Pointwise ( L,M )-fuzzy uniformities induced by ( L,M )-fuzzy pseudo-metric and pointwise pseudo-metric chains

Abstract

In this paper, the relations among (L, M)-fuzzy pseudo-metrics, pointwise pseudo-metric chains and pointwise (L, M)-fuzzy uniformities are investigated. It is shown that an (L, M)-fuzzy pseudo-metric can induce a pointwise (L, M)-fuzzy uniformity, and an (L, M)-fuzzy pseudo-metric and its induced pointwise (L, M)-fuzzy uniformity generate the same (L, M)-fuzzy topology. Also, it is proved that a pointwise pseudo-metric chain can induce a pointwise (L, M)-fuzzy uniformity, and a pointwise pseudo-metric chain and its induced (L, M)-fuzzy pseudo-metric generate the same pointwise (L, M)-fuzzy uniformity. Moreover, a pointwise pseudo-metric chain and its induced pointwise (L, M)-fuzzy uniformity generate the same (L, M)-fuzzy topology.

Keywords

Pointwise pseudo-metric chain

1 Introduction

With the development of fuzzy set theory, many mathematical structures have been generalized to the fuzzy case, such as fuzzy topology [1 , 35], fuzzy convergence [9 , 34] and fuzzy convexity [19 , 32]. In the framework of fuzzy topology, many researchers have defined several kinds of fuzzy metrics [3 , 14] and fuzzy uniformities [6 , 13]. In [29], Shi presented the concept of pointwise pseudo-metrics and studied its relations with pointwise quasi-uniformities [28]. Motivated by the notion of fuzzy real lines [7], Shi introduced M-fuzzy non-negative real number and used it to define (L, M)-fuzzy metrics [30]. This kind of fuzzy metric can be considered as a generalization of KM metrics [11], Morsi’s fuzzy metrics [14] and Shi’s pointwise metrics. Later, Yue and his coauthors introduced the concepts of pointwise (L, M)-fuzzy quasi-uniformities [36] and pointwise (L, M)-fuzzy uniformities [37]. Moreover, it has been shown that (L, M)-fuzzy pseudo-metric spaces and pointwise (L, M)-fuzzy uniform spaces both can be endowed with (L, M)-fuzzy topologies in the sense of [12, 27]. Recently, Pang and Shi [17] proposed the concept of pointwise pseudo-metric chains, which provided an approach to fuzzifications of pointwise pseudo-metrics. And there are close relations between (L, M)-fuzzy pseudo-metrics and pointwise pseudo-metric chains.

(L, M)-fuzzy pseudo-metrics and pointwise pseudo-metric chains are both fuzzy counterparts of pseudo-metrics, while pointwise (L, M)-fuzzy uniformities are fuzzy counterparts of uniformities. As we all know, there are close relations between pseudo-metrics and uniformities in general topology. This motivates us to consider the relations among (L, M)-fuzzy pseudo-metrics, pointwise pseudo-metric chains and pointwise (L, M)-fuzzy uniformities. Concretely, we will consider how to equip an (L, M)-fuzzy pseudo-metric and a pointwise pseudo-metric chain with pointwise (L, M)-fuzzy uniformities, respectively. Then we will show that they all have compatible relations with (L, M)-fuzzy topologies.

2 Preliminaries

Throughout this paper, both L and M denote completely distributive lattices and ^′ is an order-reversing involution on L. The smallest element and the largest element in L (M) are denoted by ⊥_L (⊥_M) and ⊤_L (⊤_M), respectively. For a, b ∈ L, we say that a is wedge below b in L, in symbols a ≺ b, if for every subset D ⊆ L, ⋁D ⩾ b implies d ⩾ a for some d ∈ D [2, 24]. A complete lattice L is completely distributive if and only if b = ⋁ {a ∈ L ∣ a ≺ b} for each b ∈ L. It is easy to see that a ≺ ⋀ _j∈Jb_j implies a ≺ b_j for every j ∈ J, whereas a ≺ ⋁ _j∈Jb_j is equivalent to a ≺ b_j for some j ∈ J. An element a in L is called co-prime if a ⩽ b ∨ c implies a ⩽ b or a ⩽ c [5]. The set of non-zero co-prime elements in L and M is denoted by J (L) and J (M), respectively. The set of all real numbers and the unit interval [0, 1] are denoted by $ℝ$ and I, respectively.

For a completely distributive lattice L and a nonempty set X, L^X denotes the set of all L-subsets on X. L^X is also a complete lattice when it inherits the structure of lattice L in a natural way, by defining ∨, ∧, ⩽ pointwise. The smallest element and the largest element in L^X are denoted by $⊥_{L}^{X}$ and $⊤_{L}^{X}$ , respectively. The set of non-zero co-prime elements in L^X is denoted by J(L^X). Each member in J (L^X) is also called a point. It is easy to see that J (L^X) is exactly the set of all fuzzy points x_λ (λ ∈ J (L)). Let φ : X ⟶ Y be a map. Define φ^→ : L^X ⟶ L^Y and φ^← : L^Y ⟶ L^X (see [25, 26]) by φ^→ (A) (y) = ⋁ _φ(x)=yA (x) for A ∈ L^X and y ∈ Y, and φ^← (B) = B ∘ φ for B ∈ L^Y.

Based on the concept of fuzzy real lines in the sense of Hutton [7], Shi introduced the following definition.

Definition 2.1. ([30]) An M-fuzzy non-negative real number is an equivalence class [λ] of antitone maps $λ : ℝ ⟶ M$ satisfying $λ (0 -) ≜ ⋀_{t < 0} λ (t) = ⊤_{M}, λ (+ \infty) ≜ ⋀_{t \in ℝ} λ (t) = ⊥_{M},$ where the equivalence identifies two such maps λ, μ if and only if ∀t ∈ I, λ (t -) = μ (t -).

We shall not distinguish an M-fuzzy real number [λ] and its representative function λ being left continuous. The set of all M-fuzzy non-negative real numbers is denoted by [0, + ∞) (M).

Definition 2.2 ([30]). An (L, M)-fuzzy pseudo-metric on X is a map d : J (L^X) × J (L^X) ⟶ [0, + ∞) (M) satisfying: ∀a, b, c, u, v ∈ J (L^X), ∀r, s > 0,

(LMD1) a⩽ b ⇒ d (a, b) (0 +) = ⊥;

(LMD2) d (a, c) (r + s) ⩽ d (a, b) (r) ∨ d (b, c) (s);

(LMD3) d (a, b) = ⋀ _c≺bd (a, c);

(LMD4) ⋀_{anotleqslantu′}d (a, v) = ⋀ _{bnotleqslantv′}d (b, u). For an (L, M)-fuzzy pseudo-metric d on X, the pair (X, d) is called an (L, M)-fuzzy pseudo-metric space.

Definition 2.3 ([29]). A pointwise pseudo-metric on X is a map $ℳ : J (ℒ) \times J (ℒ) [0, + \infty)$ satisfying: ∀a, b, c ∈ J (L^X),

(PM1) $ℳ (a, a) = 0$ ;

(PM2) $M (a, c) ⩽ M (a, b) + M (b, c)$ ;

(PM3) $M (a, b) = ⋀_{c ≺ b} M (a, c);$

(PM4) $a ⩽ b \Rightarrow M (a, c) ⩽ M (b, c);$

(PM5) Given u, v ∈ J (L^X), there exists a point anotleqslantu′ such that $ℳ (a, v) < r$ if and only if there exists a point bnotleqslantv′ such that $M (b, u) < r$ . For a pointwise pseudo-metric $M$ on X, the pair $(X, M)$ is called a pointwise pseudo-metric space.

The set of all pointwise pseudo-metrics on X is denoted by $ℳ (X) . Φ : J (M) ℳ_{L} (X)$ is called a pointwise pseudo-metric chain if Φ (α) ⩽ Φ (β) whenever α ⩾ β.

Definition 2.5 (12, 27]). An (L, M)-fuzzy topology on X is a map τ : L^X ⟶ M which satisfies:

(LFT1) $τ (⊥_{L}^{X}) = τ (⊤_{L}^{X}) = ⊤_{M}$ ;

(LFT2) τ (A ∧ B) ⩾ τ (A) ∧ τ (B);

(LFT3) τ (⋁ _j∈JA_j) ⩾ ⋀ _j∈Jτ (A_j). For an (L, M)-fuzzy topology τ on X, the pair (X, τ) is called an (L, M)-fuzzy topological space.

Let $D (ℒ)$ be the set of all maps from J (L^X) to L^X such that anotleqslantd (a) for all a ∈ mhdj. For any $f, g \in D (L^{X})$ , we define

(1) f ⩽ g if and only if ∀a ∈ mhdj, f (a) ⩽ g (a).

(2) (f ∨ g) (a) = f (a) ∨ g (a).

(3) (f ◊ g) (a) = ⋀ {f (b) ∣ bnotleqslantg (a)}. Then d₀ defined by $d_{0} (e) = ⊥_{L}^{X}$ for all e ∈ mhdj is the smallest element of $D (ℒ) f \lor g, f g \in D (L^{X}), f g ⩽ f, f g ⩽ g$ and the operations “∨” and “◊” satisfy the associate law. An element $f \in D (L^{X})$ is said to be symmetric if ∀λ, μ ∈ mhdj, there exists an anotleqslantλ′ such that μnotleqslantf (a) implies that there exists a bnotleqslantμ′ such that λnotleqslantf (b). If f, g are symmetric, then so are f ∨ g and f ◊ g. Let $D (ℒ)$ denote the set of all symmetric maps in $D (ℒ)$ .

Definition 2.6 ([28]). A pointwise quasi-uniformity on X is a nonempty subset scripfontU of $D (ℒ)$ satisfying:

(PU1) $f \in U, g \in D (L^{X}), g ⩽ f$ implies g ∈ scripfontU.

(PU2) f, g ∈ scripfontU implies f ∨ g ∈ scripfontU.

(PU3) f ∈ scripfontU implies ∃g ∈ scripfontU such that g ◊ g ⩾ f. For a pointwise quasi-uniformity scripfontU on X, the pair (X, scripfontU) is called a pointwise quasi-uniform space.

scripfontA ⊆ scripfontU is called a basis of scripfontU if for each f ∈ scripfontU, there exists a g ∈ scripfontA such that f ⩽ g. A pointwise quasi-uniformity is called a pointwise uniformity if it has a basis of symmetric elements.

Definition 2.7 ([36]) A pointwise (L, M)-fuzzy quasi-uniformity on X is a map $U : D (ℒ) ℳ$ such that

(LMU1) $U (_{0}) = ⊤$ ;

(LMU2) $U (f \lor g) = U (f) \land U (g)$ ;

(LMU3) $U (f) = ⋁_{g ◊ g ⩾ f} U (g)$ . For a pointwise (L, M)-fuzzy quasi-uniformity $U$ on X, the pair $(X, U)$ is called a pointwise (L, M)-fuzzy quasi-uniform space. A map $φ : (X, U_{X}) ⟶ (Y, U_{Y})$ is called uniformly continuous if $U (φ^{\leftarrow} ()) ⩾ U ()$ , where φ^← (f) = φ^← ∘ f ∘ φ^→.

3 Pointwise (L, M)-fuzzy uniformity induced by (L, M)-fuzzy pseudo-metric

In general topology, a pseudo-metric can induce a uniformity. In this section, we will show a pointwise (L, M)-fuzzy pseudo-metric can induce a pointwise (L, M)-fuzzy uniformity. For this, we first recall the following definitions. Definition 3.1 ([36]). A map $ℬ : D (ℒ) ℳ$ is called a base of one pointwise (L, M)-fuzzy quasi-uniformity if it satisfies: (LMB1) $ℬ (_{0}) = ⊤$ ; (LMB2) $B (f \lor g) ⩾ B (f) \land B (g)$ ; (LMB3) $B (f) ⩽ ⋁_{g ◊ g ⩾ f} B (g)$ . Definition 3.2 ([37]). Let $U$ be a pointwise (L, M)-fuzzy quasi-uniformity on X. $U$ is called a pointwise (L, M)-fuzzy uniformity if there exists a map $ℬ : D (ℒ) ℳ$ satisfying (LMB1)–(LMB3) and (LMB0) $ℬ () = ⊥$ , $\forall f \in D (L^{X}) - D_{s} (L^{X})$ ; (LMB4) $U (f) = ⋁_{g ⩾ f} B (g) .$ In this case, we call $B$ a symmetric base of $U$ . If $B$ satisfies (LMB1)–(LMB4), we call $B$ a base of $U$ .

Lemma 3.3 ([18]). Let (X, d) be an B (a, ∊, α) for each a ∈ mhdj, ∊ > 0 and α ∈ M as follows: $B (a,, α) = {b \in ∣ d (a, b) () ⩾ α} .$ Then $e ⩽ B (a, ∊, α) \Leftrightarrow d (a, e) (∊) ⩾ α .$

Theorem 3.4. Let (X, d) be an (L, M) space and define $U : D (ℒ) ℳ$ as follows: $U_{d} (f) = ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (r) .$ Then $U_{d}$ is a pointwise (L, M)-fuzzy uniformity on X.

Proof. The verifications of the fact that $U_{d}$ is a pointwise (L, M)-fuzzy quasi-uniformity on X can be referred to Theorem 6.13 in [18]. Next we show $U_{d}$ is a pointwise (L, M)-fuzzy uniformity on X. Define $ℬ : D (ℒ) ℳ$ as follows: $\begin{matrix} B_{d} (f) = {\begin{matrix} U_{d} (f), & f \in D_{s} (L^{X}), \\ ⊥_{M}, & f \notin D_{s} (L^{X}) . \end{matrix} \end{matrix}$ Then we claim that $ℬ U$ . (LMB0) and (LMB1) are obvious. (LMB2) By the definition of $ℬ$ , it suffices to prove that $ℬ (\lor) ⩾ ℬ () \land ℬ ()$ whenever $f, g \in D_{s} (L^{X})$ . Since $f, g \in D_{s} (L^{X})$ implies $f \lor g \in D_{s} (L^{X})$ , we have $\begin{matrix} B_{d} (f) \land B_{d} (g) = U_{d} (f) \land U_{d} (g) = U_{d} (f \lor g) \\ = B_{d} (f \lor g) . \end{matrix}$ (LMB3) For any $f \in D_{s} (L^{X}),$ let $α ≺ B_{d} (f) = U_{d} (f)$ . That is, $α ≺ ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (r) .$ Then there exists r > 0 such that ${(D 1) \forall a \in, \forall b ⩽ f (a), α ⩽ d (a, b) (r) .$ Define g_r : mhdj ⟶ L^X as follows: $g_{r} (a) = ⋁ {b \in ∣ d (a, b) (\frac{r}{2}) ⩾ α} .$ Then by Lemma 3.3, we have ${(D 2) \forall c \in, c ⩽ g_{r} (a) \Leftrightarrow d (a, c) (\frac{r}{2}) ⩾ α .$ Obviously, $g_{r} \in D (L^{X}) .$ Next we prove that g_r ◊ g_r ⩾ f, i.e., $g_{r} ◊ g_{r} (a) = ⋀ {g_{r} (b) ∣ b g_{r} (a)} ⩾ f (a) .$ Take any e ∈ mhdj with enotleqslantg_r ◊ g_r (a). Then there exists b ∈ mhdj such that bnotleqslantg_r (a) and enotleqslantg_r (b). By (D2), we have $d (a, b) (\frac{r}{2}) α$ and $d (b, e) (\frac{r}{2}) α$ . Further, $α d (a, b) (\frac{r}{2}) \lor d (b, e) (\frac{r}{2}) ⩾ d (a, e) (r)$ . By (D1), we obtain enotleqslantf (a). By the arbitrariness of e, g_r ◊ g_r ⩾ f is proved. Further, by (D2), we have $d (a, b) (\frac{r}{2}) ⩾ α$ for each a ∈ mhdj and b ⩽ g_r (a). This implies $\begin{matrix} α & ⩽ & ⋀_{a \in} ⋀_{b ⩽ g_{r} (a)} d (a, b) (\frac{r}{2}) \\ ⩽ & ⋁_{s > 0} ⋀_{a \in} ⋀_{b ⩽ g_{r} (a)} d (a, b) (s) \\ = & U_{d} (g_{r}) . \end{matrix}$ Moreover, by (D2) and (LMD4), it is easy to verify that $g_{r} \in D_{s} (L^{X})$ . Then it follows that $α ⩽ U_{d} (g_{r}) = B_{d} (g_{r}) ⩽ ⋁_{g ◊ g ⩾ f} B_{d} (g) .$ This proves $ℬ () ⩽_{⩾} ℬ () .$ (LMB4) By the definition of $ℬ$ , we know that $U () ⩾_{⩾} ℬ ()$ . In order to prove the inverse, take $α ≺ U_{d} (f)$ . Similar to (LMB3), there exists $g_{r} \in D_{s} (L^{X})$ such that g_r ⩾ f and $α ⩽ U_{d} (g_{r}) = B_{d} (g_{r}) .$ This shows $α ⩽ B_{d} (g_{r}) ⩽ ⋁_{g ⩾ f} B_{d} (g),$ as desired. Therefore, $U$ is a pointwise (L, M)-fuzzy uniformity on X. Example 3.5. Let X be any set and M ={⊥, ν₀, ⊤}. Then J (M) = {ν₀, ⊤}. Define d : J (L^X) × J (L^X) ⟶ [0, + ∞) (M) as follows: for each a, b ∈ J (L^X), (1) If a ⩽ b, then $\begin{matrix} d (a, b) (t) = {\begin{matrix} ⊤, & t = 0; \\ ⊥, & t > 0 . \end{matrix} \end{matrix}$ (2) If anotleqslantb, then $\begin{matrix} d (a, b) (t) & = & {\begin{matrix} ⊤, & t = 0; \\ ν_{0}, & 0 < t ⩽ 2; \\ ⊥, & t > 2 . \end{matrix} \end{matrix}$ Then it is trial to check that d is an (L, M)-fuzzy pseudo-metric. For each $f \in D (L^{X})$ , if f = d₀, then $U () =_{> 0}_{\in} \emptyset = ⊤ .$ If f ¬ = d₀, then for each a ∈ J (L^X), there exists b ∈ J (L^X) such that b ⩽ f (a). Further, if follows from anotleqslantf (a) that anotleqslantb. This implies that $\begin{matrix} U_{d} (f) & = & ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (r) \\ = & ⋁_{0 < r ⩽ 2} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (r) \\ \lor ⋁_{r > 2} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (r) \\ = & ν_{0} \land T_{M} \\ = & ν_{0} . \end{matrix}$ Thus, we obtain the concrete form of $U_{d}$ as follows: $\begin{matrix} U_{Φ} (f) = {\begin{matrix} ⊤_{M}, & f = d_{0}; \\ ν_{0}, & f \neg = d_{0} . \end{matrix} \end{matrix}$ Definition 3.6. A map φ : (X, d_X) ⟶ (Y, d_Y) is called continuous provided that for each ∊ > 0, there exists δ_∊ > 0 such that d_X (a, b) (δ_∊) ⩾ d_Y (φ^→ (a), φ^→ (b)) (∊). Theorem 3.7. Let (X, d_X) and (Y, d_Y) be two (L, M)-fuzzy pseudo-metric spaces. If φ : (X, d_X) ⟶ (Y, d_Y) is continuous, then $φ : (X, U_{d_{X}}) ⟶ (Y, U_{d_{Y}})$ is uniformly continuous. Proof. Since φ : (X, d_X) ⟶ (Y, d_Y) is continuous, it follows that for each ∊ > 0, there exists δ_∊ > 0 such that $d_{X} (a, b) (δ_{∊}) ⩾ d_{Y} (φ^{\to} (a), φ^{\to} (b)) (∊) .$ Then for each $f \in D (L^{Y}),$ we have $\begin{matrix} U_{d_{Y}} (f) & = & ⋁_{∊ > 0} ⋀_{c \in J (L^{Y})} ⋀_{e ⩽ f (c)} d_{Y} (c, e) (∊) \\ ⩽ & ⋁_{∊ > 0} ⋀_{φ^{\to} (a) \in J (L^{Y})} \\ ⋀_{φ^{\to} (b) ⩽ f (φ^{\to} (a))} d_{Y} (φ^{\to} (a), φ^{\to} (b)) (∊) \\ ⩽ & ⋁_{∊ > 0} ⋀_{a \in J (L^{X})} ⋀_{b ⩽ φ^{\leftarrow} \circ f \circ φ^{\to} (a)} d_{X} (a, b) (δ_{∊}) \\ ⩽ & ⋁_{r > 0} ⋀_{a \in J (L^{X})} ⋀_{b ⩽ φ^{\leftarrow} (f) (a)} d_{X} (a, b) (r) \\ = & U_{d_{X}} (φ^{\leftarrow} (f)) . \end{matrix}$ Therefore, $φ : (X, U_{d_{X}}) ⟶ (Y, U_{d_{Y}})$ is uniformly continuous. In [30, 36], the authors showed that an (L, M) and a pointwise (L, M)-fuzzy quasi-uniformity can induce an (L, M)-fuzzy topology, respectively.

Theorem 3.8 ([30]). Let (X, d) be (L, M) space. For any A ∈ L^X, define τ_d (A) = ⋀ _{anotleqslantA′} ⋁ _r>0 ⋀ _c⩽A′d (a, c) (r), $Q (A) =_{> 0}_{⩽ A^{'}} (,) () .$ Then τ_d is an (L, M)-fuzzy topology on X, and $Q = {Q ∣ \in}$ is the (L, M)-fuzzy quasi-coincident neighborhood system with respect to (X, τ_d) in the sense of Fang [4].

Theorem 3.9 ([36]). Let $(X, U)$ be a pointwise (L, M)-fuzzy quasi-uniform space and define $τ_{U} : L^{X} ⟶ M$ as follows: $\forall A \in L^{X}, τ_{U} (A) = ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} U (f) .$ Then $τ_{U}$ is an (L, M)-fuzzy topology on X.

Based on these results, we will show an (L, M)-fuzzy pseudo-metric and its induced pointwise (L, M)-fuzzy uniformity generate the same (L, M)-fuzzy topology. For this, the following lemma is necessary.

Lemma 3.10. Let (X, d) be an (L, M)-fuzzy pseudo-metric space. Then $Q (A) =_{A^{'} ⩽ ()} U ()$ for all a ∈ mhdj and A ∈ L^X. Proof. By Theorems 3.4, 3.8 and 3.9, we need only show that $⋁_{r > 0} ⋀_{c ⩽ A^{'}} d (a, c) (r) = ⋁_{A^{'} ⩽ f (a)} ⋁_{r > 0} ⋀_{b \in} ⋀_{c ⩽ f (b)} d (b, c) (r) .$ On one hand, $\begin{matrix} ⋁_{A^{'} ⩽ f (a)} ⋁_{r > 0} ⋀_{b \in} ⋀_{c ⩽ f (b)} d (b, c) (r) \\ ⩽ & ⋁_{A^{'} ⩽ f (a)} ⋁_{r > 0} ⋀_{c ⩽ f (a)} d (a, c) (r) \\ ⩽ & ⋁_{A^{'} ⩽ f (a)} ⋁_{r > 0} ⋀_{c ⩽ A^{'}} d (a, c) (r) \\ = & ⋁_{r > 0} ⋀_{c ⩽ A^{'}} d (a, c) (r), \end{matrix}$ i.e., $Q (A) ⩾_{A^{'} ⩽ ()} U ()$ . On the other hand, we divide into two cases: Case 1: a ⩽ A′. By (LMD1), $Q (A) = ⊥ ⩽_{A^{'} ⩽ ()} U ()$ . Case 2: anotleqslantA′. Let α ∈ J (M) such that $α ≺ ⋁_{r > 0} ⋀_{c ⩽ A^{'}} d (a, c) (r) .$ Then there exists r > 0 such that α ⩽ d (a, c) (r) for all c ⩽ A′. Define f_a : mhdj ⟶ L^X as follows: $\begin{matrix} f_{a} (b) = {\begin{matrix} A^{'}, & b ⩾ a, \\ ⊥_{L}^{X}, & otherwise . \end{matrix} \end{matrix}$ It is easy to check that $f_{a} \in D_{L} (X)$ and f_a (a) = A′. Thus, for all b ∈ mhdj and c ⩽ f_a (b), it follows that f_a (b) = A and b ⩾ a. By b ⩾ a, we have α ⩽ d (a, c) (r) ⩽ d (b, c) (r). This shows that there exist $f_{a} \in D_{L} (X)$ and r > 0 such that f_a (a) = A′ and α ⩽ d (b, c) (r) for all b ∈ mhdj and c ⩽ f_a (b). Therefore, $α ⩽ ⋁_{A^{'} ⩽ f (a)} ⋁_{r > 0} ⋀_{b \in} ⋀_{c ⩽ f (b)} d (b, c) (r) .$ By the arbitrariness of α, we obtain $Q (A) ⩽_{A^{'} ⩽ ()} U () .$ Theorem 3.11. If (X, d) is an (L, M)-fuzzy pseudo-metric space, then $τ_{d} = τ U$ . Proof. Take any A ∈ L^X. Then $\begin{array}{l} τ_{d} (A) & = \underset{a ≰ A^{'}}{\land} Q_{a}^{d} (A) & (by Theorem 3 .8) \\ = \underset{a ≰ A^{'}}{\land} \underset{A^{'} ⩽ (a)}{\lor} U_{d} (f) & (by Lemma 3 .10) \\ τ_{U_{d}} (A) . & (by Theorem 3 .9) \end{array}$ Definition 3.12. Let $(X, U)$ be a pointwise (L, M)-fuzzy uniform space. $U$ is called (L, M)-fuzzy pseudo-metrizable if there exists an (L, M)-fuzzy pseudo-metric d such that $U = U .$

Theorem 3.13. Let $(X, U)$ be a pointwise (L, I)-fuzzy uniform space. If $(X, U)$ is (L, I)-fuzzy pseudo-metrizable, then $U$ has a countable base. Proof. Since $(X, U)$ is (L, I)-fuzzy pseudo-metrizable, there exists an (L, I)-fuzzy pseudo-metric d on X such that $U = U .$ Let $A = {f_{(n, r)} ∣ n \in N, r \in Q},$ where $ℕ$ denotes the set of all positive integers, $ℚ$ denotes the set of all rational numbers and f_(n,r) : mhdj ⟶ L^X is defined by $f_{(n, r)} (a) = ⋁ {b \in ∣ d (a, b) (\frac{1}{n}) ⩾ r} .$ Then $A$ is a countable set. By Lemma 3.3 and (LMD4), we know $f_{(n, r)} \in D_{s} (L^{X})$ . Now define $V : D (ℒ) � ℐ$ as follows: $\begin{matrix} V (f) = {\begin{matrix} 1, & f = d_{0}, \\ U_{d} (f), & f \in A, \\ 0, & others . \end{matrix} \end{matrix}$ Then we assert that $V$ is a countable base of $U .$ We need only to show that $V$ satisfies (LMB1)–(LMB4). The verifications of (LMB1) and (LMB2) are trivial. (LMB3) Suppose $V (f) \neg = 0$ , let t be any rational number in I such that $t < V (f) = U_{d} (f) = ⋁_{n \in ℕ} ⋀_{a \in} ⋀_{b ⩽ f (a)} d (a, b) (\frac{1}{n}) .$ Then there exists $n_{0} \in ℕ$ such that $t < d (a, b) (\frac{1}{n_{0}})$ for all a ∈ mhdj and b ⩽ f (a). By the definition of f_(n₀,t), we have b ⩽ f_(n₀,t) (a). Further, by the arbitrariness of a and b, it follows that f ⩽ f_(n₀,t). By Lemma 3.3, we can check f_(2n₀,t) ◊ f_(2n₀,t) ⩾ f_(n₀,t). Thus, it follows that $\begin{matrix} V (f_{(2 n_{0}, t)}) = U_{d} (f_{(2 n_{0}, t)}) \\ = ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f_{(2 n_{0}, t)} (a)} d (a, b) (r) \\ ⩾ ⋀_{a \in} ⋀_{b ⩽ f_{(2 n_{0}, t)} (a)} d (a, b) (\frac{1}{2 n_{0}}) (by Lemma 3.3) \\ ⩾ t . \end{matrix}$ Hence, $t ⩽ V (f_{(2 n_{0}, t)}) ⩽ ⋁_{g ◊ g ⩾ f_{(n_{0}, t)}} V (g) ⩽ ⋁_{g ◊ g ⩾ f} V (g)$ . This proves $U_{d} (f) ⩽ \lor_{g g ⩾ f} V (g)$ (LMB4) Since $U_{d} (f) ⩾ \lor_{g ⩾ f} V (g)$ is obvious, it suffices to show that $U () ⩽_{⩾} V () .$ Let t be any rational number in I with $t < U_{d} (f)$ . Similar to (LMB3), there exists $f_{(n_{0}, t)} \in A$ such that f_(n₀,t) ⩾ f and $V (f_{(n_{0}, t)}) ⩾ t$ . By the arbitrariness of t, $U () ⩽_{⩾} V ()$ is proved. Therefore, $V$ is a countable base of $U$ , as desired. □

4 Pointwise (L, M)-fuzzy uniformity induced by pointwise pseudo-metric chain

In [29], Shi showed there are close relations between pointwise pseudo-metrics and pointwise uniformities. Pointwise pseudo-metric chains and pointwise (L, M)-fuzzy uniformities are generalizations of pointwise pseudo-metrics and pointwise uniformities, respectively. In this section, we will consider the relations between pointwise pseudo-metric chains and pointwise (L, M)-fuzzy uniformities.

Definition 4.1 ([29]). Let $M$ be a pointwise pseudo-metric on X. For each r ∈ (0, + ∞), define a map P_r : J (L^X) ⟶ L^X by $P_{r} (a) = ⋁ {b \in J (L^{X}) ∣ M (a, b) ⩾ r} .$ Then {P_r ∣ r ∈ (0, + ∞)} is called the family of remote-neighborhood maps (or R-nbd maps for short) of $M$ . The pointwise uniformity generated by the base {P_r ∣ r ∈ (0, + ∞)} is denoted by $U_{M}$ .

For a pointwise pseudo-metric chain Φ, let ${P_{r}^{α} ∣ r \in [0, + \infty)}$ denote the family of R-nbd maps of Φ (α), i.e., $P_{r}^{α} (a) = ⋁ {b \in ∣ Φ (α) (a, b) ⩾ r}$ . Let ξ_Φ(α) denote the L-topology induced by Φ (α) and ${\bar{A}}^{α}$ the closure of A in ξ_Φ(α). Then for all a ∈ J (L^X), $a ⩽ {\bar{A}}^{α}$ if and only if for all $r > 0, A P_{r}^{α} (a)$ .

Let scripfontU_Φ(α) denote the pointwise uniformity induced by the pointwise pseudo-metric Φ (α), i.e., $U_{Φ (α)} = {f \in D (L^{X}) ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}}$ . Then the following lemmas hold.

Lemma 4.2. If Φ is a pointwise pseudo-metric chain, then scripfontU_Φ(β) ⊆ scripfontU_Φ(α) for α ⩽ β.

Proof. For each α ⩽ β, it follows that $\begin{matrix} P_{r}^{α} (a) & = & ⋁ {b ∣ Φ (α) (a, b) ⩾ r} \\ ⩾ & ⋁ {b ∣ Φ (β) (a, b) ⩾ r} = P_{r}^{β} (a) . \end{matrix}$ □

Theorem 4.3. Let Φ be a pointwise pseudo-metric chain and define $U_{Φ} : D (ℒ) ℳ$ as follows: $U_{Φ} (f) = ⋁ {α \in J (M) ∣ f \in U_{Φ (α)}} .$ Then $U_{Φ}$ is a pointwise (L, M)-fuzzy uniformity on X.

Proof. (LMU1) Obviously.

(LMU2) By the definition of $U_{Φ}$ , we know that $U_{Φ} (f \lor g) ⩽ U_{Φ} (f) \land U_{Φ} (g) .$ Conversely, let $α ≺ U_{Φ} (f) \land U_{Φ} (g) .$ Then there exists β, γ ∈ J (M) such that α ⩽ β, α ⩽ γ, f ∈ scripfontU_Φ(β) and g ∈ scripfontU_Φ(γ). By Lemma 4, scripfontU_Φ(β) ⊆ scripfontU_Φ(α) and scripfontU_Φ(γ) ⊆ scripfontU_Φ(α). Then it follows that f ∨ g ∈ scripfontU_Φ(α). So $α ⩽ U_{Φ} (f \lor g),$ as desired.

(LMU3) By (LMU2), we know that $U_{Φ} (f) ⩾ ⋁_{g ◊ g ⩾ f} U_{Φ} (g)$ . It needs to prove $U_{Φ} (f) ⩽ ⋁_{g ◊ g ⩾ f} U_{Φ} (g)$ . Let α ∈ J (M) with f ∈ scripfontU_Φ(α). By (PU3), we know there exists g ∈ scripfontU_Φ(α) such that g ◊ g ⩾ f. Then $U_{Φ} (g) = ⋁ {β \in J (M) ∣ g \in U_{Φ (β)}} ⩾ α .$ Therefore $α ⩽ ⋁_{g ◊ g ⩾ f} U_{Φ} (g) .$ This proves $U_{Φ} (f) ⩽ ⋁_{g ◊ g ⩾ f} U_{Φ} (g)$ .

By (LMU1)–(LMU3), we know that $U_{Φ}$ is a pointwise (L, M)-fuzzy quasi-uniformity on X. Define $ℬ_{Φ} : D (ℒ) ℳ$ as follows: $\begin{matrix} B_{Φ} (f) = {\begin{matrix} U_{Φ} (f), & f \in D_{s} (L^{X}), \\ ⊥_{M}, & f \notin D_{s} (L^{X}) . \end{matrix} \end{matrix}$

Then we verify that $B_{Φ}$ is a symmetric base of $U_{Φ}$ .

(LMB0) and (LMB1) are straightforward.

(LMB2) By the definition of $B_{Φ}$ , it suffices to prove that $B_{Φ} (f \lor g) ⩾ B_{Φ} (f) \land B_{Φ} (g)$ whenever $f, g \in D_{s} (L^{X})$ . Since $f, g \in D_{s} (L^{X})$ implies $f \lor g \in D_{s} (L^{X})$ , so we have $\begin{matrix} B_{Φ} (f) \land B_{Φ} (g) & = & U_{Φ} (f) \land U_{Φ} (g) \\ = & U_{Φ} (f \lor g) = B_{Φ} (f \lor g) . \end{matrix}$ (LMB3) For any $f \in D_{s} (L^{X}),$ let $α ≺ B_{Φ} (f) = U_{Φ} (f)$ . Then there exists β ⩾ α and f ∈ scripfontU_Φ(β) ⊆ scripfontU_Φ(α). By (PU3), there exists g ∈ scripfontU_Φ(α) such that g ◊ g ⩾ f. Since scripfontU_Φ(α) is a pointwise uniformity, there exsits h ∈ scripfontU_Φ(α) such that $h \in D_{s} (L^{X})$ and h ⩾ g. So h ◊ h ⩾ g ◊ g ⩾ f and $U_{Φ} () = {β \in J (ℳ) ∣ \in U_{Φ (β)}} ⩾ α .$ Therefore, $α ⩽ U_{Φ} (h) = B_{Φ} (h) ⩽ ⋁_{g ◊ g ⩾ f} B_{Φ} (g) .$ This proves $B_{Φ} (f) ⩽ ⋁_{g ◊ g ⩾ f} B_{Φ} (g)$ .

(LMB4) By the definition of $B_{Φ}$ , we know that $U_{Φ} (f) ⩾ ⋁_{g ⩾ f} B_{Φ} (g)$ . In order to prove $U_{Φ} (f) ⩽ ⋁_{g ⩾ f} B_{Φ} (g)$ , let $β ≺ U_{Φ} (f) = ⋁ {α \in J (M) ∣ f \in U_{Φ (α)}} .$ Then there exists α ∈ J (M) such that α ⩾ β and f ∈ scripfontU_Φ(α) ⊆ scripfontU_Φ(β). Since scripfontU_Φ(β) is a pointwise uniformity, there exists h ∈ scripfontU_Φ(β) such that $h \in D_{s} (L^{X})$ and h ⩾ f. This implies $β ⩽ U_{Φ} (h) = B_{Φ} (h) ⩽ ⋁_{g ⩾ f} B_{Φ} (g) .$ By the arbitrariness of β, we have $U_{Φ} (f) ⩽ ⋁_{g ⩾ f} B_{Φ} (g) .$ Therefore, $U_{Φ}$ is a pointwise (L, M)-fuzzy uniformity on X. □

Example 4.4. Let X be any set and M = {⊥, ν₀, ⊤}. Then J (M) = {ν₀, ⊤}. Define $Φ : J (M) \to {PM}_{L} (X)$ as follows: for each a, b ∈ mhdj,

(1) If a ⩽ b, then Φ (ν₀) (a, b) = Φ (⊤) (a, b) =0;

(2) If anotleqslantb, then Φ (ν₀) (a, b) =2, Φ (⊤) (a, b) =0.

Then it is easy to check that Φ is a pointwise pseudo-metric chain. By Theorem 4, it follows that $\begin{matrix} U_{Φ} (f) & = & ⋁ {α \in J (M) ∣ f \in U_{Φ (α)}} \\ ⩽ & {α \in J (M) ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}} . \end{matrix}$

If f = d₀, then $f ⩽ P_{r}^{α}$ for all r > 0 and all α ∈ J (M). This means $U_{Φ} (_{0}) = ⊤$ .

If f ¬ = d₀, let $Λ_{f} = {α \in J (M) ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}} .$ Take any α ∈ Λ_f. Then there exists r > 0 such that for each a, b ∈ J (L^X) satisfying b ⩽ f (a), we have $b ⩽ P_{r}^{α} (a)$ . By the property of $P_{r}^{α}$ , we obtain Φ (α) (a, b) ⩾ r. In this case, if α = ⊤ _M, then Φ (α) (a, b) =0notgeqslantr for all r > 0. This means ⊤_M ∉ Λ_f. If α = ν₀, put r = 2. Then for each a, b ∈ J (L^X) such that b ⩽ f (a), it follows from anotleqslantf (a) that anotleqslantb. By the definition of Φ, we have Φ (ν₀) (a, b) =2 ⩾ r. This means $b ⩽ P_{r}^{α} (a)$ . By the arbitrariness of a and b, we obtain $f ⩽ P_{r}^{α}$ . This means ν₀ ∈ Λ_f. Then we obtain $U_{Φ} (f) = ν_{0}$ . Thus, we get the concrete form of $U_{Φ}$ , $\begin{matrix} U_{Φ} (f) = {\begin{matrix} ⊤_{M}, & f = d_{0}; \\ ν_{0}, & f \neg = d_{0} . \end{matrix} \end{matrix}$

In [17], Pang and Shi established the relations between (L, M)-fuzzy pseudo-metrics and pointwise pseudo-metric chains as follows:

Theorem 4.5 ([17]). (1) Let (X, d) be an (L, M) space and define $Φ_{d} : J (M) ⟶ {PM}_{L} (X)$ as follows: for each a, b ∈ mhdj and α ∈ J (M), $Φ_{d} (α) (a, b) = {s ∣ d (a, b) (s) α} .$ Then Φ_d is a pointwise pseudo-metric chain.

(2) Let Φ be a pointwise pseudo-metric chain and define d_Φ : mhdj × mhdj ⟶ [0, ∞) (M) as follows: for each a, b ∈ mhdj and t ∈ [0, ∞), $d_{Φ} (a, b) (t) = ⋁ {α ∣ Φ (α) (a, b) ⩾ t} .$ Then d_Φ is an (L, M) on L^X.

(3) If d is an (L, M) on X, then d_{Φ
_d} = d.

Next we show a pointwise pseudo-metric chain and its induced (L, M)-fuzzy pseudo-metric generate the same pointwise (L, M)-fuzzy uniformity.

Theorem 4.6. If Φ is a pointwise pseudo-metric chain, then $U_{Φ} = U_{d_{Φ}} .$

Proof. We need prove $U_{Φ} (f) = U_{d_{Φ}} (f),$ i.e., $\begin{matrix} ⋁ {α \in J (M) ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}} \\ = ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} ⋁ {β \in J (M) ∣ Φ (β) (a, b) ⩾ r} . \end{matrix}$

On one hand, take any α ∈ J (M) with $f ⩽ P_{r}^{α}$ for some r > 0. Then it follows that Φ (α) (a, b) ⩾ r for each a, b ∈ mhdj with $b ⩽ f (a) ⩽ P_{r}^{α} (a) .$ So α ⩽ ⋁ _r>0 ⋀ _a∈mhdj ⋀ _b⩽f(a) ⋁ {β ∣ Φ (β) (a, b) ⩾ r}. By the arbitrariness of α, we have $\begin{matrix} ⋁ {α ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}} \\ ⩽ & ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} ⋁ {β ∣ Φ (β) (a, b) ⩾ r} . \end{matrix}$

On the other hand, let $γ ≺ ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} ⋁ {β ∣ Φ (β) (a, b) ⩾ r} .$ Then there exist r > 0 and β ⩾ γ with Φ (β) (a, b) ⩾ r for all a ∈ mhdj and b ⩽ f (a). Further, Φ (γ) (a, b) ⩾ r. By the definition of $P_{r}^{γ}$ , we have $b ⩽ P_{r}^{γ} (a)$ . By the arbitrariness of b and a, we have $f ⩽ P_{r}^{γ}$ . This shows $γ ⩽ ⋁ {α ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}}$ . Therefore $\begin{matrix} ⋁_{r > 0} ⋀_{a \in} ⋀_{b ⩽ f (a)} ⋁ {β ∣ Φ (β) (a, b) ⩾ r} \\ ⩽ & ⋁ {α ∣ \exists r > 0, s . t . f ⩽ P_{r}^{α}} . \end{matrix}$

This completes the proof. □

By Theorems 4.5 and 4.6, we obtain

Theorem 4.7. If d is an (L, M)-fuzzy pseudo-metric, then $U = U Φ .$

Theorem 4.8 ([17]). Let Φ be a pointwise pseudo-metric chain and define τ_Φ : L^X ⟶ M as follows: $\forall A \in L^{X}, τ_{Φ} (A) = ⋁ {α \in J (M) ∣ A \in ξ_{Φ (α)}} .$ Then τ_Φ is an (L, M)-fuzzy topology on X.

Theorem 4.9. If Φ is a pointwise pseudo-metric chain, then $τ_{U_{Φ}} = τ_{Φ}$ .

Proof. By Theorems 3.9, 4.3, 4.8, and 4, we need only verify the following equality. $\begin{matrix} ⋁ {α \in J (M) ∣ A \in ξ_{Φ (α)}} \\ = & ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} ⋁ {β \in J (M) ∣ f \in U_{Φ (β)}} . \end{matrix}$

On one hand, take any α ∈ J (M) such that A ∈ ξ_Φ(α). Then $A^{'} = {\bar{A^{'}}}^{α}$ . This implies that $a {\bar{A^{'}}}^{α} \Leftrightarrow \exists r > 0, A^{'} ⩽ P_{r}^{α} (a) .$ Clearly, $P_{r}^{α} \in U_{Φ (α)}$ . This implies $α ⩽ ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} ⋁ {β \in J (M) ∣ f \in U_{Φ (β)}} .$ By the arbitrariness of α, we have $\begin{matrix} ⋁ {α \in J (M) ∣ A \in ξ_{Φ (α)}} \\ ⩽ & ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} ⋁ {β \in J (M) ∣ f \in U_{Φ (β)}} . \end{matrix}$

On the other hand, let α ∈ J (M) such that $α ≺ ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} ⋁ {β \in J (M) ∣ f \in U_{Φ (β)}} .$ Then for each anotleqslantA′, there exists $f \in D (L^{X})$ and β ∈ J (M) such that A′ ⩽ f (a), β ⩾ α and f ∈ scripfontU_Φ(β). By Lemma 4.2, it follows that f ∈ scripfontU_Φ(α).

In order to prove A ∈ ξ_Φ(α), i.e., ${\bar{A^{'}}}^{α} ⩽ A^{'}$ , let anotleqslantA′. Then there exists r > 0 such that $f ⩽ P_{r}^{α}$ . This implies that $A^{'} ⩽ f (a) ⩽ P_{r}^{α} (a)$ . Then we obtain $a {\bar{A^{'}}}^{α}$ . This proves ${\bar{A^{'}}}^{α} ⩽ A^{'}$ . Hence, A ∈ ξ_Φ(α). Therefore, α ⩽ ⋁ {β ∈ J (M) ∣ A ∈ ξ_Φ(β)}. By the arbitrariness of α, we obtain $\begin{matrix} ⋀_{a A^{'}} ⋁_{A^{'} ⩽ f (a)} ⋁ {β \in J (M) ∣ f \in U_{Φ (β)}} \\ ⩽ & ⋁ {α \in J (M) ∣ A \in ξ_{Φ (α)}}, \end{matrix}$ as desired.

5 Conclusion

In this paper, we mainly discussed the relations among (L, M)-fuzzy pseudo-metrics, pointwise pseudo-metric chains and pointwise (L, M)-fuzzy uniformities. Concretely, we showed that an (L, M)-fuzzy pseudo-metric and a pointwise pseudo-metric chain can induce a pointwise (L, M)-fuzzy uniformity, respectively. And they all had compatible relations with (L, M)-fuzzy topologies.

In the classical case, for a uniform space, it is pseudo-metrizable if and only if it has a countable base. In the framework of pointwise (L, M)-fuzzy uniform space, we only obtained the corresponding result in Theorem 2. On one hand, in this theorem, M is restricted to the unit interval I. On the other hand, the sufficiency of this conclusion is uncertain. Therefore, we will consider these two problems as our future work.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers for their constructive comments. This work is supported by the National Natural Science Foundation of China (No. 11701122), Beijing Institute of Technology Research Fund Program for Young Scholars, the Natural Science Foundation of Hunan Province (No. 2018JJ3192), the Educational Commission Foundation of Hunan Province (No. 15C0586) and the Scientific Research Foundation of CUIT (KYTZ201631, CRF201611, 2017Z056).

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