The L -lim-inf-convergence in fuzzy posets

Abstract

In domain theory, the lim-inf-convergence in posets was introduce to characterize continuous posets. In this paper, the concept of the L-lim-inf-convergence of nets (x_i) _i∈I in fuzzy posets is proposed and its relationship with continuous fuzzy posets is studied. It is shown that for an arbitrary fuzzy poset, the L-lim-inf-convergence is topological if and only if the fuzzy poset is continuous.

Keywords

Fuzzy poset continuity topology

1 Introduction

In classical domain theory, the network convergence has been a research focus, it is closely related with the convergence of topological spaces. In the literature [4], Scott introduced the S-convergence and lower limit of network convergence on directed complete posets, and proved a directed complete poset is continuous if and only if S-convergence of the network is topological. In the literature [10], Bin Zhao and Dongsheng Zhao extended these results to continuous posets.

As the partial order is an important mathematical structure and is useful in many areas, many people have endeavored to extend basic notions in domain theory to the many valued setting. Recently, Fan and Zhang [1 –3] introduced the fuzzy partial order and fuzzy poset. Then, they [7 –9] proposed the notions of fuzzy directed-complete poset, fuzzy domain, generalized Scott topology and fuzzy complete lattice and obtained some good results. Motivated by the above works, this paper is devoted to extending lim-inf-convergence in posets to many-valued setting.

The content of the paper is arranged as follows. In Section 2, we recall some notions and properties known. In Section 3, we define the L-lim-inf-convergence and exploit its relationship with continuous fuzzy posets. It is shown that for an arbitrary fuzzy poset, the L-lim-inf-convergence is topological if and only if the fuzzy poset is continuous. Finally, some conclusions are proposed in Section 4.

2 Preliminaries

Suppose that L is a complete lattice and p, q ∈ L. As defined in [4], p is said to well way below q, denoted by p ⋘ q, if for any subset A ⊆ L, q ≤ ∨ A implies p ≤ r for some r ∈ A. The relation ⋘ is called multiplicative if for any p, q, r ∈ L, p ⋘ q and p ⋘ r imply p ⋘ q ∧ r.

Suppose that L is a completely distributive lattice and p ∈ L. If p = a ∨ b implies p = a or p = b for any a, b ∈ L, then p is said to be ∨-irreducible.

Suppose that L is a frame (or complete Heyting algebra) and a, b ∈ L. We define a → b = ⋁ {c ∈ L|a ∧ c ≤ b}.

Throughout this paper, L denotes a frame. The following definitions and theorems can be found in [7 –9].

2.1 Fuzzy posets

Definition 2.1. A fuzzy poset is a pair (X, e) such that X is a set and e : X × X ⟶ L is a mapping, called a fuzzy order, that satisfies for every x, y, z ∈ X,

e (x, x) =1;

e (x, y) ∧ e (y, z) ≤ e (x, z);

e (x, y) = e (y, x) =1 implies x = y.

Dual to the logical correspondence in [5], let (X, e) be a fuzzy poset, x, y ∈ X, a, b ∈ L, {p_i|i ∈ I} a family of elements of L, and A ∈ L^X. Then: $e (x, y) = [x \leq y], A (x) = [x \in A], 1 = [True],$ $(a \land b) = [a & b], a \to b = [a \Rightarrow b],$ $⋀_{i \in I} p_{i} = [\forall i : I . p_{i}],$ $⋁_{i \in I} p_{i} = [\exists i : I . p_{i}], (a \leq b) = [a ⊢ b] .$

Definition 2.2. Let (X, e) be a fuzzy poset. φ ∈ L^X is called a fuzzy ideal on X if

for any x, y ∈ X, φ (x) ∧ e (y, x) ≤ φ (y);

there exists x ∈ X such that 0 ⋘ φ (x);

for any x₁, x₂ ∈ X, a₁, a₂, a ∈ L with a₁ ⋘ φ (x₁) , a₂ ⋘ φ (x₂) and a ⋘ 1, there is x ∈ X such that a ⋘ φ (x), a₁ ⋘ e (x₁, x), and a₂ ⋘ e (x₂, x).

The set of all fuzzy ideals on X is denoted by I_L (X).

Definition 2.3. [7] Let (X, e) be a fuzzy poset, x₀ ∈ X and φ ∈ L^X. Consider the following conditions:

for any x ∈ X, φ (x) ≤ e (x, x₀);

for any y ∈ X, ⋀ _x∈Xφ (x) → e (x, y) ≤ e (x₀, y).

x₀ is called a join of φ, denoted by ⊔φ, if it satisfies (1) and (2).

In a same way as for fuzzy directed-complete posets [7], we can define the ⪡_L and continuity on fuzzy posets.

Definition 2.4. Let (X, e) be a fuzzy poset, x, y ∈ X and a ∈ L. If for any φ ∈ I_L (X) with ⊔φ existing, a ⋘ e (y, ⊔ φ) implies that a ⋘ φ (z) and a ⋘ e (x, z) for some z ∈ X, then x is called L_a-way below y, denoted by x ⪡ _ay. x is said to be L-way below y, denoted by x ⪡ _Ly, if for any a ⋘ 1, x ⪡ _ay.

Definition 2.5. A fuzzy poset (X, e) is said to be continuous if for any x ∈ X, there exists φ ∈ I_L (X) with ⊔φ existing, such that x = ⊔ φ and for any a ⋘ 1, σ_a (φ) = ⇓ _ax, where ⇓_ax = {y ∈ X | y ⪡ _ax}, σ_a (φ) = {x ∈ X | a ⋘ φ (x)}.

Theorem 2.6. Let L be a completely distributive lattice in which 1 is ∨- irreducible, ⋘ is multiplicative, (X, e) a continuous fuzzy poset, 0 ≠ a ⋘ 1 and x, y ∈ X with x ⪡ _ay. Then there exists a z ∈ X such that x ⪡ _az ⪡ _ay.

Remark 2.7. Definitions 2.4, 2.5 and Theorem 2.6 construct the ⪡_L, continuity and interpolation property on fuzzy posets, which are the generalization on fuzzy directed-complete posets. For details, the readers can refer to the literature [7].

3 L-Lim-inf-convergences

In this section, we propose the L-lim-inf-convergence in a fuzzy poset and discuss its relationship with the continuous fuzzy poset.

Definition 3.1. Let (X, e) be a fuzzy poset, a net (x_i) _i∈I ⊆ X, x ∈ X and a ∈ L. If there is φ ∈ I_L (X) with ⊔φ existing such that

a ⋘ e (x, ⊔ φ);

for any y ∈ X, φ (y) ≤ e (y, x_i) holds eventually, that is, there exists k ∈ I such that φ (y) ≤ e (y, x_i) for all i ≥ k,

then (x_i) _i∈I is said to L_a-lim-inf-converge to x and write briefly x ≡ _alim-inf x_i. x is said to be L-lim-inf-converge to x, denoted by x ≡ _Llim-inf x_i, if for any a ⋘ 1, x ≡ _alim-inf x_i.

It is clear that for any constant net (x_i) _i∈I in a fuzzy poset (X, e) with value x, we have x ≡ _Llim-inf x_i. If (x_i) _i∈I L-lim-inf-converges to x, then it L-lim-inf-converges to any y ∈ X with e (y, x) =1. Thus the L-lim-inf-limits of a net are generally not unique.

Example 1. Let X = {T} ∪ {x₁, x₂, . . .} ∪ {y₁, y₂, . . .} ∪ {z} and L = {0, 1}. We define e : X × X ⟶ L as follows: e (z, T) = e (x_i, T) = e (y_i, z) = e (z, x_i) = e (y_i, x_j) =1 for any i, j = 1, 2, . . .; e (y_i, y_j) =1 whenever i ≤ j; otherwise e (x, y) =0. Then we can check that (X, e) is a fuzzy poset. Define φ ∈ L^X by φ (x) =1 if x = y_i, and otherwise, φ (x) =0. Then it is clear that ⊔φ = z and φ ∈ I_L (X). Since e (y_j, x_i) =1 and e (z, x_i) =1 for any i, j = 1, 2, 3, . . ., we have that the net (x_i) _i∈I L-lim-inf-converges to z.

Let (X, ≤) be a poset. We define e_≤ : X × X ⟶ L = {0, 1} as e_≤ (x, y) =1 if x ≤ y, and otherwise, e_≤ (x, y) =0 for any x, y ∈ X. Then (X, e_≤) is a fuzzy poset by Remark made just after Proposition 3.3 in [7].

Moreover, we have the following result.

Theorem 3.2. Let (X, ≤) be a poset. Then x≡lim-inf x_i (on (X, ≤)) iff x ≡ _Llim-inf x_i (on (X, e_≤)).

Proof. Necessity. Suppose x≡lim-inf x_i. Then there exists a directed set D of eventual lower bounds of net (x_i) _i∈I such that x ≤ supD. By Remark made just after Proposition 3.3 in [7], we have e_≤ (x, ⊔ χ_D) =1 and χ_D ∈ D_L (X). Next we show for any y ∈ X, there exists k ∈ I such that χ_D (y) ≤ e_≤ (y, x_i) for all i ≥ k. In fact, for any y ∈ X, it follows y ∈ D or y ∉ D. When y ∉ D, i.e., χ_D (y) =0, it obviously holds. If y ∈ D, i.e., χ_D (y) =1, then there exists k ∈ I such that y ≤ x_i for any i ≥ k, which implies e_≤ (y, x_j) =1 for any k ≥ i. Then we have that χ_D (y) ≤ e_≤ (y, x_i) for all i ≥ k. Therefore, we have x ≡ _Llim-inf x_i.

Sufficiency. Suppose x ≡ _Llim-inf x_i. Then there exists φ ∈ D_L (X) such that e (x, ⊔ φ) =1 and for any y ∈ X, there exists k ∈ I such that φ (y) ≤ e_≤ (y, x_i) for all i ≥ k. By Remark made just after Proposition 3.3 in [7], it follows that D = σ₁ (φ) = {y ∈ X|φ (y) =1} is a directed set and ⊔φ = ∨ D. Then x ≤ ∨ D. Next we show that D is a set of eventual lower bounds of the net (x_i) _i∈I. Let y ∈ D, i.e., φ (y) =1, which implies there exists k ∈ I such that e_≤ (y, x_i) =1 for any i ≥ k. Then it follows y ≤ x_i for any i ≥ k, which implies y is an eventual lower bound of net (x_i) _i∈i. By the arbitrariness of y, D is a set of eventual lower bounds of the net (x_i) _i∈i. Therefore, it follows x≡lim-inf x_i.

Remark 3.3. Theorem 3.2 means that the L-lim-inf-convergence in fuzzy posets generalizes the lim-inf-convergence in posets.

Lemma 3.4. Let (X, e) be a fuzzy poset, a ∈ L and x, y ⊆ X. Then x ⪡ _ay iff for any net (x_i) _i∈I ⊆ X which L_a-lim-inf-converges to y, a ⋘ e (x, x_i) holdseventually.

Proof. Necessity. Suppose x ⪡ _ay and y ≡ _alim-inf x_i. Then there exists φ ∈ I_L (X) such that a ⋘ e (y, ⊔ φ) and for any z ∈ X, φ (z) ≤ e (z, x_i) holds eventually. Since x ⪡ _ay, there is z ∈ X such that a ⋘ φ (z) and a ⋘ e (x, z). Hence a ⋘ e (z, x_i) ∧ e (x, z) ≤ e (x, x_i) holds eventually.

Sufficiency. Suppose the condition is satisfied. If φ ∈ I_L (X) with a ⋘ e (y, ⊔ φ), then let I = {z ∈ X | a ⋘ φ (z)} and define a binary relation ≤_I on I by z₁ ≤ _Iz₂ iff a ⋘ e (z₁, z₂). It is easy to check that (I, ≤ _I) is directed. For i = z_i ∈ I, let x_i = z, then (x_i) _i∈I is a net and it L_a-lim-inf-converges to y by the directionality of I. By the assumption, there exists z ∈ X such that a ⋘ φ (z) and a ⋘ e (x, z). Therefore, x ⪡ _ay.

By the above lemma, we obtain the followingcorollaries.

Corollary 3.5. Let (X, e) be a fuzzy poset and x, y ⊆ X. Then x ⪡ _Ly iff for any a ⋘ 1 and any net (x_i) _i∈I ⊆ X which L_a-lim-inf-converges to y, a ⋘ e (x, x_i) holds eventually.

Corollary 3.6. Let (X, e) be a continuous fuzzy poset and a ∈ L, then a net (x_i) _i∈I ⊆ X L_a-lim-inf-converges to y iff for any x ⪡ _ay, a ⋘ e (x, x_i) holdseventually.

Corollary 3.7. Let (X, e) be a continuous fuzzy poset, then a net (x_i) _i∈I ⊆ X L-lim-inf-converges to y iff for any x ⪡ _Ly, a ⋘ e (x, x_i) holds eventually for any a ⋘ 1.

For an arbitrary class ℒℐ of those pairs ((x_i) _i∈I, x) with x ≡ _Llim-inf x_i, the class ℒℐ is called topological if there is a topology T on X such that $((x_{i})_{i \in I}, x) \in LI$ iff the net (x_i) _i∈I converges to x with respect to the topology T. By [6], we have the following results:

Proposition 3.8. Let ℒℐ be a class of pairs ((x_i) _i∈I, x) with x ≡ _Llim-inf x_i. Then ℒℐ is topological iff it satisfies the following four conditions:

(CONSTANTS) If (x_i) _i∈I is a constant net with x_i = x, for any i ∈ I, then $((x_{i})_{i \in I}, x) \in LI$ .

(SUBNETS) If $((x_{i})_{i \in I}, x) \in LI$ and (y_j) _j∈J is a subnet of (x_i) _i∈I then $((y_{j})_{j \in J}, x) \in LI$ .

(DIVERGENCE) If $((x_{i})_{i \in I}, x) \notin LI$ , then there is a subnet (y_j) _j∈J of (x_i) _i∈I which has no subnet (z_k) _k∈K such that $((z_{k})_{k \in K}, x) \in LI$ .

(ITERATED LIMITS) If $((x_{i})_{i \in I}, x) \in LI$ and $((x_{i, j})_{j \in J (i)}, x_{i}) \in LI$ for any i ∈ I, then $((x_{i, f (i)})_{(i, f) \in I \times M}, x) \in LI,$ where M = Π_i∈IJ (i).

Lemma 3.9. (1) For every fuzzy poset the class LI satisfies the axioms (CONSTANTS) and (SUBNETS).

(2) Let L be a completely distributive lattice in which 1 is ∨- irreducible and ⋘ is multiplicative. If (X, e) is a continuous fuzzy poset then ℒℐ also satisfies the axioms (DIVERGENCE) and (ITERATEDLIMITS).

Proof. (1) The axiom (CONSTANTS) is clearly satisfied.

(SUBNETS) Suppose that $((x_{i})_{i \in I}, x) \in LI$ . Then for any a ⋘ 1, x ≡ _alim-inf x_i and so there is φ ∈ I_L (X) such that a ⋘ e (x, ⊔ φ) and for any y ∈ X, φ (y) ≤ e (y, x_i) holds eventually. Thus for any subnet (y_j) _j∈J of (x_i) _i∈I and for any y ∈ X, φ (y) ≤ e (y, x_j) holds eventually. Therefore, x ≡ _alim-inf y_j and so $((y_{j})_{j \in J}, x) \in LI$ by the arbitrariness of a.

(2) Suppose that (X, e) is continuous.

(DIVERGENCE) Assume that $((x_{i})_{i \in I}, x) \notin LI$ , then there is a ⋘ 1 such that x_notequivalim-inf x_i. By the continuity of (X, e), there is φ ∈ I_L (X) such that x = ⊔ φ and for any b ⋘ 1, σ_b (φ) = ⇓ _bx and thus there is z ∈ X such that for any i ∈ I there is j ∈ I with j ≥ i and φ (z) ≰ e (z, x_j). Let J be the subset of I consisting of all k ∈ I such that φ (z) ≰ e (z, x_k). Then J is co-final in I and (x_j) _j∈J is a subnet of (x_i) _i∈I. In addition, by Lemma 3.4 it follows that there is no subnet (z_k) _k∈k of (x_j) _j∈J such that $((z_{k})_{k \in k}, x) \in LI$ . Hence axiom (DIVERGENCE) holds.

(ITERATED LIMITS) Suppose that $((x_{i})_{i \in I}, x) \in LI$ and $((x_{i, j})_{j \in J (i)}, x_{i}) \in LI$ for any i ∈ I. By corollary 3.5, it is enough to show that for any a ⋘ 1 and any z ⪡ _ax, a ⋘ e (z, x_i,f(i)) holds eventually. In fact, for any a ⋘ 1 and any z ⪡ _ax, choose w ∈ X such that z ⪡ _aw ⪡ _ax. There exists i₀ ∈ I such that a ⋘ e (w, x_i) for any i ≥ i₀. Thus z ⪡ _ax_i for any i ≥ i₀. Again as $((x_{i, j})_{j \in J (i)}, x_{i}) \in LI$ , so for any i ≥ i₀ there exists g (i) ∈ J (i) such that if j ∈ J (i) and j ≥ g (i) then a ⋘ e (z, x_i,j). Define h ∈ Π_i∈IJ (i) such that h (i) = g (i) if i ≥ i₀ and h (i) is any element in J (i) otherwise. Now if (i, f) ∈ I × M and (i, f) ≥ (i₀, h), then a ⋘ e (z, x_i,f(i)).

Lemma 3.10. Let (X, e) be a fuzzy poset. If there exists φ ∈ I_L (X) such that e (x, ⊔ φ) =1 and σ_a (φ) ⊆ ⇓ _ax for any a ⋘ 1, then σ_a (φ) = ⇓ _ax and x = ⊔ φ.

Proof. Firstly, we only need to show ⇓_ax ⊆ σ_a (φ). Let y ∈ ⇓ _ax. Then there exists z ∈ X such that a ⋘ φ (z) and a ⋘ e (y, z). Then we have a ⋘ φ (z) ∧ e (y, z) ≤ φ (y), which implies y ∈ σ_a (φ). Hence, it follows ⇓_ax ⊆ σ_a (φ).

We further show x = ⊔ φ. Let y ∈ X and a ⋘ φ (y). Since σ_a (φ) ⊆ ⇓ _ax, we have y ∈ ⇓ _ax, which implies a ⋘ e (y, x). By the arbitrariness of a, it follows φ (y) ≤ e (y, x), i.e., φ (y) → e (y, x) =1. Hence, we have e (⊔ φ, x) = ⋀ _y∈Xφ (y) → e (y, x) =1 by the arbitrariness of y. Therefore, it follows that x = ⊔ φ.

Proposition 3.11. Let L be a completely distributive lattice in which 1 is ∨- irreducible and ⋘ is multiplicative. Then in a fuzzy poset (X, e), if the class ℒℐ satisfies the condition (ITERATED LIMITS) then (X, e) is continuous.

Proof. Let x ∈ X and D_x={ϕ_i∈I_L(X) | ∃_ai∈ L such that a_i⋘e(x,⊔ ϕ_i)}. Then let I be a index set corresponding to elements in D_x, that is, every i ∈ I corresponds to $φ_{i} \in D_{x}$ , and and define an order on I by i ≤ k holds iff a_i ⋘ a_k ∧ e (⊔ φ_i, ⊔ φ_k). Then ≤ is reflexive and transitive. Obviously, I is nonempty. Suppose i, k ∈ I. Choose $a_{1}^{'}, a_{2}^{'}, b ⋘ 1$ such that $a_{i}^{'} ⋘ e (x, ⊔ φ_{i})$ , $a_{i} ⋘ a_{i}^{'} \land b$ , $a_{k}^{'} ⋘ e (x, ⊔ φ_{k})$ and $a_{k} ⋘ a_{k}^{'} \land b$ . Choose $φ \in D_{x}$ such that b ⋘ e (x, ⊔ φ), $a_{i}^{'}$ ⋘e (⊔ φ_i, ⊔ φ) and $a_{k}^{'} ⋘ e (⊔ φ_{k}, ⊔ φ)$ . Then (⊔ φ, b) ∈ J (i) and (⊔ φ_i, a_i) , (⊔ φ_k, a_k) ≤ (⊔ φ, b). Hence, (I, ≤) is a directed set. For every i ∈ I, let x_i = ⊔ φ_i. Then (x_i) _i∈I is a net and for every i ∈ I, there is a_i ∈ L such that a_i ⋘ e (x, x_i). Moreover, since $χ_{{x}} \in D_{x}$ , it is easy to check that x ≡ _Llim-inf x_i. For each i ∈ I, let J (i) = {(z, a) | a ⋘ φ_i (z)} and define an order on J (i) by j_i = (z₁, a₁) ≤ j₂ = (z₂, a₂) if a₁ ⋘ a₂ ∧ e (z₁, z₂). Then J (i) is directed set. Let x_i,j = z for each j = (z, a) ∈ J (i), then for each i ∈ I, (x_i,j) _j∈J(i) is a net in X, which obviously L-lim-inf-converges to x_i. Since the condition (ITERATED LIMITS) is satisfied, the net (x_i,f(i)) _(i,f)∈I×M L-lim-inf-converges to x, where M = Π_i∈IJ (i). By the definition of L-lim-inf limit, there is Φ ∈ I_L (X) such that 1 = e (x, ⊔ Φ) and for any z ∈ X, Φ (z) ≤ e (z, x_i,f(i)) holds eventually. Next we show that for any a ⋘ 1, σ_a (Φ) ⊆ ⇓ _ax. Let a ⋘ Φ (z). For any φ_m ∈ I_L (X) with a ⋘ e (x, ⊔ φ_m), let h (m) = {(z, a) | z ∈ σ_a (φ_m)} = {(z, a) | a ⋘ φ_m (z)} ⊆ J (m). Then (x_m,j) _j∈h(m) is a subnet of (x_i,f(i)) _(i,f)∈I×M. Hence, a ⋘ e (z, x_m,j), which implies that there is (m, f_z (m)) such that a ⋘ e (z, x_{m,f_z(m)}). Note that x_{m,f_z(m)} ∈ σ_a (φ_m), thus z ⪡ _ax. Thus Φ ∈ I_L (X) such that 1 = e (x, ⊔ Φ) and for any a ⋘ 1, σ_a (φ_m) ⊆ ⇓ _ax and so x = ⊔ Φ for any a ⋘ 1, σ_a (φ_m) = ⇓ _ax by Lemma 3.10. Therefore, (X, e) is continuous by the arbitrariness of x.

By Proposition 3.8, 3.11 and Lemma 3.9, we can obtain the following result, which is the main conclusion of this paper.

Theorem 3.12. Let L be a completely distributive lattice in which 1 is ∨- irreducible and ⋘ is multiplicative. Then for any fuzzy poset (X, e), the L-lim-inf-convergence is topological iff (X, e) is continuous.

Proof. Necessity. Suppose for any fuzzy poset (X, e), the L-lim-inf-convergence is topological. Then the class ℒℐ satisfies the condition (ITERATED LIMITS) by Proposition 3.8 and thus (X, e) is continuous by Proposition 3.11.

Sufficiency. Suppose (X, e) is continuous. Then the class ℒℐ satisfies the conditions (CONSTANTS), (SUBNETS), (DIVERGENCE) and (ITERATEDLIMITS) by Lemma 3.9 and thus the L-lim-inf-convergence is topological by Proposition 3.8.

4 Conclusions

Taking the frame as the structure of truth value, we propose in fuzzy posets the L-lim-inf-convergence, which is a generalization of lim-inf-convergence in crisp posets. In particular, it is shown that for an arbitrary fuzzy poset, the L-lim-inf-convergence is topological if and only if the fuzzy poset is continuous.

Acknowledgements

This work was supported by the National Social Science Foundation of China (NO. 12BJY122) and the Science Foundation of Education Committee of Jiangxi (NO. GJJ14481).

References

10.