The L -lim-inf-convergence in fuzzy posets

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].

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);

It is clear that for any constant net (x _i ) _i∈I in a fuzzy poset (X, e) with value x, we have x ≡  _L lim-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 ≡  _L lim-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 ≡  _L lim-inf x _i .

Sufficiency. Suppose x ≡  _L lim-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 ⪡  _a y 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 ⪡  _a y and y ≡  _a lim-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 ⪡  _a y, 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 ≤  _I z₂ 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 ⪡  _a y.

By the above lemma, we obtain the followingcorollaries.

Corollary 3.5. Let (X, e) be a fuzzy poset and x, y ⊆ X. Then x ⪡  _L y 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 ⪡  _a y, 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 ⪡  _L y, 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 ≡  _L lim-inf x _i , the class ℒℐ is called topological if there is a topology T on X such that ( ( x i ) i ∈ I , x ) ∈ 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 ≡  _L lim-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 ∈ I , x ) ∈ LI.

(SUBNETS) If ( ( x i ) i ∈ I , x ) ∈ LI and (y _j ) _j∈J is a subnet of (x _i ) _i∈I then ( ( y j ) j ∈ J , x ) ∈ LI.

(DIVERGENCE) If ( ( x i ) i ∈ I , x ) ∉ 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 ∈ K , x ) ∈ LI.

(ITERATED LIMITS) If ( ( x i ) i ∈ I , x ) ∈ LI and ( ( x i , j ) j ∈ J ( i ) , x i ) ∈ LI for any i ∈ I, then ( ( x i , f ( i ) ) ( i , f ) ∈ I × M , x ) ∈ 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 ∈ I , x ) ∈ LI. Then for any a ⋘ 1, x ≡  _a lim-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 ≡  _a lim-inf y _j and so ( ( y j ) j ∈ J , x ) ∈ LI by the arbitrariness of a.

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

(DIVERGENCE) Assume that ( ( x i ) i ∈ I , x ) ∉ 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) = ⇓  _b x 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 ∈ k , x ) ∈ LI. Hence axiom (DIVERGENCE) holds.

(ITERATED LIMITS) Suppose that ( ( x i ) i ∈ I , x ) ∈ LI and ( ( x i , j ) j ∈ J ( i ) , x i ) ∈ LI for any i ∈ I. By corollary 3.5, it is enough to show that for any a ⋘ 1 and any z ⪡  _a x, a ⋘ e (z, x_i,f(i)) holds eventually. In fact, for any a ⋘ 1 and any z ⪡  _a x, choose w ∈ X such that z ⪡  _a w ⪡  _a x. There exists i₀ ∈ I such that a ⋘ e (w, x _i ) for any i ≥ i₀. Thus z ⪡  _a x _i for any i ≥ i₀. Again as ( ( x i , j ) j ∈ J ( i ) , x i ) ∈ 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) ⊆ ⇓  _a x for any a ⋘ 1, then σ _a  (φ) = ⇓  _a x and x = ⊔ φ.

Proof. Firstly, we only need to show ⇓ _a x ⊆ σ _a  (φ). Let y ∈ ⇓  _a x. 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 ⇓ _a x ⊆ σ _a  (φ).

We further show x = ⊔ φ. Let y ∈ X and a ⋘ φ (y). Since σ _a  (φ) ⊆ ⇓  _a x, we have y ∈ ⇓  _a x, 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 ∈ 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 ′ ∧ b, a k ′ ⋘ e ( x , ⊔ φ k ) and a k ⋘ a k ′ ∧ b. Choose φ ∈ 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 } ∈ D x, it is easy to check that x ≡  _L lim-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) ⊆ ⇓  _a x. 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 ⪡  _a x. Thus Φ ∈ I _L  (X) such that 1 = e (x, ⊔ Φ) and for any a ⋘ 1, σ _a  (φ _m ) ⊆ ⇓  _a x and so x = ⊔ Φ for any a ⋘ 1, σ _a  (φ _m ) = ⇓  _a x 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).