Fuzzy filters on equality algebras with applications

1 Introduction

Fuzzy type theory [9] was developed as a fuzzy counterpart of the classical higher-order logic. Since the truth values for algebra is no longer a residuated lattice, a specific algebra called an EQ-algebra [8] was proposed by Novák and De Baets. Viewing the axioms of EQ-algebras with a purely algebraic eye it appears that unlike in the case of residuated lattices where the adjointness condition ties product with implication, the product in EQ-algebras is quite loosely which can be replaced by any other smaller binary operation. Furthermore, the freedom in choosing the product might prohibit to find deep related algebraic results. Based on the above reasons, a new algebraic structure was introduced by Jenei in [4], called equality algebra, which consisting of two binary operations, meet and equivalence, and constant 1.

Filters are an important concept in logical deductive systems and logical algebra systems. From a logical point of view, various filters correspond to various sets of provable formulas. Jenei [4] introduced the notion of filters on equality algebras. Then, Zarean etc. [11] studied the prime filters and maximal filters, and gave some important results. Borzooei etc. [1] introduced some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters on equality algebra and discuss the relations among these filters. The concept of fuzzy sets was introduced by Zadeh [10]. At present, these ideas have been applied to other algebraic structures such as residuated lattices [13], EQ-algebras [7] and so on. Since each residuated lattice is an equality algebra, and each good EQ-algebra is an equality algebra, it is natural to extend some fuzzy filters of residuated lattices and EQ-algebras to equality algebras, and we shall obtain some general results of fuzzy filters on equality algebras.

The paper is organized as follows: In section 2, we recall some facts about equality algebras, residuated lattices, and fuzzy set theories. In section 3, we introduce the notion of fuzzy filters on equality algebras and investigate some properties of fuzzy filters. Also, we give a method about generating a fuzzy filter by a fuzzy set. In [6] Kadji, Lele and Tonga presented an open problem as follows: “Let L be a residuated lattice, where lattice is a completely meet-distributive, and FF (L) be a set of all fuzzy filters of L. Is it possible that (FF (L) , ∧ , ⊔) is a completely meet-distributive lattice?" Thus, for the more generalized form of the open problem in an equality algebra, we give a positive answer, and hence, it obviously holds in residuated lattices. Moreover, we prove that the algebra E/f, which is the set of all cosets of a fuzzy filter f, is an equality algebra, and is isomorphic to the E/f_f(1), where f_f(1) = {x ∈ E|f (x) = f (1)}. In section 4, we present the concept of fuzzy congruences on equality algebras and show that there is one-to-one correspondence between fuzzy filters and fuzzy congruences. In Section 5, we establish uniform structures by the special family of extreme fuzzy filters on equality algebras, and then induce uniform topologies. Finally, we show that equality algebras with uniform topologies are topological equality algebras, and also some properties of them are investigated.

2 Preliminaries

In this section, we summarize some definitions and results about equality algebras which will be used in the following sections.

Definition 2.1. [3, 4] An algebra structure (E, ∧ , ∼ , 1) of type (2, 2, 0) is called an equality algebra, if it satisfies the following conditions: for all x, y, z ∈ E, (E1) (E, ∧ , ∼ , 1) is a commutative idempotent integral monoid (i.e. meet semilattice with top element 1), (E2) x ∼ y = y ∼ x, (E3) x ∼ x = 1, (E4) x ∼ 1 =1, (E5) x ≤ y ≤ z implies x ∼ z ≤ y ∼ z and x ∼ z ≤ x ∼ y, (E6) x ∼ y ≤ (x ∧ z) ∼ (y ∧ z), (E7) x ∼ y ≤ (x ∼ z) ∼ (y ∼ z).

Where x ≤ y if and only if x ∧ y = x, for all x, y ∈ E.

The operation ∧ is called meet (infimum) and ∼ is an equality operation. Also, other two operations are defined, called implication and equivalence operation, respectively:       x → y = x ∼ (x ∧ y)                            (Q₁)

x ↔ y = (x → y) ∧ (y → x)                     (Q₂)

In what follows, we let E = (E, ∧ , ∼ , 1) be an equality algebra.

Proposition 2.2. [3] Let E be an equality algebra. Then the following statements hold: for all x, y, z ∈ E, (1) x ∼ y ≤ x ↔ y ≤ x → y, (2) x → y = 1 if and only if x ≤ y, (3) 1 → x = x, x → 1 =1, x → x = 1, (4) x ≤ y → x, (5) x ≤ (x ∼ y) ∼ y ≤ (x → y) → y, (6) x → y ≤ (y → z) → (x → z), (7) x ≤ y → z if and only if y ≤ x → z, (8) x → (y → z) = y → (x → z), (9) y ≤ x implies x ↔ y = x → y = x ∼ y.

Proposition 2.3. [12] Let E be an equality algebra. Then the following properties hold: for all x, y, z ∈ E, (1) x, y ∈ E implies y → z ≤ x → z, z → x ≤ z → y, (2) x → y = x → (x ∧ y), (3) x ∼ y ≤ (z → x) ∼ (z → y), (4) x ∼ y ≤ (z → x) → (z → y), (5) x → y ≤ (z → x) → (z → y), (6) x → y ≤ (x ∧ z) → (y ∧ z), (7) x → y = ((x → y) → y) → y.

Definition 2.4. [12] A lattice equality algebra is an equality algebra which is lattice.

Proposition 2.5. [12] Let E be a lattice equality algebra. Then the following hold: for all x, y ∈ E, (1) for all indexed families {x _i } _i∈E on E, we have (⋁ _i∈Ex _i ) → y = ⋀ _i∈E (x _i  → y), provided that the infimum and suprimum of {x _i } _i∈E exist in E, (2) (x ∨ y) → z = (x → z) ∧ (y → z), (3) x → y = (x ∨ y) → y.

Now, we recall that some facts about filters of equality algebras.

Definition 2.6. [4] Let E be an equality algebra and F is a nonempty set of E. F is called a filter of E if for all x, y ∈ E, (F₁) 1 ∈ F, (F₂) x ∈ F, x ≤ y ⇒ y ∈ F, (F₃) x, x ∼ y ∈ F ⇒ y ∈ F.

Theorem 2.7. [4] Let E be an equality algebra. Then F is a filter of E iff for all x, y ∈ E, ( F 1 ′ ) 1 ∈ F, ( F 2 ′ ) x, x → y ∈ F ⇒ y ∈ F.

For a nonempty subset X of an equality algebra E, the smallest filter of E which contains X, i.e., ⋂ {F ∈ F (E) : X ⊆ F} is said to be a filter of E generated by X and will be denoted by 〈X〉. If a ∈ E and X = {a}, we denote by 〈a〉 the filter generated by {a}.

The following theorem gives a characterization of a filter generated by a set.

Theorem 2.8. [11] Let X be a nonempty subset of an equality algebra E. Then 〈X〉 = {a ∈ E|x₁ → (x₂ → … (x _n  → a) …) =1, for some x _i  ∈ X and n ≥ 1}.

Definition 2.9. [2, 4] Let E be an equality algebra. A subset μ ⊆ E × E is called a congruence of E if it is an equivalence relation on E and for all x₁, x₂, y₁, y₂ ∈ E such that (x₁, y₁) , (x₂, y₂) ∈ μ the following hold,

(1) (x₁ ∧ x₂, y₁ ∧ y₂) ∈ μ,

(2) (x₁ ∼ x₂, y₁ ∼ y₂) ∈ μ.

Let F be a filter of E. Define the congruence ≡ _F on E by x ≡  _F y if and only if x ∼ y ∈ F if and only if {x → y, y → x} ⊆ F ([4]).

In order to solve the open problem, which was presented in [6] by A. Kadji etc, we must review some facts about residuated lattices.

Theorem 2.10. [13] An algebraic structure (L, ∧ , ∨ , ⊗ , →, 0, 1) of type (2, 2, 2, 2, 0, 0) is called a residuated lattice if it satisfies the following conditions: (1) (L, ∧ , ∨ , 0, 1) is a bounded lattice; (2) (L, ⊗ , 1) is a commutative monoid; (3) x ⊗ y ≤ z if and only if x ≤ y → z, for all x, y, z ∈ L, where ≤ is the partial order of the lattice (L, ∧ , ∨ , 0, 1).

Definition 2.11. [13] Let (L, ∧ , ∨ , ⊗ , → , 0, 1) be a residuated lattice. A nonempty fuzzy subset f of L is called a fuzzy filter of L if satisfies: for all x, y ∈ L, (1) x ≤ y ⇒ f (x) ≤ f (y), (2) f (x) ∧ f (y) ≤ f (x ⊗ y).

Proposition 2.12. [13] Let (L, ∧ , ∨ , ⊗ , → , 0, 1) be a residuated lattice and f be a nonempty fuzzy set. Then the following hold: for all x, y ∈ L, (1) f is a fuzzy filter of L if and only if f (x) ≤ f (1) and f (x) ∧ f (x → y) ≤ f (y). (2) 〈 f 〉 ( x ) = ⋁ { ⋀ k = 1 n f ( a k ) | a 1 , a 2 , ⋯, a_n ∈ L, a₁ ⊗a₂ ⊗ ⋯ ⊗ a _n  ≤ x}.

In what follows, we review some notions about fuzzy sets, cuts and fuzzy equivalences which will be necessary in the following section.

A fuzzy set on a nonempty set X, is a map f : X → [0, 1]. Let f be a fuzzy set of X. Then, for all α ∈ [0, 1], the set f _α  = {x ∈ E|f (x) ≥ α} is called a cut of f. A fuzzy subset R on E × E is called a fuzzy relation on E; A fuzzy relation R is called a fuzzy equivalence if it satisfies R (x, x) =1 (reflexivity), R (x, y) = R (y, x) (symmetry) and R (x, y) ∧ R (y, z) ≤ R (x, z) (transitivity) [10].

For fuzzy sets f, g of E, we define f ⊆ g if and only if f ≤ g if and only if f (x) ≤ g (x) for all x ∈ E.

The fuzzy set f ∧ g is defined as follows: for all x ∈ E, f ∧ g = f ∩ g, and (f ∧ g) (x) = f (x) ∧ g (x).

3 Fuzzy filters of equality algebras

In this section, we introduce the notion of fuzzy filters on equality algebras and investigate some their properties. Also we study a fuzzy filter generated by a fuzzy set on equality algebras, and solve an open problem, which was presented in [6] by Kadji, Lele and Tonga.

Definition 3.1. Let E be an equality algebra. A fuzzy set f of E is said to be a fuzzy filter of E, if satisfies for all x, y ∈ E, (1) f (x) ∧ f (x ∼ y) ≤ f (y), (2) if x ≤ y, then f (x) ≤ f (y).

Example 3.2. Let E = [0, 1]. For all x, y ∈ E, define x ∧ y = min {x, y} and x ∼ y = 1 - |x - y|. By routine calculations, we can see that (E, ∧ , ∼ , 1) is an equality algebra. Let f be a fuzzy set on E given by: f ( x ) = { α , x ≠ 1 β , x = 1

where 0 < α < β < 1. Then it is routine to verify that f is a fuzzy filter of E.

Then by routine calculations, we can see that (E, ∧ , ∼ , 1) is an equality algebra. Let f be a fuzzy set in E given by: f ( x ) = { t 1 , x = 0 , a t 2 , x = b t 3 , x = 1

where 0 ≤ t₁ < t₂ < t₃ ≤ 1. Then it is routine to verify that f is a fuzzy filter of E.

Proposition 3.4. Let f be a fuzzy set of E. Then f is a fuzzy filter if and only if for all α ∈ [0, 1] such that f _α ≠ ∅ is a filter of E.

Proof. It is clear.

Proposition 3.5. Let E be the equality algebra, which is given in Example 3.2. (1) Then E has exactly two filters: E and {1}. (2) For each fuzzy filter g of E, we have Im (g) = {α}, or Im (g) = {α, β}, where 0 < α < β < 1.

Proof. (1) By (F₂), we can get that F has only a form, which is F = [a, 1]. When a = 1 and a = 0, we obtain F = {1} and F = E are filters of E. We wish to show that E has exactly two filters: E and {1}. In other words, for all F = [a, 1] (0 < a < 1) is not a filter of E. We proceed by cases.

Case1. 1/2 ≤ a < 1. Assume that F is a filter of E. Then taking x = a and y = 2a - 1, we have x ∼ y = a ∈ F. Since 2a - 1 < a, we obtain y ∉ F. Thus x, x ∼ y ∈ F but y ∉ F, it contradicts with F₃.

Case2. 0 < a < 1/2. Suppose that F is a filter of E. Similarly, taking x = a and y = a/2, we have x ∼ y = 1 - a/2 ∈ F. Since a/2 < a, we get that y ∉ F. Thus x, x ∼ y ∈ F but y ∉ F, it contradicts with F₃.

Hence F = [a, 1] (0 < a < 1) is not a filter of E. (2) Assume that Im (g) has at least three elements. We let α, β, γ ∈ Im (g), where 0 < α < β < γ < 1. Then there exist x _i  ∈ E (i = 1, 2, 3), x₁ < x₂ < x₃ such that g (x₁) = α, g (x₂) = β, g (x₃) = γ. Then x₁, x₂, x₃ ∈ g _α , x₁ ∉ g _β and x₁, x₂ ∉ g _γ . By Proposition 3.4, we can get that g _α , g _β , g _γ are filters, which contradicts with (1). Therefore, Im (g) = {α}, or Im (g) = {α, β}.

Theorem 3.6. Let f be a fuzzy set of E. Then the following statements are equivalent: (1) f is a fuzzy filter; (2) f satisfies f (x) ≤ f (1) and f (x) ∧ f (x → y) ≤ f (y), for all x, y ∈ E.

Proof. (1) ⇒ (2) Suppose that f is a fuzzy filter. Trivially f (x) ≤ f (1) holds. Let x, y ∈ E. By Definition 3.1 and (Q₁), we get f (x) ∧ f (x → y) = f (x) ∧ f (x ∼ (x ∧ y)) ≤ f (x ∧ y) ≤ f (y), hence f (x) ∧ f (x → y) ≤ f (y). (2) ⇒ (1) Suppose that (2) holds. Let x, y ∈ E. If x ≤ y, then x → y = 1. By f (x) ≤ f (1) and f (x) ∧ f (x → y) ≤ f (y), we can get f (x) ∧ f (x → y) = f (x) ∧ f (1) = f (x) ≤ f (y), and so f (x) ≤ f (y). By f (x) ∧ f (x → y) ≤ f (y) and Proposition 2.2 (1), we get f (x ∼ y) ≤ f (x → y), and hence f (x) ∧ f (x ∼ y) ≤ f (x) ∧ f (x → y) ≤ f (y). Therefore, f is a fuzzy filter of E by Definition 3.1.

Theorem 3.7. Let f be a fuzzy set of E. Then the following statements are equivalent: (1) f is a fuzzy filter; (2) (∀x, y, z ∈ E) x ≤ y → z ⇒ f (x) ∧ f (y) ≤ f (z).

Proof. (1) ⇒ (2) Suppose that f is a fuzzy filter. Let x, y, z ∈ E such that x ≤ y → z. Then f (x) ≤ f (y → z), and so f (z) ≥ f (y) ∧ f (y → z) ≥ f (y) ∧ f (x). (2) ⇒ (1) Let f be a fuzzy set of E satisfying (2). Since x ≤ 1 = x → 1 for all x ∈ E, it follows from (2) that f (1) ≥ f (x) ∧ f (x) = f (x) for all x ∈ E. Since x → y ≤ x → y for all x, y ∈ E, we have f (y) ≥ f (x → y) ∧ f (x) for all x, y ∈ E. Hence f is a fuzzy filter of E by Theorem 3.6.

Next, we present some properties of fuzzy filters on equality algebras.

Proposition 3.8. Let f be a fuzzy filter of E. Then the following hold: for all x, y, z, s, t ∈ E, (1) If f (x → y) = f (1), then f (x) ≤ f (y), (2) f (x) ∧ f (x → y) ≤ f (x ∧ y), (3) f (x) ∧ f (y) = f (x ∧ y), (4) f (x ∼ y) ∧ f (y ∼ z) ≤ f (x ∼ z), (5) f (x → y) ∧ f (y → z) ≤ f (x → z).

Proof. (1) By Theorem 3.6, then f (x) ∧ f (x → y) = f (x) ∧ f (1) = f (x) ≤ f (y). (2) By Definition 3.1 and (Q₁), we have f (x) ∧f (x → y) = f (x) ∧ f (x ∼ (x ∧ y)) ≤ f (x ∧ y). (3) Obviously, f (x ∧ y) ≤ f (x) ∧ f (y). Conversely, applying Proposition 2.2 (4) and item(2), we obtain f (x) ∧ f (y) ≤ f (x) ∧ f (x → y) ≤ f (x ∧ y). (4) By Definition 2.1 (E7) and Definition 3.1, it follows that f (x ∼ y) ∧ f (y ∼ z) ≤ f ((x ∼ y) ∼ (y ∼ z)) ∧ f (y ∼ z) ≤ f (x ∼ z). (5) By Proposition 2.2 (6), Definition 3.1 and Theorem 3.6, we have f (x → y) ∧ f (y → z) ≤ f ((y → z) → (x → z)) ∧ f (y → z) ≤ f (x → z).

Proposition 3.9. Let f _i  (i ∈ I) be fuzzy filters. Then ⋀_i∈If _i is a fuzzy filter.

Proof. Assume that x ≤ y → z, for any x, y, z ∈ E. Since f _i are fuzzy filters, we have f _i  (z) ≥ f _i  (x) ∧ f _i  (y) by Theorem 3.7. Hence, ⋀_i∈If _i  (z) ≥ ⋀ _i∈I (f _i  (x) ∧ f _i  (y)) = ⋀ _i∈If _i  (x) ∧ ⋀ _i∈If _i  (y). Therefore, ⋀_i∈If _i is a fuzzy filter by Theorem 3.7.

In what follows, we give a characterization of a fuzzy filter generated by a fuzzy set.

Definition 3.10. Let f be a fuzzy set of an equality algebra E. A fuzzy filter g of E is said to be generated by f, if f ≤ g and for any fuzzy filter h of E, f ≤ h implies g ≤ h. The fuzzy filter generated by f will be denoted by 〈f〉.

Lemma 3.11. Let E be an equality algebra and f be a fuzzy set of E. Then

( ⋁ { ⋀ i = 1 m f ( a i ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 } ) ∧ ( ⋁ { ⋀ j = 1 p f ( b j ) | b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ) = ⋁ { ( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 , b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } .

Proof. Obviously, we obtain ⋁ { ( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 , b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ≤ ( ⋁ { ⋀ i = 1 m f ( a i ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 } ) ∧ ( ⋁ { ⋀ j = 1 p f ( b j ) | b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ) .

Conversely, x ∈ E, let l = ⋁ { ⋀ i = 1 m f ( a i ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 } , m = ⋁ { ⋀ j = 1 p f ( b j ) | b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } and n = ⋁ { ( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) | a 1 → ( a 2 → … ( a m → x ) ⋯ ) = 1 , b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } . Hence, for any σ > 0, there exist a₁, …, a _m , b₁, …, b _p , s.t.

⋀ i = 1 m f ( a i ) > l - σ , a 1 → ( a 2 → … ( a m → x ) ⋯ ) = 1 ;

⋀ j = 1 p f ( b j ) > m - σ , b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 .

Thus, we have

( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) > min { ( l - σ ) , ( m - σ ) } , a₁ → (a₂ → … (a _m  → x) ⋯) =1, b₁ → (b₂ → … (b _p  → (x → y)) …) =1 . Therefore, n > min {(l - σ) , (m - σ)}. Since σ is a arbitrary, we obtain n ≥ min {l, m}, that is, ⋁ { ( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) | a 1 → ( a 2 → … ( a m → x ) ⋯ ) = 1 , b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ≥ ( ⋁ { ⋀ i = 1 m f ( a i ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 } ) ∧ ( ⋁ { ⋀ j = 1 p f ( b j ) | b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ) .

Theorem 3.12. Let E be an equality algebra and f be a nonempty fuzzy set of E. Then for any x ∈ E, 〈 f 〉 ( x ) = ⋁ { ⋀ k = 1 n f ( a k ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 , a i ∈ E , i ≥ 1 } .

Proof. Let g ( x ) = ⋁ { ⋀ k = 1 n f ( a k ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 , a i ∈ E , i ≥ 1 } .

Firstly, we prove that g is a fuzzy filter of E. (1) For all x ∈ E, we have, g ( x ) = ⋁ { ⋀ k = 1 n f ( a k ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 } ≤ ⋁ { ⋀ k = 1 n f ( a k ) | a 1 → ( a 2 → … ( a n → 1 ) … ) = 1 } = g ( 1 ) . (2) It follows from Lemma 3.11 thatg (x) ∧ g (x → y) = ( ⋁ { ⋀ i = 1 m f ( a i ) | a 1 → ( a 2 → … ( a m → x ) … ) = 1 } )     ∧ ( ⋁ { ⋀ j = 1 p f ( b j ) | b 1 → ( b 2 → … ( b p → ( x → y ) ) … ) = 1 } ) = ⋁ { ( ⋀ i = 1 m f ( a i ) ) ∧ ( ⋀ j = 1 p f ( b j ) ) | a 1 → ( a 2 → … ( a m →    x) ⋯) =1, b₁ → (b₂ → … (b _p  → (x → y)) …) =1} ≤ ⋁ { ⋀ k = 1 n f ( c k ) | c 1 → ( c 2 → … ( c n → y ) … ) = 1 } = g (y).

Therefore g is a fuzzy filter of E by Theorem 3.6.

Next, since x ≤ 1 = x → x, we have x → (x → x) =1, then g (x) ≥ f (x) ∧ f (x) = f (x) by Theorem 3.7.

Finally, suppose that h is a fuzzy filter with f ≤ h. Then for all x ∈ E, g ( x ) = ⋁ { ⋀ i = 1 n f ( a i ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 } ≤ ⋁ { ⋀ i = 1 n h ( a i ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 } ≤ ⋁ { h ( x ) } = h ( x ) .

By Definition 3.10, we have that g is a fuzzy filter generated by f, that is, g = 〈f〉. This completes the proof.

We will denote the set of all fuzzy filters of an equality algebra E by FF (E).

Recall that a lattice (L, ∧ , ∨) is called completely meet-distributive, if it is complete and the identity x ∧ (⋁ _i∈Ly _i ) = ⋁ _i∈L (x ∧ y _i ) holds in L, for all x ∈ L, and {y _i , i ∈ I} ⊆ L.

In [6], Kadji, Lele and Tonga presented an open problem as follows: Let L be a residuated lattice, where lattice is a completely meet-distributive, and FF (L) be a set of all fuzzy filters of L. Is (FF (L) , ∧ , ⊔) a completely meet-distributive lattice?

In order to solve the problem, we must study relations between residuated lattices and equality algebras. It was proved in [12] that any residuated lattice (L, ⊙ → , ∨ , ∧ , 0, 1), the structure E _L  = (L, ∼ , ∧ , ∨ , 1) is a lattice equality algebra, where the operation ∼ defined on L as follows, x ∼ y = (x → y) ∧ (y → x). Therefore, equality algebras are a generalization of residuated lattices.

In what follows, by L we denote the residuated lattice (L, ⊙ → , ∨ , ∧ , 0, 1) and by E _L we denote the lattice equality algebra (L, ∼ , ∧ , ∨ , 1).

Lemma 3.13. Let L be a residuated lattice. Then FF (L) = FF (E _L ).

Proof. By Proposition 2.12 (1) and Theorem 3.6, we obtain FF (L) = FF (E _L ).

Now, we solve the above problem in equality algebras, which are a generalization of residuated lattices. As a consequence, it also holds in residuated lattices. So we give the following theorem.

Theorem 3.14. Let E be a lattice equality algebra, where lattice is a completely meet-distributive, and f _i  ∈ FF (E). The operation ⊔ on FF (E) is definded by: ⨆_i∈If _i  = 〈 ⋁ _i∈If _i 〉, Then (FF (E) , ∧ , ⊔) is a completely meet-distributive lattice.

Proof. Obviously, (FF (E) , ∧ , ⊔) is a completely lattice.

Let f, g _i  ∈ FF (E). In the following, we prove the meet-distributive law: f ∧ (⨆ _i∈Ig _i ) = ⨆ _i∈I (f ∧ g _i ) Obviously, ⨆_i∈I (f ∧ g _i ) ≤ f ∧ (⨆ _i∈Ig _i ) holds.

On the other hand, note that a₁ → (a₂ → … (a _n  → x) …) =1 if and only if (a₁ ∨ x) → ((a₂ ∨ x) → … ((a _n  ∨ x) → x) …) =1 by Proposition 2.5 (2). Therefore,

(f ∧ (⨆ _i∈Ig _i )) (x) = f (x) ∧ (⨆ _i∈Ig _i ) (x) = f ( x ) ∧ ( ⋁ { ⋀ k = 1 n ( ⋁ i ∈ I g i ) ( a k ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 } ) = ⋁ { f ( x ) ∧ ( ⋀ k = 1 n ( ⋁ i ∈ I g i ) ( a k ) ) | a 1 → ( a 2 → … ( a n → x ) … ) = 1 } ≤ ⋁ { ⋀ k = 1 n ( f ( x ∨ a k ) ∧ ( ⋁ i ∈ I g i ) ( x ∨ a k ) ) | a 1 → ( a 2 → … ( a n → x) …) =1} ≤ ⋁ { ⋀ k = 1 n ( f ( b k ) ∧ ( ⋁ i ∈ I g i ) ( b k ) ) | b 1 → ( b 2 → … ( b n → x ) … ) = 1 } = ⋁ { ⋀ k = 1 n ( ⋁ ( f ∧ g i ) ( b k ) ) | b 1 → ( b 2 → … ( b n → x ) … ) = 1 } = (⨆ _i∈I (f ∧ g _i )) (x). Thus, we can get f ∧ (⨆ _i∈Ig _i ) = ⨆ _i∈I (f ∧ g _i ).

In the following corollary, we solve the open problem in residuated lattices, which was presented in [6] by Kadji, Lele and Tonga.

Corollary 3.15. Let L be a residuated lattice, where lattice is a completely meet-distributive. Then (FF (L) , ∧, ⊔) is a completely meet-distributive lattice.

Proof. Let L be a residuated lattice, where lattice is a completely meet-distributive. Then E _L  = (L, ∼ , ∧ , ∨ , 1) is a lattice equality algebra, in which the lattice also is a completely meet-distributive. Moreover, we can obtain that (FF (E _L ) , ∧ , ⊔) is a completely meet-distributive lattice by Theorem 3.14. Using Lemma 3.13, we can get that (FF (L) , ∧ , ⊔) is a completely meet-distributive lattice.

In what follows, we construct a quotient algebra of an equality algebra by using a fuzzy filter. A fuzzy subset defined by f ^x  (z) = f (x ∼ z) is called the fuzzy coset determined by x and f.

Lemma 3.16. Let E be an equality algebra and f be a fuzzy filter of E. Then for any x, y ∈ E, f ^x  = f ^y if and only if f (x ∼ y) = f (1).

Proof. Suppose that for all z ∈ E, f ^x  (z) = f ^y  (z) holds, that is, f (x ∼ z) = f (y ∼ z). Particularly, let z = y, then we have f (x ∼ y) = f (y ∼ y) = f (1). Conversely, assume that f (x ∼ y) = f (1) holds. Applying Proposition 3.8 (4), we get f (y ∼ z) = f (x ∼ y) ∧ f (y ∼ z) ≤ f (x ∼ z). Analogously, we obtain f (x ∼ z) ≤ f (y ∼ z). This completes the proof.

Lemma 3.17. Let E be an equality algebra and f be a fuzzy filter of E. Then for any x, y, u, v ∈ E, f ^x  = f ^y and f ^u  = f ^v imply f^x⋄u = f^y⋄v, where ⋄ ∈ {∧ , ∼}.

Proof. From Lemma 3.16, we have f ^x  = f ^y and f ^u  = f ^v imply f (x ∼ y) = f (1) and f (u ∼ v) = f (1). Since f is a fuzzy filter, then it follows from Definition 2.1 (E6)(E7) and Definition 3.1 (2) that f (1) = f (x ∼ y) ∧ f (u ∼ v) ≤ f ((x ⋄ u) ∼ (y ⋄ v)). By Lemma 3.16 again, we get f^x⋄u = f^y⋄v.

Let f be a fuzzy filter of E. Suppose that E/f denotes the set of all fuzzy cosets of f. For any f ^x , f ^y  ∈ E/f, we define f ^x  ∧ f ^y  = f^x∧y, f ^x  ∼ f ^y  = f^x∼y.

From Lemma 3.17, we can show that the operations on E/f are well-defined. The binary relation on E/f is defined by the derived meet operation in the following way: f ^x  ≤ f ^y iff f ^x  ∧ f ^y  = f ^x iff f^x∧y = f ^x iff f ((x ∧ y) ∼ x) = f (1) iff f (x → y) = f (1). It is clear that ≤ is a partial order.

Theorem 3.18. Let E be an equality algebra and f be a fuzzy filter of E. Then E/f = (E/f, ∧ , ∼ , f¹) is an equality algebra.

Proof. It is trivial to verify that E/f is an equality algebra.

Theorem 3.19. Let E be an equality algebra and f be a fuzzy filter of E. Define a mapping μ : E ↦ E/f, μ (x) = f ^x . Then (1) μ is a surjective homomorphism, (2) Ker(μ) = f_f(1), (3) E/f is isomorphic to E/f_f(1).

Proof. (1) It is trivial.

(2) For all x∈ Ker(μ) if and only if μ (x) = f¹ if and only if f ^x  = f¹ if and only if f (x ∼ 1) = f (1) if and only if f (x) = f (1) if and only if x ∈ f_f(1).

(3) By (1) and (2), we can get that E/f is isomorphic to E/f_f(1).

4 Fuzzy congruences of equality algebras

In this section, we introduce the concept of fuzzy congruences on equality algebras, and show that there is one-to-one correspondence between fuzzy filters and fuzzy congruences.

Definition 4.1. Let E be an equality algebra. A fuzzy relation ρ on E is called a fuzzy congruence if ρ is a fuzzy equivalent and satisfies ρ (x,y) ≤ ρ (x ⋄ z,y ⋄ z), for all x, y, z ∈ E, where ⋄ ∈ {∧ , ∼}.

Let ρ be a fuzzy congruence, the fuzzy subset ρ ^a  : E → [0, 1], which is defined by ρ ^a  (x) = ρ (a, x), is said to be the fuzzy congruence class including x.

We will denote the set of all fuzzy congruences on an equality algebra E by FC (E).

Corollary 4.2. Let E be an equality algebra. If ρ is a fuzzy congruence on E, then ρ (x, y) ≤ ρ (x → z, y → z) ∧ ρ (z → x, z → y), for all x, y, z ∈ E.

Proof. It is clear that ρ (x, y) ≤ ρ (z → x, z → y). By Definition 4.1, we have ρ (x, y) ≤ ρ (x ∧ z, y ∧ z) ≤ ρ (x ∼ (x ∧ z) , x ∼ (y ∧ z)) and ρ (x, y)≤ ρ (x ∼ (y ∧ z) , y ∼ (y ∧ z)). By transitivity of ρ, it follows that ρ (x, y)≤ ρ (x ∼ (x ∧ z) , x ∼ (y ∧ z)) ∧ ρ (x ∼ (y ∧ z) , y ∼ (y ∧ z)) ≤ ρ (x ∼ (x ∧ z) , y ∼ (y ∧ z)) = ρ (x → z, y → z). Thus, ρ (x, y) ≤ ρ (x → z, y → z) ∧ ρ (z → x, z → y).

Definition 4.3. A fuzzy filter f of an equality algebra E is called an extreme fuzzy filter of E if f (1) =1.

In the following, we use extreme fuzzy filters of an equality algebra to induce fuzzy congruences on it.

Theorem 4.4. Let E be an equality algebra and f be an extreme fuzzy filter of E. Define a fuzzy binary relation ρ on E as follows: for all x, y ∈ E, ρ (x, y) = f (x → y) ∧ f (y → x), Then ρ is a fuzzy congruence on E, and ρ¹ = f.

Proof. For all x, y ∈ E, it is clear that ρ is reflexive and symmetry. For the transitivity, by Proposition 3.8 (5), we have ρ (x, y) ∧ ρ (y, z) = f (x → y) ∧ f (y → z) ∧ f (z → y) ∧ f (y → x) ≤ f (x → z) ∧ f (z → x) = ρ (x, z). Thus, ρ is a fuzzy equivalent relation. By (E5) (E7) and Proposition 2.2 (1), we have x → y ≤ (x ∼ z) → (y ∼ z). So it follows that ρ (x, y) ≤ ρ (x ∼ z, y ∼ z). And since Proposition 2.3 (6), we obtain ρ (x, y) ≤ ρ (x ∧ z, y ∧ z). Hence ρ is a fuzzy congruence on E, and so ρ¹ (x) = ρ (1, x) = f (1 → x) ∧ f (x → 1) = f (x). Therefore, ρ¹ = f.

The fuzzy congruence ρ which is given in Theorem 4.4 is called the fuzzy congruence induced by fuzzy filter f and denoted by ρ _f .

Corollary 4.5. Let E be an equality algebra and f be a fuzzy filter of E. Then ρ f 1 = f .

Theorem 4.6. Let E be an equality algebra and ρ be a fuzzy congruence on E. Then ρ¹ is a fuzzy filter.

Proof. By Corollary 4.2, we have ρ¹ (x) = ρ (1, x) ≤ ρ (x → 1, x → x) = ρ (1, 1) = ρ¹ (1). According to Corollary 4.2 and the transitivity of ρ, it follows that ρ¹ (x) ∧ ρ¹ (x → y) = ρ (1, x) ∧ ρ (1, x → y) ≤ ρ (y, x → y) ∧ ρ (1, x → y) ≤ ρ (1, y) = ρ¹ (y). Thus, by Theorem 3.6, ρ¹ is a fuzzy filter.

Lemma 4.7. Let E be an equality algebra and ρ be a fuzzy congruence on E. Then ρ (x, y) = ρ (1, x ∼ y), for any x, y ∈ E.

Proof. It is clear that ρ (x, y) ≤ ρ (1, x ∼ y). By Proposition 2.2 (5) and ρ is a fuzzy congruence, we have ρ (1, x ∼ y) = ρ (1, y ∼ x) ≤ ρ (1 ∼ x, (y ∼ x) ∼ x) ∧ ρ (y, y) ≤ ρ (x ∧ y, ((y ∼ x) ∼ x) ∧ y) = ρ (x ∧ y, y). Similarly, we obtain ρ (1, x ∼ y) ≤ ρ (x ∧ y, x). So ρ (1, x ∼ y) ≤ ρ (x, y). Thus, ρ (x, y) = ρ (1, x ∼ y).

Theorem 4.8. Let E be an equality algebra and ρ be a fuzzy congruence on E. Then ρ _{ρ ¹}  = ρ.

Proof. According to Corollary 4.2, we have ρ (x, y) ≤ ρ (x → x, x → y) ∧ ρ (x → x, y → x) = ρ (1, x→ y) ∧ ρ (1, y → x) = ρ¹ (x→ y) ∧ ρ¹ (y → x) = ρ _{ρ ¹}  (x, y). On the other hand, by Lemma 4.7 and the transitivity of ρ, it follows that ρ _{ρ ¹}  (x, y) = ρ¹ (x → y) ∧ ρ¹ (y → x) = ρ (1, (x ∧ y) ∼ x) ∧ ρ (1, (x ∧ y) ∼ y) = ρ (x ∧ y, x) ∧ρ (x ∧ y, y) ≤ ρ (x, y). Therefore, ρ _{ρ ¹}  = ρ.

Theorem 4.9. Let E be an equality algebra. Then there is one-to-one correspondence between FF (E) and FC (E).

Proof. Define two maps Ψ : FF (E) ↦ FC (E) by f ↦ ρ _f , and Φ : FC (E) ↦ FF (E) by ρ ↦ ρ¹. By Corollary 4.5 and Theorem 4.8, it follows that Φ ∘ Ψ = id_FF(E) and Ψ ∘ Φ = id_FC(E). This completes the proof.

5 Uniform topologies on equality algebras

In this section, we construct the uniform structures by the special family of extreme fuzzy filters, and then induce uniform topologies. Moreover, we show that equality algebras with uniform topologies are topological equality algebras, and also some properties are investigated.

Let f be an extreme fuzzy filter of an equality algebra E. According to Theorem 4.4, we have that ρ _f is a fuzzy congruence on E, where ρ _f  (x, y) = f (x → y) ∧ f (y → x). Obviously, ρ _f  (x, y) =1 if and only if f (x → y) = f (y → x) =1.

Let X be a nonempty set and A, B be any subset of X × X. We have the following notation: (1) A ∘ B = {(x, y) ∈ X × X : (x, z) ∈A, (z, y) ∈ B, for some z ∈ X}; (2) A^-1 = {(x, y) ∈ X × X : (y, x) ∈ A}; (3) △; = {(x, x) ∈ X × X : x ∈ X}.

Definition 5.1. [5] A nonempty collection K of subsets of X × X is called an uniformity on X, which satisfies the following conditions: (A1) △; ⊆ A for any A ∈ K ; (A2) if A ∈ K , then A - 1 ∈ K ; (A3) if A ∈ K , then there exists B ∈ K such that B ∘ B ⊆ A; (A4) if A , B ∈ K , then A ∩ B ∈ K ; (A5) if A ∈ K and A ⊆ B ⊆ X × X, then B ∈ K .

Then the pair ( X , K ) is called an uniform structure(uniform space).

From now on, we assume that f is a family of extreme fuzzy filter of an equality algebra E, which is closed under intersection.

In the following, we use f to induce uniform structures.

Theorem 5.2. Let E be an equality algebra, f ∈ f , A _f  = {(x, y) ∈ E × E : ρ _f  (x, y) =1} and K ∗ = { A f : f ∈ f } . Then K ∗ satisfies the conditions (A1) - (A4).

Proof. (A1): Since f is an extreme fuzzy filter of E, we have ρ _f  (x, x) =1 for all x ∈ E. Hence △; ⊆ A _f for all A f ∈ K ∗ . (A2): For all A f ∈ K ∗ , we have (x, y) ∈ (A _f ) ^-1 ⇔ (y, x) ∈ A _f  ⇔ ρ _f  (y, x) =1 ⇔ ρ _f  (x, y) =1 ⇔ (x, y) ∈ A _f . (A3): For any A f ∈ K ∗ , by the transitivity of ρ _f , we imply that A _f  ∘ A _f  ⊆ A _f . (A4): For any A f , A g ∈ K ∗ , we will show that A _f  ∩ A _g  = A_f∩g. If (x, y) ∈ A _f  ∩ A _g , then f (x → y) = f (y → x) =1 and g (x → y) = g (y → x) =1. It follows that (f ∩ g) (x → y) = (f ∩ g) (y → x) =1. Hence (x, y) ∈ A_f∩g. Conversely, let (x, y) ∈ A_f∩g. Then (f ∩ g) (x → y) = (f ∩ g) (y → x) =1. It follows that f (x → y) ∧ g (x → y) = f (y → x) ∧ g (y → x) =1. We can get that 1 = f (x → y) ∧ g (x → y) ≤ f (x → y), which implies that f (x → y) =1. Analogously, we get f (y → x) =1. Hence (x, y) ∈ A _f . Similarly, we can get (x, y) ∈ A _g . Therefore, A_f∩g ⊆ A _f  ∩ A _g . Since f , g ∈ f , we have f ∩ g ∈ f , and so A f ∩ A g ∈ K ∗ .

Theorem 5.3. Let E be an equality algebra and K = { A ⊆ E × E : ∃ A f ∈ K ∗ s . t . A f ⊆ A } , where K ∗ comes from Theorem 5.2. Then K is an uniformity on E and the pair ( E , K ) is an uniform structure.

Proof. By Theorem 5.2, the collection K satisfies conditions (A1) - (A4). So we prove that (A5). Let A ∈ K and A ⊆ B ⊆ E × E. Then there exists A _f  ⊆ A ⊆ B, which means that B ∈ K .

Let x ∈ L and A ∈ K . Define A [x] : = {y ∈ E : (x, y) ∈ A}. Obviously, if B ⊆ A, then B [x] ⊆ A [x].

Theorem 5.4. Let E be an equality algebra. Then T = { G ⊆ E : ( ∀ x ∈ G ) ( ∃ A ∈ K ) s.t. A [x] ⊆ G}is a topology on E, where K comes from Theorem 5.3.

Proof. Obviously, ∅ and E belong to T . It is clear that T is closed under arbitrary union. Finally, we will show that T is closed under finite intersection. Let G , H ∈ T and assume that x ∈ G ∩ H. Then there exist A , B ∈ K such that A [x] ⊆ G and B [x] ⊆ H. If C = A ∩ B, then C ∈ K . Thus, C [x] ⊆ A [x] ∩ B [x] and so C [x] ⊆ G ∩ H, hence G ∩ H ∈ T . Therefore, T is a topology on E.

Note that for all x ∈ E, U [x] is a neighborhood of x.

Let f be an arbitrary family of extreme fuzzy filters of an equality algebra E which is closed under intersection. Then the topology T f comes from Theorem 5.4 is called an uniform topology on E induced by f . If f = f , we denote it by T f .

In what follows, we show that equality algebras with uniform topologies are topological equality algebras, and also study some properties of them.

Definition 5.5. Let E be an equality algebra and T be a topology on E. ( E , T ) is called a topological equality algebra (TE-algebra for short) if the operations ∧, ∼ are continuous on T .

Note that the operation ★ ∈ {∧ , ∼} is continuous if and only if for any x, y ∈ E and any neighborhood C of x ★ y there exist two neighborhoods A and B of x and y, respectively, such that A ★ B ⊆ C.

Theorem 5.6. Let f be a family of extreme fuzzy filters of an equality algebra E, which is closed under intersection. Then the space ( E , T f ) is a TE-algebra.

Proof. By Definition 5.5, we will show that ★ is continuous, where ★ ∈ {∧ , ∼}. Indeed, suppose that x ★ y ∈ G, where x, y ∈ E and G is an open subset of E. Then there exist A ∈ K , A [x ★ y] ⊆ G, and an extreme fuzzy filter f such that A f ∈ K ∗ and A _f  ∈ A. We prove that the following relation holds: A _f  [x] ★ A _f  [y] ⊆ A _f  [x ★ y] ⊆ A [x ★ y].

Let h, k ∈ A _f  [x] ★ A _f  [y]. Then h ∈ A _f  [x] and k ∈ A _f  [y], it follows that ρ _f  (x, h) =1 and ρ _f  (y, k) =1. Thus ρ _f  (x ★ y, h ★ k) =1, and we get (x ★ y, h ★ k) ∈ A _f  ⊆ A. Hence h ★ k ∈ A _f  [x ★ y] ⊆ A [x ★ y]. Then h ★ k ∈ G. Obviously, A _f  [x] and A _f  [y] are neighborhoods of x and y, respectively. Therefore, the operation ★ is continuous.

Lemma 5.7. Let f and g be extreme fuzzy filters of an equality algebra E. If g ≤ f, then A _g  ⊆ A _f .

Proof. It is clear.

Theorem 5.8. Let f and g be extreme fuzzy filters of an equality algebra E. If g ≤ f, then T f ⊆ T g .

Proof. Let f = f , K ∗ = A f and K = { A ⊆ E × E : A f ⊆ A } . Let O ∈ T f . Then for all x ∈ O , there exists A ∈ K such that A [ x ] ⊆ O and A f [ x ] ⊆ A [ x ] ⊆ O . By Lemma 5.7, we have A _g  ⊆ A _f . It follows that A g [ x ] ⊆ A f [ x ] ⊆ O . Therefore, O ∈ T g , and so T f ⊆ T g .

Theorem 5.9. Let f be a family of extreme fuzzy filters of an equality algebra E, which is closed under intersection. If g = ⋂ { f : f ∈ f } , then T f = T g .

Proof. Let f 1 = g , K ∗ 1 = A g and K 1 = { A ⊆ E × E : A g ⊆ A } . Let O ∈ T g . Then for all x ∈ O , there exists A ∈ K 1 such that A [ x ] ⊆ O . Thus A g [ x ] ⊆ A [ x ] ⊆ O . Since f is closed under intersection, so we have g ∈ f . It follows that A g ∈ K , and O ∈ T f . So T g ⊆ T f . Conversely, it directly follows from Theorem 5.8.

6 Conclusions

Since Zadeh [10] introduced the concept of fuzzy sets, his ideas have been applied to various fields. For example, artificial intelligence, information science, computer science, and so on. In this paper, we apply these ideas to propose the notions of fuzzy filters and fuzzy congruences on equality algebras, and investigate some properties of them. In order to discuss the algebra structure about the set of all fuzzy filters of an equality algebra, and solve the open problem in [6] by Kadji, Lele and Tonga, we must study a method about generating a fuzzy filter by a fuzzy set. Also, we prove that the quotient algebra E/f, induced by a fuzzy filter f, is an equality algebra. We know that the quotient algebra E/F, induced by a filter F, also is an equality algebra. But what is the relationship between the two quotient algebras? we give a positive answer, that is, E/f is isomorphic to the E/f_f(1), where f_f(1) = {x ∈ E|f (x) = f (1)} is a filter of E. Moreover, we prove that there exists one-to-one correspondence between fuzzy filters and fuzzy congruences. Finally, we construct uniform structures by extreme fuzzy filters on equality algebras, and uniform topologies are induced. Moreover, we prove that equality algebras with uniform topologies are topological equality algebras, and also investigate some properties of them.

Future research will focus on studying fuzzy prime filters, and establishing fuzzy prime spectrums of equality algebras.