min {f (y₁) , f (y₂)} +1 - f (1)

= min {f (y₁) +1 - f (1) , f (y₂) +1 - f (1)}

= min {f⁺ (y₁) , f⁺ (y₂)} , for x, y, y₁, y₂ ∈ H.

Since f⁺ (1) = f (1) +1 - f (1) =1, f⁺ is normal from Proposition 5.1. Moreover, f ⊆ f⁺ since f (x) ≤ f (x) +1 - f (1) = f⁺ (x), for all x ∈ H.

Proposition 5.3. Let f be a fuzzy subset of H and f⁺ be a fuzzy filter of H defined by f⁺ (x) = f (x) +1 - f (1), for all x ∈ H. Then f is a fuzzy filter of H.

Proof. (FH1): Since 1 = f (1) +1 - f (1) = f⁺ (1) ≥ f⁺ (x) = f (x) +1 - f (1) , it follows f (1) ≥ f (x).

(FH2): Since f (y) +1 - f (1) = f⁺ (y) ≤ f⁺ (x| (y|y)) = f (x| (y|y)) +1 - f (1) , it is obtained f (y) ≤ f (x| (y|y)).

(FH3): Since

min {f (y₁) , f (y₂)} +1 - f (1)

= min {f (y₁) +1 - f (1) ,

f (y₂) +1 - f (1)}

= min {f⁺ (y₁) , f⁺ (y₂)}

≤f⁺ ((x| (y₁|y₂)) | (y₁|y₂))

= f ((x| (y₁|y₂)) | (y₁|y₂)) +1 - f (1) ,

we have min {f (y₁) , f (y₂)} ≤ f ((x| (y₁|y₂)) | (y₁|y₂)) , for x, y, y₁, y₂ ∈ H.

Corollary 5.1. If f⁺ (x) =0, then f (x) =0.

Proof. 0 = f⁺ (1) = f (x) +1 - f (1), and so f (1) = f (x) +1. Then f (x) =0.

Corollary 5.2. Let F be a filter of H. Then the characterictic function χ _F of F is a normal fuzzy filter of H.

Proof. Since F is a filter of H, χ F ( x ) = { 1 if x∈ F 0 otherwise is a fuzzy filter of H from Proposition 4.1. Also, χ _F is normal by Proposition 5.1.

Proposition 5.4. A fuzzy filter f of H is normal if and only if f⁺ = f.

Proof. Let f be a normal fuzzy filter of H. By Proposition 5.2, we get f ⊆ f⁺. Then it is obtained from Proposition 5.1 that f⁺ (x) = f (x) +1 - f (1) = f (x) +1 - 1 = f (x), for all x ∈ H. Hence, f⁺ = f.

Conversely, it is obvious that f is normal by the fact that f⁺ is normal and f⁺ = f.

Corollary 5.3. Let f be a fuzzy filter of H. Then (f⁺) ⁺ = f⁺. Also, if f is normal, then (f⁺) ⁺ = f.

Proposition 5.5. Let f and g be fuzzy filters of H such that f ⊆ g and f (1) = g (1). Then H _f  ⊆ H _g .

Proof. Let f and g be fuzzy filters of H such that f ⊆ g and f (1) = g (1), and x ∈ H _f . Since g (1) = f (1) = f (x) ≤ g (x), it follows that g (x) = g (1). Thus, x ∈ H _g which means H _f  ⊆ H _g .

Corollary 5.4. Let f and g be normal fuzzy filters of H such that f ⊆ g. Then H _f  ⊆ H _g .

However, the converse of Proposition 5.5 and Corollary 5.4 is not true.

Example 5.2. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Define fuzzy filters f and g of H by f ( x ) = { 1 if x=1, p 0.28 if x=q, 0 and g ( x ) = { 1 if x=1, p 0.26 if x=q, 0 respectively. Then H _f  = {p, 1} = {p, 1} = H _g but f ⊈ g since f (0) > g (0).

Proposition 5.6. Let f be fuzzy filter of H. Then there exists a fuzzy filter g of H such that g⁺ ⊆ f if and only if f is normal.

Proof. Assume that there exists a fuzzy filter g of H such that g⁺ ⊆ f. Since 1 = g⁺ (1) ≤ f (1), it follows that f (1) =1. Then f is normal by Proposition 5.1.

Conversely, let f be a normal fuzzy filter of H. Since f⁺ = f from Proposition 5.4, there exists a fuzzy filter g of H such that g⁺ ⊆ f, if we choose g = f.

Lemma 5.1. Let f be a fuzzy filter of H and g : [0, f (1)] ⟶ [0, 1] be an increasing function. Then a fuzzy subset f _g  : H ⟶ [0, 1] defined by f _g  (x) : = g (f (x)), for all x ∈ H, is a fuzzy filter of H. Particularly, if f _g  (1) =1, then f _g is normal. Also, f ⊆ f _g if t ≤ g (t) for all t ∈ [0, f (1)].

Proof. Let f be a fuzzy filter of H and g : [0, f (1)] ⟶ [0, 1] be an increasing function. Then

(FH1): f _g  (x) = g (f (x)) ≤ g (f (1)) = f _g  (1),

(FH2): f _g  (y) = g (f (y)) ≤ g (f (x| (y|y))) = f _g  (x| (y|y)) and

(FH3): min { f g ( y 1 ) , f g ( y 2 ) } = min { g ( f ( y 1 ) ) , g ( f ( y 2 ) ) } ≤ g ( min { f ( y 1 ) , f ( y 2 ) } ) = g ( f ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) ) ) = f g ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) )

for x, y, y₁, y₂ ∈ H. Thus, f _g is a fuzzy filter of H.

If f _g  (1) = g (f (1)) =1, then f _g is a normal fuzzy filter of H by Proposition 5.1.

Let t ≤ g (t) for all t ∈ [0, f (1)]. Then f (x) ≤ g (f (x)) = f _g  (x), for all x ∈ H, which means f ⊆ f _g .

Lemma 5.2. Let N S ( H ) be the set of all normal fuzzy filters of H. Then ( N S ( H ) , ⊆ ) is a poset.

Proof. Let N S ( H ) be the set of all normal fuzzy filters of H. We show that ( N S ( H ) , ⊆ ) is a poset.

Reflective: since f (x) ≤ f (x) for all x ∈ H, we have f ⊆ f, for all fuzzy filters f of H.

Antisymmetric: let f ⊆ g and g ⊆ f. Then f (x) ≤ g (x) and g (x) ≤ f (x), i.e., f (x) = g (x), for all x ∈ H. Thus, f = g.

Transitive: let f ⊆ g and g ⊆ h. Then f (x) ≤ g (x) and g (x) ≤ h (x), for all x ∈ H. Hence, f (x) ≤ h (x), for all x ∈ H, which implies f ⊆ h.

It is demonstrated that N S ( H ) does not form a lattice via the following example.

Example 5.3. Consider the Sheffer stroke Hilbert algebra in Example 2.1. Define normal fuzzy filters f and g of H by f ( a ) = { 1 if a=u, 1 0.2 otherwise and g ( b ) = { 1 if b=y, v, t, 1 0.3 otherwise respectively. However, ( f ∨ g ) ( a ) = max { f ( a ) , g ( a ) } = { 0.3 if a=0, x, z 1 otherwise is not a fuzzy filter of H since (f ∨ g) ((0| (y|u)) | (y|u)) = (f ∨ g) (0) =0.3 < min {(f ∨ g) (y) , (f ∨ g) (u)} = min {1, 1} =1.

Theorem 5.1. Let f ∈ N S ( H ) be a non-constant such that it is a maximal element of ( N S ( H ) , ⊆ ) . Then f takes only the values 0 and 1.

Proof. It is known that f (1) =1 since f ∈ N S ( H ) . Let x ∈ H be such that f (x) ≠1. Assume that f (x) =0. If it is not, then there exists a₀ ∈ H such that 0 < f (a₀) <1. Let g be a fuzzy subset of H defined by g ( x ) = 1 2 ( f ( x ) + f ( a 0 ) ) for all x ∈ H. Then it is obvious that g is well-defined. Now, it is shown that g is a fuzzy filter of H. For all x, y, y₁, y₂ ∈ H,

(FH1): g ( 1 ) = 1 2 ( f ( 1 ) + f ( a 0 ) ) ≥ 1 2 ( f ( x ) + f ( a 0 ) ) = g ( x ) ,

(FH2): g ( x | ( y | y ) ) = 1 2 ( f ( x | ( y | y ) ) + f ( a 0 ) ) ≥ 1 2 ( f ( y ) + f ( a 0 ) ) = g ( y ) and

(FH3):

g ((x| (y₁|y₂)) | (y₁|y₂))

= 1 2 ( f ( ( x | ( y 1 | y 2 ) ) | ( y 1 | y 2 ) ) + f ( a 0 ) )

≥ 1 2 ( min { f ( y 1 ) , f ( y 2 ) } + f ( a 0 ) )

= 1 2 min { f ( y 1 ) + f ( a 0 ) , f ( y 2 ) + f ( a 0 ) }

= min { 1 2 ( f ( y 1 ) + f ( a 0 ) ) , 1 2 ( f ( y 2 ) + f ( a 0 ) ) }

= min {g (y₁) , g (y₂)} .

Also, g⁺ is normal by Proposition 5.2. Since g + ( x ) = g ( x ) + 1 - g ( 1 ) = 1 2 ( f ( x ) + f ( a 0 ) ) + 1 - 1 2 ( f ( 1 ) + f ( a 0 ) ) = 1 2 ( f ( x ) + 1 ) ≥ f ( x ) , for all x ∈ H, f (a₀) < g⁺ (a₀) and g⁺ (a₀) < g⁺ (1) =1. Hence, g⁺ is non-constant and f ⊆ g⁺, and so, f is not maximal which is a contradiction. Thus, f (x) =0, for all x ∈ H.

Definition 5.2. A non-constant fuzzy filter f of H is called maximal if f⁺ is a maximal element of ( N S ( H ) , ⊆ ) .

Example 5.4. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Then f ( x ) = { 1 if x=p, 1 0 if x=q, 0 is a maximal fuzzy filter of H.

Lemma 5.3. Let f ∈ N S ( H ) . Let A and B be filters of H. Then A ⊆ B if and only if f _A  ⊆ f _B .

Proof. Let f ∈ N S ( H ) , and A and B be filters of H such that A ⊆ B. Define f A ( x ) = { 1 if x∈ A 0 if x∉ A and f B ( y ) = { 1 if y∈ B 0 if y∉ B for all x, y ∈ H. Then f _A and f _B are fuzzy filters of H by Proposition 4.1.

Let x ∈ A. Then x ∈ B, and so 1 = f _A  (x) ≤ f _B  (x) =1. Also, let x ∉ A. Thus, 0 = f _A  (x) ≤ f _B  (x), for all x ∈ H. Hence, f _A  (x) ≤ f _B  (x) for all x ∈ H, that is, f _A  ⊆ f _B .

Conversely, let f _A  ⊆ f _B . Then f _A  (x) ≤ f _B  (x), for all x ∈ H. Since there is no x ∈ H such that x ∈ A but x ∉ B, it follows that A ⊆ B.

Theorem 5.2. Let f be a maximal fuzzy filter of H. Then

f is normal,

Proof. Let f be a maximal fuzzy filter of H.

(1) and (2): Then f⁺ is a non-constant maximal elemant of N S ( H ) . f⁺ takes only the values 0 and 1 by Lemma 5.3. Thus, f⁺ (x) =1 if and only if f (x) = f (1), and f⁺ (x) =0 if and only if f (x) = f (1) -1. Hence, f (x) =0 by Corollary 5.1, and so f (1) =1 which proves that f is normal.

(3): Since f H f ( x ) = { 1 if x∈ Hf 0 if x∉ Hf, is defined, it follows f _{H f}  ⊆ f, and so f _{H f} takes only the values 0 and 1. Let x ∈ H. If f (x) =0, then f ⊆ f _{H f} . Also, if f (x) =1 = f (1), then x ∈ H _f , and so f _{H f}  (x) =1. Hence, f ⊆ f _{H f} . Therefore, f _{H f}  = f.

(4): Since f is non-constant, H _f is a proper filter of H from Proposition 4.1. Let F be a filter of H such that H _f  ⊆ F. Then f = f _{H f}  ⊆ f _F by Lemma 5.3 and (3). Since f and f _F are normal and f = f⁺ is a maximal element of N S ( H ) , f = f _F and f _F  = g where g : H ⟶ [0, 1] is a fuzzy subset of H defined by g (x) =1, for all x ∈ H. If f _F  = g, then F = H, and if f = f _F , then H _f  = H _{f F}  = F from Proposition 4.1. Thus, H _f is a maximal filter of H.

However, any normal fuzzy filter of a Sheffer stroke Hilbert algebra H is generally not maximal.

Example 5.5. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Then f ( x ) = { 1 if x=1 0.32 otherwise is a non-constant normal fuzzy filter of H while it is not maximal since there exists a non-constant normal fuzzy filter g of H defined by g ( x ) = { 1 if x=1 0.411 if x=p 0.41 if x=0, q and f ⊆ g.

Corollary 5.5. Let f be a non-constant maximal element of N S ( H ) . Then f is a maximal fuzzy filter of H.

6 Cartesian product of fuzzy filters

In this section, we state the Cartesion product of fuzzy filters of Sheffer Stroke Hilbert algebras and some properties.

Definition 6.1. Let g be a fuzzy subset of H. The strongest fuzzy relation on H is a fuzzy relation S _g on H defined by S g ( x , y ) = min { g ( x ) , g ( y ) } , for all x, y ∈ H.

Example 6.1. Consider the Sheffer stroke Hilbert algebra H in Example 2.2. Let a fuzzy subset g : H ⟶ [0, 1] be defined by g (0) =0.001, g (p) =0.002, g (q) =0.003 and g (1) =0.004. Then S _g  : H × H ⟶ [0, 1] defined by S g ( x , y ) = { 0.001 if x=0 or y=0 g ( x ) if x=y g ( x ) if x=1 or y=1 0.002 otherwise is the strongest fuzzy relation on H. Lemma 6.1. Let S _g be the strongest fuzzy relation on H for a fuzzy subset g of H. Then (S _g )  _t  = g _t  × g _t for all t ∈ [0, 1].

Proof. Let S _g be the strongest fuzzy relation on H for a fuzzy subset g of H, and (x, y) ∈ (S _g )  _t . Then min {g (x) , g (y)} = S _g  (x, y) ≥ t, and so g (x) ≥ t and g (y) ≥ t. Thus, x, y ∈ g _t which implies (x, y) ∈ g _t  × g _t . Hence, (S _g )  _t  ⊆ g _t  × g _t .

Conversely, let (x, y) ∈ g _t  × g _t . Then x, y ∈ g _t , and so g (x) ≥ t and g (y) ≥ t. Thus, S _g  (x, y) = min {g (x) , g (y)} ≥ min {t, t} = t which means (x, y) ∈ (S _g )  _t . Hence, g _t  × g _t  ⊆ (S _g )  _t .

Therefore, (S _g )  _t  = g _t  × g _t for all t ∈ [0, 1].

Theorem 6.1. Let g be a fuzzy subset of H and S _g be the strongest fuzzy relation on H. Then g is a fuzzy filter of H if and only if S _g is a fuzzy filter of H × H.

Proof. It is known that H × H is a Sheffer stroke Hilbert algebra from Theorem 2.1. Let g be a fuzzy filter of H. Then S _g  (x, y) = min {g (x) , g (y)} ≤ min {g (1) , g (1)} = S _g  (1, 1) for all x, y ∈ H.

S _g  ((x₁, x₂) |_H×H ((y₁, y₂) |_H×H (y₁, y₂)))

= S _g  (x₁| (y₁|y₁) , x₂| (y₂|y₂))

= min {g (x₁| (y₁|y₁)) , g (x₂| (y₂|y₂))}

≥min {g (y₁) , g (y₂)}

= S _g  (y₁, y₂)

and

S _g  (((x₁, x₂) |_H×H ((y₁, y₂) |_H×H (y₃,

y₄))) |_H×H ((y₁, y₂) |_H×H (y₃, y₄)))

= S _g  ((x₁| (y₁|y₃)) | (y₁|y₃) , (x₂| (y₂|y₄)) | (y₂|y₄))

= min {g ((x₁| (y₁|y₃)) | (y₁|y₃)) ,

g ((x₂| (y₂|y₄)) | (y₂|y₄))}}

≥min {min {g (y₁) , g (y₃)} , min {g (y₂) , g (y₄)}}

= min {min {g (y₁) , g (y₂)} , min {g (y₃) , g (y₄)}}

= min {S _g  (y₁, y₂) , S _g  (y₃, y₄)} ,

for all x₁, x₂, y₁, y₂, y₃, y₄ ∈ H. Thus, S _g is a fuzzy filter of a Sheffer stroke Hilbert algebra H × H.

Conversely, let S _g be a fuzzy filter of a Sheffer stroke Hilbert algebra H × H. Then min {g (x) , g (y)} = S _g  (x, y) ≤ S _g  (1, 1) = min {g (1) , g (1)}, for all x, y ∈ H. So, it follows g (x) ≤ g (1), for all x ∈ H. It follows from Lemma 2.1 (ii) that g ( x | ( y | y ) ) = min { g ( x | ( y | y ) ) , g ( 1 ) } = min { g ( x | ( y | y ) ) , g ( x | ( 1 | 1 ) ) } = S g ( x | ( y | y ) , x | ( 1 | 1 ) ) = S g ( ( x , x ) | H × H ( ( y , 1 ) | H × H ( y , 1 ) ) ) ≥ S g ( y , 1 ) = min { g ( y ) , g ( 1 ) } = g ( y ) and

g ((x| (y₁|y₂)) | (y₁|y₂))

= min {g ((x| (y₁|y₂)) | (y₁|y₂)) , g (1)}

= min {g ((x| (y₁|y₂)) | (y₁|y₂)) , g ((x|(1|1)) |(1|1))}

= S _g  ((x| (y₁|y₂)) | (y₁|y₂) , (x|(1|1)) |(1|1))

= S _g  (((x, x) |_H×H ((y₁, 1) |_H×H (y₂,

1))) |_H×H ((y₁, 1) |_H×H (y₂, 1)))

≥min {S _g  (y₁, 1) , S _g  (y₂, 1)}

= min {min {g (y₁) , g (1)} , min {g (y₂) , g (1)}}

= min {g (y₁) , g (y₂)} ,

for all x, y, y₁, y₂ ∈ H. Hence, g is a fuzzy filter of H.

Corollary 6.1. If g is a fuzzy filter of H, then the level filters are given by (S _g )  _t  = g _t  × g _t , for all t ∈ [0, 1].

Proposition 6.1. Let f and g be fuzzy filters of H. Then f × g is a fuzzy filter of H × H. Moreover, if f and g are normal, then f × g is normal.

Theorem 6.2. Let f and g be fuzzy subsets of H such that f × g is a fuzzy filter of H × H. Then

(1) either f (x) ≤ f (1) or g (x) ≤ g (1) for all x ∈ H,

(2) if f (x) ≤ f (1) for all x ∈ H, then either f (x) ≤ g (1) or g (x) ≤ g (1),

(3) if g (x) ≤ g (1) for all x ∈ H, then either f (x) ≤ f (1) or g (x) ≤ f (1), and

(4) either f or g is a fuzzy filter of H.

Proof. Let f and g be fuzzy subsets of H such that f × g is a fuzzy filter of H × H. Then

(1) Let f (x) > f (1) and g (y) > g (1) for some x, y ∈ H. Then (f × g) (x, y) = min {f (x) , g (y)} > min {f (1) , g (1)} = (f × g) (1, 1) which is a contradiction. Thus, either f (x) ≤ f (1) or g (x) ≤ g (1) for all x ∈ H.

(2) Let f (x) ≤ f (1) for all x ∈ H. Suppose that f (x) > g (1) or g (y) > g (1) for some x, y ∈ H. Since g (1) < f (x) ≤ f (1), it is obtained (f × g) (1, 1) = min {f (1) , g (1)} = g (1), and so (f × g) (x, y) = min {f (x) , g (y)} > g (1) = (f × g) (1, 1) which is a contradiction. Then either f (x) ≤ g (1) or g (x) ≤ g (1) if f (x) ≤ f (1) for all x ∈ H.

(3) Let g (x) ≤ g (1) for all x ∈ H. Assume that f (x) > f (1) or g (y) > f (1) for some x, y ∈ H. Since f (1) < g (y) ≤ g (1), it follows (f × g) (1, 1) = min {f (1) , g (1)} = f (1), and so (f × g) (x, y) = min {f (x) , g (y)} > f (1) = (f × g) (1, 1) which is a contradiction. Then either f (x) ≤ f (1) or g (x) ≤ f (1) if g (x) ≤ g (1) for all x ∈ H.

(4) By (1), either f (x) ≤ f (1) or g (x) ≤ g (1) for all x ∈ H. Assume that g (x) ≤ g (1) for all x ∈ H. It is obtained from (3) that either f (x) ≤ f (1) or g (x) ≤ f (1), for all x ∈ H.

We show that g is a fuzzy filter of H.

Case I. let g (x) ≤ f (1), for all x ∈ H.(FH1) holds by the assumption. Since g ( x | ( y | y ) ) = min { f ( 1 ) , g ( x | ( y | y ) ) } = ( f × g ) ( 1 , x | ( y | y ) ) = ( f × g ) ( x | ( 1 | 1 ) , x | ( y | y ) ) = ( f × g ) ( ( x , x ) | H × H ( ( 1 , y ) | H × H ( 1 , y ) ) ) ≥ ( f × g ) ( 1 , y ) = min { f ( 1 ) , g ( y ) } = g ( y ) from Lemma 2.1 (ii), (FH2) is satistied for all x, y ∈ H. We have from Lemma 2.1 (ii) that g ((x| (y₁|y₂)) | (y₁|y₂))

= min {f (1) , g ((x| (y₁|y₂)) | (y₁|y₂))}

= (f × g) (1, (x| (y₁|y₂)) | (y₁|y₂))

= (f × g) ((x|(1|1)) |(1|1) , (x| (y₁|y₂)) | (y₁|y₂))

= (f × g) (((x, x) |_H×H ((1, y₁) |_H×H

(1, y₂))) |_H×H ((1, y₁) |_H×H (1, y₂)))

≥min {(f × g) (1, y₁) , (f × g) (1, y₂)}

= min {min {f (1) , g (y₁)} , min {f (1) , g (y₂)}}

= min {g (y₁) , g (y₂)} ,

i.e., (FH3) holds for all x, y₁, y₂ ∈ H.

Now, it is shown that f is a fuzzy filter of H.

Case II. let f (x) ≤ f (1), for all x ∈ H. Then (FH1) is satisfied. Assume that g (y) > f (1), for some y ∈ H. Since g (1) ≥ g (y) > f (1) ≥ f (x), it follows g (1) > f (x) for all x ∈ H. Thus, (f × g) (x, 1) = min {f (x) , g (1)} = f (x), for all x ∈ H. It is obtained from Lemma 2.1 (ii) that f ( x | ( y | y ) ) = ( f × g ) ( x | ( y | y ) , 1 ) = ( f × g ) ( x | ( y | y ) , x | ( 1 | 1 ) ) = ( f × g ) ( ( x , x ) | H × H ( ( y , 1 ) | H × H ( y , 1 ) ) ) ≥ ( f × g ) ( y , 1 ) = min { f ( y ) , g ( 1 ) } = f ( y ) , i.e., (FH2) holds for all x, y ∈ H. Since