Extreme fuzzy filters and its application on BL -algebras

Abstract

In this paper, using a special family of extreme fuzzy filters $F$ on a BL-algebra L, we construct a uniform structure $(L, K)$ , and then the part $K$ induce a uniform topology $T_{F}$ in L. Also, we prove that the pair $(L, T_{F})$ is a topological BL-algebra, and some properties of $(L, T_{F})$ are investigated. In particular, we show that $(L, T_{F})$ is a first-countable, zero-dimensional, disconnected and completely regular space. At the same time, we give some characterizations of topological properties of $(L, T_{F})$ , for example, $(L, T_{F})$ is a discrete space if and only if g₁ = {1}, and $(L, T_{F})$ is a Hausdorff space if and only if g₁ = {1}, where $g = \cap F$ . Finally, we naturally contact the relationship between isomorphism (algebraic invariant) and homeomorphism (topological invariant) in topological BL-algebras.

Keywords

Extreme fuzzy filter uniform space

1 Introduction

In 1998, Hájek [3] introduced a very general many-valued logic, called Basic Logic (or BL), with the idea to formalize the many-valued semantics induced by a continuous t-norm on the unit real interval [0, 1]. This Basic Logic turns to be a fragment common to three important many-valued logics: Łukasiewicz logic, Gödel logic and Product logic. The Lindenbaum-Tarski algebras for Basic Logic are called BL-algebras. Apart from their logic interest, BL-algebras have important algebraic properties and they have intensively studied from an algebraic point of view. Filters theory play an important role in studying these algebras. From logical point of view, various filters correspond to various sets of provable formulae. Up to now, the filter theory of BL-algebras has been widely studied, and some important results have been obtained. In particular, in [3 , 18] some types of filters in BL-algebras were introduced, and some of their characterizations and relations were investigated. The concept of fuzzy sets was introduced by Zadeh [20] in 1965. Fuzzification of special types of filters on several different algebras of many-valued logics has been popular in recent years (see [7 , 10–15]).

The concept of a uniform space can be considered either as axiomatizations of some geometric notions, close to but quite independent of the concept of a topological space, or as convenient tools for an investigation of topological space. Recently, several researchers have studied some interesting properties of logic algebras with uniform topologies (see [1, 2 , 19]). How to give fuzzified versions of consequences mentioned above? As observed above, to induce uniform topology in their corresponding algebraic structure, authors used special family of filters which is closed under intersection. However, this approach failed when we use the family of fuzzy filters of BL-algebras. More explicitly, given a BL-algebra L, F, G are filters of L and f, g are fuzzy filters of L, respectively. Then θ_F ∩ θ_G = θ_F∩G hold but θ_f ∩ θ_g = θ_f∩g failed (see Example 3.8), where θ_F is a congruence induced by F, and θ_f is a congruence induced by f. Because this difference, unlike crisp situation, using the same approach, the family of fuzzy filters of a BL-algebra can not construct a uniform topology on it. In order to overcome this difficulty, we give the concept of extreme fuzzy filters. The congruence θ_f induced by the extreme fuzzy filter f satisfies (x, y) ∈ θ_f if and only if f (x → y) = f (y → x) =1, and it follows that θ_f ∩ θ_g = θ_f∩g. This property is very important to construct uniform topology on BL-algebras. On the other hand, we prove that (x, y) ∈ θ_f if and only if x → y, y → x ∈ f^-1 (1) and f is an extreme fuzzy filter if and only if f_t is a filter in BL-algebra for all t ∈ [0, 1]. Because these formal consistency between crisp filters and fuzzy filters, from the fuzzy viewpoint, the notion of extreme fuzzy filters of BL-algebras maybe more fuzzified. In addition, we also obtain some new and interesting results about uniform topologies on BL-algebras which will be meaningful for further study topological BL-algebras.

This paper is organized as follows: In Section 2, we recall some facts about BL-algebras, topologies and fuzzy filters, which will be needed in the sequel. In Section 3, we define a new class of fuzzy filters, i.e., extreme fuzzy filters, and give some characterizations of extreme fuzzy filters. Using the family of extreme fuzzy filters of a BL-algebra L, we construct uniform structures on it, and then induce uniform topology on L. In Section 4, we show that BL-algebras with uniform topologies are topological BL-algebras, and also some properties of them are investigated.

2 Preliminaries

In this section, we summarize results aboutBL-algebras and topologies which will be needed in the following sections.

Definition 2.1. [3] A BL-algebra is an algebra (L, ∧ , ∨ , ⊙ , → , 0, 1) of type (2,2,2,2,0,0) satisfying the following conditions:

(L, ∧ , ∨ , 0, 1) is a bounded lattice,

(L, ⊙ , 1) is a commutative monoid,

x ⊙ y ≤ z if and only if x ≤ y → z for all x, y, z ∈ L,

x ∧ y = x ⊙ (x → y) for all x, y ∈ L,

(x → y) ∨ (y → x) =1 for all x, y ∈ L.

Throughout this paper we simply write L for a BL-algebra (L, ∧ , ∨ , ⊙ , → , 0, 1) when there is no chance for confusion.

Let L be a BL-algebra and A, B ⊆ L. We write A ∗ B for {x ∗ y : x ∈ A, y ∈ B}, and when dealing with singleton sets we shall simply write a ∗ B and A ∗ b rather than {a} ∗ B and A ∗ {b}, where ∗ ∈ {∧ , ∨ , ⊙ , →}.

For the convenience of readers, we provide some basic properties of the operations on BL-algebras in the following proposition.

Proposition 2.2. [3 , 18] The following properties hold for any BL-algebras: for any x, y, z ∈ L,

x ≤ y if and only if x → y = 1,

if x ≤ y, then y → z ≤ x → z, z → x ≤ z → y and x ⊙ z ≤ y ⊙ z,

x ⊙ (x → y) ≤ y,

x ⊙ y ≤ x ∧ y, x ≤ y → x,

x → (y → z) = (x ⊙ y) → z = y → (x → z),

x → y ≤ (y → z) → (x → z),

x → y ≤ (z → x) → (z → y),

x → y ≤ x ∘ z → y ∘ z, where ∘ ∈ {∧ , ∨ , ⊙}.

Definition 2.3. [3] A filter of a BL-algebra L is a nonempty set F ⊆ L such that for all x, y ∈ L satisfies:

x, y ∈ F implies x ⊙ y ∈ F,

x ∈ F and x ≤ y imply y ∈ F.

By a deductive system we mean a subset D of L containing 1 such that x → y ∈ D, x ∈ D imply y ∈ D for each x, y ∈ L. Note that F is a filter of L if and only if F is a deductive system.

Definition 2.4. [20] A fuzzy set in X is a mapping f : X → [0, 1].

Let f be a fuzzy set in X, t ∈ [0, 1], the set f_t = {x ∈ X : f (x) ≥ t} is called a level subset of f.

For any a, b ∈ [0, 1], we denote a ∨ b : = max {a, b} and a ∧ b : = min {a, b}. For any fuzzy sets f, g in L, we define $f \leq g if and only if f (x) \leq f (y) for all x \in L .$

The fuzzy sets f ∩ g and f ∪ g are defined as follows: (f ∪ g) (x) = f (x) ∨ g (x), (f ∩ g) (x) = f (x) ∧ g (x) for all x ∈ L.

Definition 2.5. [12] Let f be a fuzzy set of aBL-algebra L. Then f is called a fuzzy filter if f_t is either empty or a filter of L for all t ∈ [0, 1].

In general, it is not difficult to obtain that the intersection of a family of fuzzy filters of L is a fuzzy filter.

Proposition 2.6. [12] The following statements are equivalent in any BL-algebra L:

f is a fuzzy filter of L,

f (1) ≥ f (x) and f (y) ≥ f (x) ∧ f (x → y) for all x, y ∈ L,

x → (y → z) =1 implies f (z) ≥ f (x) ∧ f (y),

x ⊙ y ≤ z implies f (z) ≥ f (x) ∧ f (y) for all x, y, z ∈ L,

f is order-preserving and f (x ⊙ y) ≥ f (x) ∧ f (y) for any x, y ∈ L.

Now we give some useful propositions of fuzzy filters of BL-algebras which will be needed in the sequel.

Proposition 2.7. [12] Let f be a fuzzy filter of L. Then the following statements hold: for any x, y, z ∈ L,

if f (x → y) = f (1), then f (x) ≤ f (y),

if x ≤ y, then f (x) ≤ f (y),

if f (x → y) ∧ f (y → x) ≤ f (x → z),

f (x → y) ≤ f ((x ⊙ z) → (y ⊙ z)),

f (x → y) ≤ f ((x ∧ z) → (y ∧ z)),

f (x → y) ≤ f ((x ∨ z) → (y ∨ z)),

f (x → y) ≤ f ((y → z) → (x → z)),

f (x → y) = f (x ∨ y → y).

At the end of this section, let us recall some basic notions of general topology which will be needed in the sequel.

Recall that a set A with a family $T$ of its subsets is called a topological space, denoted by $(A, T)$ , if $A, \emptyset \in T$ , the intersection of any finite members of $T$ is in $T$ , and the arbitrary union of members of $T$ is in $T$ . The members of $T$ are called open sets of A, and the complement of an open set U, i.e., A \ U, is called a closed set. A subfamily {U_α} _α∈I of $T$ is called a base of $T$ if for each $x \in U \in T$ there is an α ∈ I such that x ∈ U_α ⊆ U. A subset P of A is a neighborhood of x ∈ A, if there exists an open set U such that x ∈ U ⊆ P. Let $T_{x}$ denote the totality of all neighborhoods of x in A, then subfamily $V_{x}$ of $T_{x}$ is a fundamental system of neighborhoods of x, if for each U_x in $T_{x}$ , there exists a V_x in $V_{x}$ such that V_x ⊆ U_x. If every point x in A has a countable fundamental system of neighborhoods, then we say that the space $(A, T)$ satisfies the first axiom of countability or is first-countable. A topological space $(A, T)$ is a zero-dimensional space if $T$ has a clopen base. A topological space $(A, T)$ is called a regular space if for any closed subset C of A and x ∈ A such that x ∉ C, then there exist disjoint open sets U, V such that x ∈ U and C ⊆ V, or equivalently, for any open subset U containing x, there exists open subset V such that $x \in V \subseteq \bar{V} \subseteq U$ . Let $(A, T)$ and $(B, V)$ be two topological spaces, a mapping f of A to B is continuous if $f^{- 1} (U) \in T$ for any $U \in V$ . A topological space $(A, T)$ is called a completely regular space, if for every x ∈ X and every closed set F ⊂ A such that x ∉ F there exists a continuous function f : A → [0, 1] such that f (x) =0 and f (y) =1 for y ∈ F. The mapping f from $(A, T)$ to $(B, V)$ is called a homeomorphism if f is bijective, and f and f^-1 are continuous, or equivalently, if f is bijective, continuous and open(closed). The mapping f from $(A, T)$ to $(B, V)$ is called a quotient map if f is surjective, and $V \in V$ if and only if $f^{- 1} (V) \in T$ . A topological space $(A, T)$ is compact if each open cover of A is reducible to a finite open subcover.

Let $(X, T)$ be a topological space. We have following separation axioms in $(X, T)$ :

T₀ : For each x, y ∈ X and x ≠ y, there is at least one in an open neighborhood excluding the other.

T₁ : For each x, y ∈ X and x ≠ y, each has an open neighborhood not containing the other.

T₂ : For each x, y ∈ X and x ≠ y, both have disjoint open neighborhoods U, V such that x ∈ U and y ∈ V .

A topological space satisfying T_i is called a T_i-space for any i = 0, 1, 2 . A T₂-space is also known as a Hausdorff space.

3 Extreme fuzzy filter and uniform topology on BL-algebras

Definition 3.1. A fuzzy filter f of a BL-algebra L is called an extreme fuzzy filter of L (EF-filter for short) if f (1) =1.

Remark 3.2. Note that F is a filter of aBL-algebra L if and only if χ_F is a fuzzy filter in L, where χ_F is the characteristic function of F. Clearly, χ_F is an EF-filter if F is a filter of L. We denote F (L), FF (L) and EFF (L) the sets of all filters, fuzzy filters and extreme fuzzy filters of L, respectively. Then Card (F (L)) ≤ Card (EFF (L)) ≤ Card (FF (L)), where Card (X) is cardinality of the set of X.

Lemma 3.3. Let f is a fuzzy filter of L. Then f_t =∅ if and only if t > f (1).

Proof. The proof is straightforward. □

In the following we give some characterizations of extreme fuzzy filters.

Theorem 3.4. Let f be a fuzzy set of a BL-algebra L. The following statements are equivalent:

f is an extreme fuzzy filter of L,

f_t≠ ∅ for all t ∈ [0, 1],

f₁≠ ∅,

f₁ is a filter of L,

f_t is a filter of L for all t ∈ [0, 1].

Proof. (i) ⇒ (ii): Suppose that (i) is satisfied. Then f_t≠ ∅ for all t ∈ [0, 1]. Otherwise there exists t ∈ [0, 1] such that f_t =∅. By Lemma 3.3, t > f (1) =1 which is a contradiction.

(ii) ⇒ (iii) and (iii) ⇒ (iv) are straightforward.

(iv) ⇒ (v): Suppose that (iv) is satisfied. It follows that ∅ ≠ f₁ ⊆ f_t for all t ∈ [0, 1]. Since f is a fuzzy filter, then f_t is a filter of L.

(v) ⇒ (i): Suppose that the v is satisfied. Since f is a fuzzy filter of L and 1 ∈ f₁. It follows that f (1) =1. □

Interestingly, in BL-algebras every fuzzy filter can be expanded into an extreme fuzzy filter (Expansion principle).

Theorem 3.5. (Expansion principle) Let f be a fuzzy filter of L. Then there exists an extreme fuzzy filter $\bar{f}$ such that $f \leq \bar{f}$ .

Proof. Let f be a fuzzy filter of L. We define $\bar{f}$ by

$\bar{f} (x) = {\begin{matrix} 1, & x = 1, \\ f (x), & otherwise . \end{matrix}$ (1)

Clearly, $f \leq \bar{f}$ . One can easily check that for all t ∈ [0, 1],

${\bar{f}}_{t} = {\begin{matrix} {1}, & f (1) < t \leq 1, \\ f_{t}, & t \leq f (1) . \end{matrix}$ (2)

As a consequence $\bar{f}$ is an extreme fuzzy filter of L. □

Unfortunately, in general, the expansion of fuzzy filter is not unique.

Example 3.6. Let L = {0, a, b, c, 1}. Define ⊙ and → as follow:

⊙	a	b	c	1
0	0	0	0	0
a	a	c	c	a
b	c	b	c	b
c	c	c	c	c
1	a	b	c	1

→	0	a	b	c	1
0	1	1	1	1	1
a	0	1	b	b	1
b	0	a	1	a	1
c	0	1	1	1	1
1	0	a	b	c	1

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a BL-algebra (see [17]). Define fuzzy set f in L by f (0) = f (c) = f (b) =0.2, f (a) = f (1) =0.5. It is easily checked that f is a fuzzy filter. We can define extreme fuzzy filters $\bar{f}$ and $\tilde{f}$ by $\bar{f} (0) = \bar{f} (c) = \bar{f} (b) = 0.2, \bar{f} (a) = 0.5, \bar{f} (1) = 1$ , and $\tilde{f} (0) = \tilde{f} (c) = \tilde{f} (b) = 0.2, \tilde{f} (a) = \tilde{f} (1) = 1$ . Routine calculation shows that $\bar{f}$ and $\tilde{f}$ are expansion of f, but $\bar{f} \neq \tilde{f}$ .

It is well known that filter F of a BL-algebra is corresponding to congruence θ_F. In the following we shall use extreme fuzzy filters of a BL-algebra to induce congruences on it.

Let f be an extreme fuzzy filter of a BL-algebra L. We define a binary relation ≡_f on L by: $x \equiv_{f} y if and only if f (x \to y) \land f (y \to x) = 1 .$

Clearly, x ≡ _fy ⇔ f (x → y) = f (y → x) =1 ⇔ x → y, y → x ∈ f^-1 (1) ⇔ (x → y) ⊙ (y → x) ∈ f^-1 (1) ⇔ (x → y) ∧ (y → x) ∈ f^-1 (1). In ([11], Lemmas 3.18, 3.19, Corollary 3.20, [12], Corollary 5.4), using a fuzzy filter f of L, authors defined a binary relation θ_f by xθ_fy if and only if x → y ∈ f_f(1) and y → x ∈ f_f(1). In view of above fact, when f is extreme we conclude that θ_f = ≡ _f. In particular, if F is a filter of L, then χ_F is an extreme fuzzy filter and ≡_{χ
_F} = θ_F, where θ_F is a congruence induced by F as usual.

Note that if f is an extreme fuzzy filter of L, then f^-1 (1) = f₁ is a filter of L. Thus like ([12], Lemma 5.6) we have the following theorem.

Theorem 3.7. Let f be an extreme fuzzy filter of L. Then ≡_f is a congruence on L.

In the following we give an example to show that, in general, θ_f ∩ θ_g ≠ θ_f∩g if f and g are fuzzy filters.

Example 3.8. Let L = {0, a, b, c, 1}. Define ⊙ and → as follow:

⊙	c	a	b	1
0	0	0	0	0
c	0	c	c	c
a	c	a	c	a
b	c	c	b	b
1	c	a	b	1

→	0	c	a	b	1
0	1	1	1	1	1
c	0	1	1	1	1
a	0	b	1	b	1
b	0	a	a	1	1
1	0	c	a	b	1

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a BL-algebra (see [16]). Define fuzzy sets f, g in L by f (0) = f (c) = f (b) =0.2, f (a) = f (1) =0.5 and g (0) = g (c) = g (a) =0.5, g (b) = g (1) =0.8. One can easily check that for all t ∈ [0, 1], $f_{t} = {\begin{matrix} \emptyset, & t > 0.5, \\ {a, 1}, & 0.2 < t \leq 0.5, \\ L, & t \leq 0.2 . \end{matrix}$ (3) $g_{t} = {\begin{matrix} \emptyset, & t > 0.8, \\ {b, 1}, & 0.5 < t \leq 0.8, \\ L, & t \leq 0.5 . \end{matrix}$ (4)

Hence f, g are fuzzy filters of L. Clearly, f ∩ g = f. Routine calculation that θ_f = {(0, 0) , (c, c) , (a, a) , (b, b) , (1, 1) , (a, 1) , (1, a)} , θ_g = {(0, 0) , (c, c) , (a, a) , (b, b) , (1, 1) , (b, 1) , (1, b)} . Hence θ_f ∩ θ_g ≠ θ_f∩g.

Let X be a nonempty set and U, V be any subsets of X × X. We have the following notation:

U ∘ V = {(x, y) ∈ X × X : (x, z) ∈ U, (z, y) ∈ V, for some z ∈ X},

U^-1 = {(x, y) ∈ X × X : (y, x) ∈ U},

Δ = {(x, x) ∈ X × X : x ∈ X}.

Definition 3.9. [6] By a uniformity on X we shall mean a nonempty collection $K$ of subsets of X × X which satisfies the following conditions:

Δ ⊆ U for any $U \in K$ ,

if $U \in K$ , then $U^{- 1} \in K$ ,

if $U \in K$ , then there exists $V \in K$ such that V ∘ V ⊆ U,

if $U, V \in K$ , then $U \cap V \in K$ ,

if $U \in K$ and U ⊆ V ⊆ X × X, then $V \in K$ .

The pair $(X, K)$ is then called a uniform structure (uniform space).

For the rest of this paper we always suppose that $F$ is a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection, unless otherwise stated.

In the following we use $F$ to induce uniform structures.

Theorem 3.10. Let L be a BL-algebra, f belong to $F$ , U_f = {(x, y) ∈ L × L : x ≡ _fy} and $K^{*} = {U_{f} : f \in F}$ . Then $K^{*}$ satisfies the conditions (U1)-(U4).

Proof. (U1): Since f is a fuzzy filter of L, we have x ≡ _Fx, for any x ∈ L. Hence Δ ⊆ U_f for all $U_{f} \in K^{*}$ .

(U2): For any $U_{f} \in K^{*}$ , we have (x, y) ∈ (U_f) ^-1 ⇔ (y, x) ∈ U_f ⇔ y ≡ _fx ⇔ x ≡ _fy ⇔ (x, y) ∈ U_f .

(U3): For any $U_{f} \in K^{*}$ , the transitivity of ≡_f implies that U_f ∘ U_f ⊆ U_f.

(U4): For any $U_{f}, U_{g} \in K^{*}$ , we claim that U_f ∩ U_g = U_f∩g. If (x, y) ∈ U_f ∩ U_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) ∈ U_f∩g. Conversely, let (x, y) ∈ U_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 (f (x → y) ∧ g (x → y)) ∨ f (x → y) =1 ∨ f (x → y), this implies that f (x → y) =1. Using similar approach, we get f (y → x) =1. Hence (x, y) ∈ U_f. Similarly, we can get (x, y) ∈ U_g. Thus U_f∩g ⊆ U_f ∩ U_g. Since $f, g \in F$ , we have $f \cap g \in F$ , $U_{f} \cap U_{g} \in K^{*}$ . □

Theorem 3.11. Let L be a BL-algebra and $K = {U \subseteq L \times L : \exists U_{f} \in K^{*} s . t . U_{f} \subseteq U}$ , where $K^{*}$ comes from Theorem 3.10. Then $K$ is a uniformity on L and the pair $(L, K)$ is a uniformstructure.

Proof. By Theorem 3.10, the collection $K$ satisfies conditions (U1)-(U4). It suffices to show that $K$ satisfies (U5). Let $U \in K$ and U ⊆ V ⊆ L × L. Then there exists U_f ⊆ U ⊆ V, which means that $V \in K$ . □

Let x ∈ L and $U \in K$ . Define U [x] : = {y ∈ E : (x, y) ∈ U}. Clearly, if V ⊆ U, then V [x] ⊆ U [x].

Theorem 3.12. Let L be a BL-algebra. Then $T = {G \subseteq L : (\forall x \in G) (\exists U \in K) s . t . U [x] \subseteq G}$ is a topology on L, where $K$ comes from Theorem 3.11.

Proof. Clearly, ∅ and the set L belong to $T$ . It is clear that $T$ is closed under arbitrary union. Finally to show that $T$ is closed under finite intersection, let $G, H \in T$ and suppose that x ∈ G ∩ H. Then there exist $U, V \in K$ such that U [x] ⊆ G and V [x] ⊆ H. If W = U ∩ V, then $W \in K$ . Also W [x] ⊆ U [x] ∩ V [x] and so W [x] ⊆ G ∩ H, hence $G \cap H \in T$ . Thus $T$ is a topology on L. □

Note that for any x in L, U [x] is a neighborhood of x.

Definition 3.13. Let $F$ be an arbitrary family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. Then the topology $T_{F}$ comes from Theorem 3.12 is called a uniform topology on L induced by $F$ .

We denote the uniform topology obtained by an arbitrary family of extreme fuzzy filters $F$ by $T_{F}$ , and if $F = {f}$ , we denote it by $T_{f}$ .

Example 3.14. In Example 3.8 we define a fuzzy set f by f (0) = f (c) = f (a) =0.5, f (b) = f (1) =1. It is easily check that f is an extreme fuzzy filters. Consider $F = {f}$ . By Theorem 3.11, we construct $K^{*} = {U_{f}} = {{(x, y) : x \equiv_{g} y}} = {{(0, 0), (a, a), (b, b), (c, c), (1, 1), (1, b), (b, 1)}}$ . We can check that $(L, K)$ is a uniform space, where $K = {U : U_{f} \subseteq U}$ . Open neighborhoods are U_f [0] = {0}, U_f [a] = {a, c}, U_f [b] = {b, 1}, U_f [c] = {a, c}, U_f [1] = {b, 1}. By simple calculation, we get $T_{f} = {\emptyset, {0}, {a, c}, {b, 1}, {0, a, c}, {0, b, 1}, {a, b, c, 1}, {0, a, b, c, 1}}$ .

4 Topological properties of the space

(L, T_{F})

Note that from Theorem 3.12, giving $F$ the family of extreme fuzzy filters of a BL-algebra L which is closed under intersection, we can induce a uniform topology $T_{F}$ on L. In this section we study topological properties of $(L, T_{F})$ .

Definition 4.1. [21] Let $T$ be a topology of L. Then $(L, T)$ is called a topological BL-algebra (TBL-algebra for short) if the operations ∧, ∨ , ⊙ , → are continuous.

Note that the operation ∗ ∈ {∧ , ∨ , ⊗ , →} is continuous if and only if for any x, y ∈ L and any neighborhood W of x ∗ y there exist two neighborhoods U and V of x and y, respectively, such that U ∗ V ⊆ W.

Theorem 4.2. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. Then the space $(E, T_{F})$ is a TBL-algebra.

Proof. By Definition 4.1, it suffices to show that ∗ is continuous, where ∗ ∈ {∧ , ∨ , ⊗ , →}. Indeed, assume that x ∗ y ∈ G, where x, y ∈ E and G is an open subset of L. Then there exist $U \in K$ , U [x ∗ y] ⊆ G, and an extreme fuzzy filter f such that $U_{f} \in K^{*}$ and U_f ⊆ U. We claim that the following relation holds: $U_{f} [x] * U_{f} [y] \subseteq U_{f} [x * y] \subseteq U [x * y] .$ Let h ∗ k ∈ U_f [x] ∗ U_f [y]. Then h ∈ U_f [x] and k ∈ U_f [y] we get that x ≡ _fh and y ≡ _fk. Hence x ∗ y ≡ _fh ∗ k. It follows that (x ∗ y, h ∗ k) ∈ U_f ⊆ U. Hence h ∗ k ∈ U_f [x ∗ y] ⊆ U [x ∗ y]. Then h ∗ k ∈ G. Clearly, U_f [x] and U_f [y] are neighborhoods of x and y, respectively. Therefore, the operation ∗ is continuous. □

Theorem 4.3. Consider the space $(L, T_{F})$ in Theorem 4.2. If x ∈ L and $f \in F$ , then U_f [x] is a clopen subset of L for the topology $T_{F}$ .

Proof. Let x ∈ L and $f \in F$ . Now we show that (U_f [x]) ^c is open. Let y ∈ (U_f [x]) ^c. We claim that U_f [y] ⊆ (U_f [x]) ^c. Assume z ∈ U_f [y]. Then f (z → y) = f (y → z) =1. If z ∈ U_f [x], then we have f (z → x) = f (x → z) =1. It follows that 1 = f (y → z) ∧ f (z → x) ≤ f (y → x) and 1 = f (x → z) ∧ f (z → y) ≤ f (x → y). Hence f (x → y) = f (y → x) =1 and y ∈ U_f [x] which is a contradiction. Hence U_f [y] ⊆ (Uf [x]) ^c for all y ∈ (U_f [x]) ^c, and so U_f [x] is closed. It is clear that U_F [x] is open. So U_F [x] is a clopen subset of L. □

A topological space X is connected if and only if X has only X and ∅ as clopen subsets. Therefore, we have the following result.

Corollary 4.4. The space $(L, T_{F})$ is not, in general, a connected space.

Proof. It directly follows from Theorem 4.3. □

Theorem 4.5. Let F be a filter of L such that $χ_{F} \in F$ . Then F is a clopen subset of L.

Proof. Let F be a filter of L such that $χ_{F} \in F$ . It suffices to show that F^c = ⋃ {U_{χ
_F} [y] : y ∈ F^c} and F = ⋃ {U_{χ
_F} [y] : y ∈ F}. To do this, suppose y ∈ F^c. Since y ∈ U_{χ
_F} [y], then y ∈ ⋃ {U_{χ
_F} [y] : y ∈ F^c}. Conversely, assume y ∈ ⋃ {U_{χ
_F} [y] : y ∈ F^c}. Then there exists z ∈ F^c such that y ∈ U_{χ
_F} [z]. Hence χ_F (y → z) = χ_F (z → y) =1. It follows that y → z, z → y ∈ F. If y ∈ F, then z ∈ F which is a contradiction. Hence y ∉ F and thus F^c = ⋃ {U_{χ
_F} [y] : y ∈ F^c}. By similar way we can show that F = ⋃ {U_{χ
_F} [y] : y ∈ F}. □

Lemma 4.6. Let f and g be extreme fuzzy filters of a BL-algebra L. If g ≤ f, then U_g ⊆ U_f.

Proof. The proof is straightforward. □

Theorem 4.7. Let f and g be extreme fuzzy filters of a BL-algebra L. If g ≤ f, then $T_{f} \subseteq T_{g}$ .

Proof. Let $F = f$ , $K^{*} = U_{f}$ and $K = {U \subseteq L \times L : U_{f} \subseteq U}$ . Let $O \in T_{f}$ . Then for all x ∈ O, there exists $U \in K$ such that U [x] ⊆ O and so U_f [x] ⊆ U [x] ⊆ O. By Lemma 4.6, U_g ⊆ U_f. It follows that U_g [x] ⊆ U_f [x] ⊆ O. Therefore, $O \in T_{g}$ and so $T_{f} \subseteq T_{g}$ . □

Theorem 4.8. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. If $g = ⋂ {f : f \in F}$ , then $T_{F} = T_{g}$ .

Proof. Let $F_{1} = {g}$ , $K_{1}^{*} = U_{g}$ and $K_{1} = {U \subseteq L \times L : U_{g} \subseteq U}$ . Let $O \in T_{g}$ . Then for all x ∈ O, there exists $U \in K_{1}$ such that U [x] ⊆ O. So U_g [x] ⊆ U [x]. Since $F$ is closed under intersection, then $g \in F$ . Thus we have $U_{g} \in K$ and hence $O \in T_{F}$ . So $T_{g} \subseteq T_{F}$ . Conversely, it directly follows from 4.7. □

Remark 4.9. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. If $g = ⋂ {f : f \in F}$ , then we have the following statements:

By Theorem 4.8, we know that $T_{F} = T_{g}$ . For any $U \in K$ , x ∈ L, we can get that U_g [x] ⊆ U [x]. Hence $T_{F}$ is equivalent to {A ⊆ E : ∀ x ∈ A, U_g [x] ⊆ A}. So A ⊆ L is an open set if and only if for all x ∈ A, U_g [x] ⊆ A if and only if A = ⋃ _x∈AU_g [x];

For all x ∈ E, by (i), we know that U_g [x] is the smallest open neighborhood of x;

Let $B_{J} = {U_{g} [x] : x \in L}$ . By (i) and (ii), it is easy to check that $B_{g}$ is a base of $T_{J}$ ;

For all x ∈ L, {U_g [x]} is a denumerable fundamental system of neighborhoods of x.

Theorem 4.10. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. If $g = ⋂ {f : f \in F}$ , then for all x ∈ LU_g [x] is a clopen compact set in the topological space $(L, T_{F})$ .

Proof. By Theorem 4.3, it is enough to show that U_g [x] is a compact set. Let U_g [x] ⊆ ⋃ _α∈IO_α, where each O_α is an open set of L. Since x ∈ U_g [x], there exists α ∈ I such that x ∈ O_α. Then U_g [x] ⊆ O_α. Hence U_g [x] is compact. Therefore U_g [x] is a clopen compact set in the topological space $(L, T_{F})$ . □

Theorem 4.11. The topological space $(L, T_{F})$ is a first-countable, zero-dimensional, disconnected and completely regular space.

Proof. By Theorem 4.8, it suffices to show that $(L, T_{g})$ is a first-countable, zero-dimensional, disconnected and completely regular space. Let x ∈ L. By Remark 4.9 (iv), {U_g [x]} is a denumerable fundamental system of neighborhoods of x, so $(L, T_{g})$ is first-countable. Let $B_{g} = {U_{g} [x] : x \in L}$ . By Remark 4.9 (iii) and Theorem 4.3, we get that $B_{g}$ is a clopen basis of $(L, T_{g})$ , hence $(L, T_{g})$ is a zero-dimensional space. By Corollary 4.4, we get $(L, T_{g})$ is a disconnected space. By Theorem 4.10 and Remark 4.9 (ii), U_g [x] is a compact neighborhood of x, hence $(L, T_{g})$ is a locally compact space. Let x ∈ L and V be an open neighborhood of x. By Remark 4.9 (ii) and Theorem 4.10, there exists closed neighborhood U_g [x] of x such that U_g [x] ⊆ V. Therefore, $(L, T_{g})$ is a regular space. Since $(L, T_{g})$ is a locally compact space, it follows that it is completely regular. □

Theorem 4.12. The space $(L, T_{F})$ is a discrete space if and only if there exists $f \in F$ such that U_f [x] = {x} for all x ∈ L.

Proof. Let $T_{F}$ be a discrete topology on L. If for any $f \in F$ , there exists x ∈ E such that U_f [x] ≠ {x}. Let $g = \cap F$ . Then $g \in F$ , there exists x₀ ∈ L such that U_g [x₀] ≠ {x₀}. It follows that there exists y₀ ∈ U_g [x₀] and x₀ ≠ y₀. By Remark 4.9 (ii), U_g [x₀] is the smallest open neighborhood of x₀. Hence {x₀} is not an open subset of L, which is a contradiction. Conversely, for any x ∈ L, there exists $f \in F$ such that U_f [x] = {x}. Hence {x} is an open set of L. Therefore, $(L, T_{F})$ is a discrete space. □

Theorem 4.13. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. If $g = \cap F$ , then the following statements are equivalent:

$(L, T_{F})$ is a discrete space,

g₁ = {1}, where g₁ is a level subset.

Proof. (i) ⇒ (ii): By Theorem 4.12, there exist $f \in F$ such that U_f [1] = {1}. Since 1 ∈ U_g [1] ⊆ U_f, we have U_g [1] = {1}. It suffices to show g₁ ⊆ U_J [1]. Let x ∈ g₁. Then g (x) =1, this implies that g (1 → x) = g (x → 1) =1. Hence x ∈ U_g [1].

(ii) ⇒ (i): Let g₁ = {1}. By Theorem 4.12, it suffices to show that U_g [1] = {1}. Suppose x ∈ U_g [1], then g (x) =1. It follows that x = 1. Clearly, {1} ⊆ U_g [1]. Therefore, U_g [1] = {1}. □

Theorem 4.14. Let $F$ be a family of extreme fuzzy filters of a BL-algebra L which is closed under intersection. If $g = \cap F$ , then the space $(L, T_{F})$ is a Hausdorff space if and only if g₁ = {1}.

Proof. Let $(L, T_{F})$ be a Hausdorff space. First we show that for any x ∈ L, U_g [x] = {x}. If not, then there exist x ≠ y ∈ U_g [x] such that y ∈ U_J [x] ∩ U_J [y]. By Remark 4.9 (ii), U_g [x] and U_g [y] are the smallest open neighborhoods of x and y, respectively. Hence for any open neighborhood U of x and open neighborhood V of y, we have that y∈ U_g [x] ∩ U_g [y] ⊆ U ∩ V ≠ ∅, which is a contradiction. Hence, by Theorems 4.12 and 4.13, g₁ = {1}. The remainder of the proof directly follows from Theorem 4.13. □

Definition 4.15. Let L₁ and L₂ be BL-algebras. A mapping φ : L₁ → L₂ is called a BL-morphism if $φ (x * y) = φ (x) * φ (y)$ for any ∗ ∈ {∧ , ∨ , ⊗ , →}. If, in addition, the mapping φ is bijective, then we call φ is a BL-isomorphism. Note that φ (1) =1 when φ is a BL-morphism.

Proposition 4.16. Let L₁ and L₂ be BL-algebras and φ : L₁ → L₂ be a BL-morphism. Then the following statements hold:

if g is an extreme fuzzy filter of L₂, then g ∘ φ is an extreme fuzzy filter of L₁,

if φ is a BL-isomorphism and f is an extreme fuzzy filter of L₁, then f ∘ φ^-1 is an extreme fuzzy filter of L₂.

Proof. (i) Let φ : L₁ → L₂ be a BL-morphism and g be an extreme fuzzy filter of L₂. Clearly, (g ∘ φ) (1) =1. By Proposition 2.6 (v), it suffices to show that g ∘ φ is order-preserving and (g ∘ φ) (x) (x ⊙ y) ≥ (g ∘ φ) (x) ∧ (g ∘ φ) (y) for all x, y ∈ L₁. Since g and φ are order-preserving, then g ∘ φ is order-preserving. We easily have that (g ∘ φ) (x ∘ y) = g (φ (x ⊙ y)) = g (φ (x) ⊙ φ (y)) ≥ g (φ (x)) ∧ g (φ (y)) = (g ∘ φ) (x) ∧ (g ∘ φ) (y) There-fore, g ∘ φ is an extreme fuzzy filter of L₁.

(ii) By similar way, we prove (ii) holds. □

Lemma 4.17. Let L₁ and L₂ be BL-algebras and g be an extreme fuzzy filter of L₂. If φ : L₁ → L₂ is a BL-morphism, then we have (a, b) ∈ U_g∘φ if and only if (φ (a) , φ (b)) ∈ U_g for any a, b ∈ L₁.

Proof. The proof is straightforward. □

Lemma 4.18. Let L₁ and L₂ be BL-algebras and g be an extreme fuzzy filter of L₂. If φ : L₁ → L₂ is a BL-morphism, then we have the following statements:

if φ is surjective, then φ (U_g∘φ [a]) = U_g [φ (a)] for all a ∈ L₁,

if φ is a BL-isomorphism, then φ^-1 (U_g [b]) = U_g∘φ [φ^-1 (b)] for all b ∈ L₂.

Proof. (i) Let b ∈ φ (U_g∘φ [a]). Then there exists c ∈ U_g∘φ [a] such that φ (c) = b. It follows that (g ∘ φ) (c → a) = (g ∘ φ) (a → c) =1. Since φ is a BL-morphism, we have g (φ (c) → φ (a)) = g (φ (a) → φ (c)) =1. Since φ (c) = b, we get g (b → φ (a)) = g (φ (a) → b) =1. Hence b ∈ U_g [φ (a)]. Conversely, let b ∈ U_g [φ (a)]. Then we have g (b → φ (a)) = g (φ (a) → b) =1. Since φ is surjective, there exists c ∈ L₁ such that φ (c) = b. It follows that g (φ (c) → φ (a)) = g (φ (a) → φ (c)) =1. Hence (g ∘ φ) (c → a) = (g ∘ φ) (a → c) =1, namely, c ∈ U_g∘φ [a] and so b = φ (c) ∈ φ (U_g∘φ [a]).

(ii)a ∈ φ^-1 (U_g [b]) ⇔ φ (a) ∈ U_g [b] ⇔ g (φ (a) → b) = g (b → φ (a)) =1 ⇔ g (φ (a) → φ (φ^-1 (b))) = g (φ (φ^-1 (b)) → φ (a)) =1 ⇔ (g ∘ φ) (a → φ^-1 (b)) = (g ∘ φ) (φ^-1 (b) → a) =1 ⇔ a ∈ U_g∘φ [φ^-1 (b)] . □

Theorem 4.19. Let L₁ and L₂ be BL-algebras and g be an extreme fuzzy filter of L₂. If φ : E₁ → E₂ is a BL-isomorphism, then φ is a continuous map from $(L_{1}, T_{g \circ φ})$ to $(L_{2}, T_{g})$ .

Proof. Let $A \in T_{g}$ . By Remark 4.9 (i), we can get that A = ⋃ _a∈AU_g [a]. It follows that φ^-1 (A) = φ^-1 (⋃ _a∈AU_g [a]) = ⋃ _a∈Aφ^-1 (U_g [a]). By Lemma 4.18, we have that φ^-1 (U_g [a]) = U_g∘φ [φ^-1 (a)]. Hence $φ^{- 1} (A) = ⋃_{a \in A} U_{g \circ φ} [φ^{- 1} (a)] \in T_{g \circ φ}$ . There-fore, φ is continuous. □

Theorem 4.20. Let L₁ and L₂ be BL-algebras and g be an extreme fuzzy filter of L₂. If φ : E₁ → E₂ is a BL-isomorphism, then φ is a quotient map from $(L_{1}, T_{g \circ φ})$ to $(L_{2}, T_{g})$ .

Proof. By Theorem 4.19, we get φ is a continuous surjective map. It is enough to show that φ is an open map. Let A be an open set of $(L_{1}, T_{g \circ φ})$ . We claim that φ (A) is an open set in $(L_{2}, T_{g})$ . Let b ∈ φ (A). We shall show that U_g [b] ⊆ φ (A). Since φ is surjective, there exists a ∈ A such that φ (a) = b. Hence U_g [b] = U_g [φ (a)] and so U_g [φ (a)] = φ (U_g∘φ [a]) by Lemma 4.18 (ii). Since A is open and a ∈ A, we get U_g∘φ [a] ⊆ A. It follows that U_g [b] = U_g [φ (a)] = φ (U_g∘φ [a]) ⊆ φ (A). Therefore, U_F [a] ⊆ φ (A). So φ is a quotient map. □

Theorem 4.21. Let L₁ and L₂ be BL-algebras and g be an extreme fuzzy filter of L₂. If φ : E₁ → E₂ is a BL-isomorphism, then φ is a homeomorphic map from $(L_{1}, T_{g \circ φ})$ to $(L_{2}, T_{g})$ .

Proof. It clearly follows from Theorem 4.20. □

5 Conclusion

It is well known that certain information processing, especially inferences based on certain information, is based on the classical logic (classical two-valued logic). Naturally, it is necessary to establish some rational logic systems as the logical foundation for uncertain information processing. For this reason, various kinds of non-classical logic systems have been extensively proposed and researched. Indeed, non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information. Recently, fuzzification of special types of filters on several different algebras of many-valued logics has been very popular. In this study, by using extreme fuzzy filters, we endowed a BL-algebra L with uniform topology $T_{F}$ . We stated and proved special properties of $(L, T_{F})$ . Especially, we proved that $(L, T_{F})$ is a first-countable, zero-dimensional, disconnected and completely regular space. From the category point of view, the role of isomorphism in algebras is the same as role of homeomorphism in topologies, and all of them are invariants in their fields, respectively. Interestingly, in this paper, we naturally contact with these invariants in topological BL-algebras.

Footnotes

Acknowledgments

The authors thank the editors and the anonymous reviewers for their valuable suggestions in improving this paper. This research is supported by Xi’an Aeronautical University research fund (2017KY1226).

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