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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 15, Issue 1

Issues

Scott convergence and fuzzy Scott topology on L-posets

Hongping Liu
  • Corresponding author
  • School of Science, Shandong Jianzhu University, Jinan 250101, China
  • College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
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/ Ling Chen
Published Online: 2017-06-22 | DOI: https://doi.org/10.1515/math-2017-0067

Abstract

We firstly generalize the fuzzy way-below relation on an L-poset, and consider its continuity by means of this relation. After that, we introduce a kind of stratified L-generalized convergence structure on an L-poset. In terms of that, L-fuzzy Scott topology and fuzzy Scott topology are considered, and the properties of fuzzy Scott topology are discussed in detail. At last, we investigate the Scott convergence of stratified L-filters on an L-poset, and show that an L-poset is continuous if and only if the Scott convergence on it coincides with the convergence with respect to the corresponding topological space.

Keywords: L-poset; Convergence structure; Scott topology; Stratified L-filter

MSC 2010: 03E72; 06A06; 54A40

1 Introduction

Ordered structure and topological structure are two basic and crucial structures in Mathematics, closely related to each other. Many works have been done to compare and combine the two structures [16]. At the beginning, classical Scott convergence and Scott topology are, in view of the theory of continuous lattices, only defined for complete lattices. Not very soon, these definitions have been found to be very fruitful for dcpos [2]. Unfortunately, they are not fit for arbitrary partially ordered sets (posets), since the join of a directed subset is involved in the definition of Scott convergence, which may not exist in a poset. Regarding this, several alternative choices have been proposed to generalize the definition of Scott convergence in posets [1, 611], and the Scott topology related to Scott convergence has also been studied.

In recent years, quantitative domain theory has attracted many people because it provides a model of concurrent systems. Wagner’s Ω-categories [12], Rutten’s generalized metric spaces [13] and Flagg’s continuity spaces [14] are examples of quantitative domain theory. Fan and Zhang [15, 16] studied quantitative domain via fuzzy set theory, where fuzzy partial order was clearly proposed. After analysis, it is easily seen that Ω-categories could be regarded as a fuzzy preordered set in [15, 16], and a fuzzy partial ordered set (an L-poset) is equivalent to an L-ordered set introduced by Bělohlávek [17, 18]. Later on, Lai and Zhang [19, 20] studied complete and directed-complete Ω-categories, and their continuity was also discussed. Following [15, 16, 19], Yao [21, 22] studied the continuity of fuzzy dcpos, and further extended the Scott convergence and Scott topology on classical dcpos to fuzzy setting. But the results in [22] do not adapt to fuzzy partially ordered sets as well as in the classical case, even the continuity needs to be modified. This provides sufficient motivations for this paper. We firstly redefine the fuzzy way-below relation on L-posets and reconsider the continuity. Then we introduce a kind of stratified L-generalized convergence structure on L-posets, and restudy fuzzy Scott topology associating with it. Finally, we establish the Scott convergence theory on L-posets, and prove that it is an effective tool to characterize the continuity.

The paper is organized as follows. In Section 2, we recall some necessary definitions and results needed later on. In Section 3, we give a fuzzy way-below relation on L-posets and based on that the continuity for L-posets is considered. In Section 4, we introduce a new stratified L-generalized convergence structure on L-posets, then study and characterize fuzzy Scott topology. In Section 5, the properties of Scott convergence are given, the description for continuous L-posets via Scott convergence is constructed. In the final section, we summarize the results and draw a conclusion.

Preliminaries

A complete residuated lattice [23] L is a structure (L, *, →, ∨, ∧, 0, 1), such that (1) (L, ∨, ∧, 0, 1) is a complete lattice with the greatest element 1 and the least element 0;(2)(L, *, 1) is a commutative monoid with the identity 1 and * is isotone at both arguments; (3) (*, →) is an adjoint pair, i.e., x * yz iff xyz for all x, y, zL. Some basic properties of complete residuated lattices are collected here ([17, 23, 24]).

  1. 1 → a = a;

  2. ab iff ab = 1;

  3. (ab)*(bc) ≤ ac;

  4. a →(bc) = b →(ac) = (a * b) → c;

  5. ab →(a * b);

  6. a *(ab) ≤ b;

  7. a(iIbi)=iI(abi),a(iIbi)iI(abi);

  8. a(iIbi)=iI(abi),(iIai)b=iI(aib);

  9. abb;

  10. bc ≤ (ab) → (ac), bc ≤ (ca) → (ba);

  11. (ab) → ba;

  12. (ab)*(cd) ≤ (a * c) → (b * d).

Let X be a nonempty set, LX denote the set of all L-subsets of X. ∀ A, BLX, define: (AB)(x)=A(x)B(x),(AB)(x)=A(x)B(x),(AB)(x)=A(x)B(x),(AB)(x)=A(x)B(x). Then (LX, *, →, ∨, ∧, 0, 1) is also a complete residuated lattice, and we never discriminate the constant value function a with a, e.g., (a * A)(x) = a * A(x) and (aA)(x) = aA(x) for every xX.

A complete residuated lattice L with * = ∧ is just a complete Heyting algebra (or a frame). Throughout this paper, L always denotes a complete Heyting algebra.

Fuzzy order was first introduced by Zadeh [25], from then on, different kinds of fuzzy order have been introduced and studied by different authors (the reader is referred to [4, 12, 15, 16, 2628]). In this paper, we adopt the definition of fuzzy order in [15, 16].

Definition 2.1

A fuzzy (partial) order e (also called an L-order) on X is an L-relation satisfying:

  1. xX, e(x, x) = 1;

  2. x, y, zX, e(x, y)∧ e(y, z) ≤ e(x, z);

  3. x, yX, e(x, y) = e(y, x) = 1 ⇒ x = y.

Then (X, e) is called a fuzzy (partially) ordered set or an L-poset for simplicity.

Remark 2.2

In [17, 18], an L-preordered set is defined to be a triple (X, R, ≈), whereis an L-equality on X and R is an L-preorder on X which is compatible with ≈. It is verified in [18, 22] that if R is compatible with ≈, it must hold that ≈ = RRop. Thus, the L-equalityis completely determined by R, so it can be omitted in the definition.

Example 2.3

([15, 22, 26]).

  1. In a complete residuated lattice L, define eL : L × LL by eL(x, y) = xy for all x, yL. Then (L, eL) is an L-poset.

  2. For all A, BLX, define sub (A, B) = xX A(x) → B(x), then (LX, sub) is an L-poset where sub called the fuzzy inclusion order, and sub (A, B) is explained as the subsethood degree or fuzzy inclusion degree of A in B.

Definition 2.4

([16, 21, 22]). Let (X, e) be an L-poset and zX, ALX. Then

  1. AuLX is defined byxX, Au(x) = ⋀yX(A(y) → e(y, x)).

  2. AlLX is defined byxX, Al(x) = ⋀yX(A(y) → e(x, y)).

  3. ALX is defined byxX, ↓ A(x) = ⋁yX(A(y) ∧ e(x, y andALX is defined dually.

  4. zLX is defined byxX, ↓ z(x) = e(x, z), andyLX is defined byxX, ↑ z(x) = e(z, x).

Moreover, A is called a lower L-set or fuzzy lower set if A(x)∧ e(y, x) ≤ A(y) for all x, yX. A is called an upper L-set or fuzzy upper set if A(x) ∧ e(x, y) ≤ A(y) for all x, yX.

Lemma 2.5

([16, 22]). Let (X, e) be an L-poset. Then for all x, yX, e(x,y)=zX(e(z,x)e(z,y))=zX(e(y,z)e(x,z)).

Definition 2.6

([19, 21, 22]). Let (X, e) be an L-poset and DLX. Then D is called a fuzzy directed set or directed L-set if

  1. xX D(x) = 1;

  2. x, yX, D(x)∧ D(y) ≤ ⋁zX(D(z)∧ e(x, z)∧ e(y, z)).

A fuzzy directed set ILX is called a fuzzy ideal if it is also a fuzzy lower set. The set of all fuzzy ideals of (X, e) is denoted by 𝓘L(X).

Definition 2.7

([16, 19, 21, 22]). Let (X, e) be an L-poset, ALX, x0X. x0 is called a fuzzy join (resp., fuzzy meet) of A denoted by x0 = ⊔ A (resp., x0 = ⊓ A) if

  1. xX, A(x) ≤ e(x, x0) (resp., A(x) ≤ e(x0, x));

  2. yX, xXA(x)e(x,y)e(x0,y)(resp.,xXA(x)e(y,x)e(y,x0)).

It is easy to see that the fuzzy join or the fuzzy meet is unique if it exists.

Theorem 2.8

([19, 21, 26]). Let (X, e) be an L-poset, ALX, x0X. Then

  1. x0 = ⊔ A ⟺ ∀ yX, e(x0, y)= xX (A(x) → e (x, y));

  2. x0=⊓ A ⟺ ∀ yX, e(y, x0) = xX (A(x)→ e(y, x)).

Lemma 2.9

[20] Let (X, e) be a complete L-lattice, and xX, {yi}iIX. Then we have e(x,iIyi)=iIe(x,yi)ande(iIyi,x)=iIe(yi,x), whereandrespectively, denote join and meet in the underlying poset X~=(X,e).

Proposition 2.10

[21], 22, 29] Let (X, e) be an L-poset. Then

  1. Al(x) = sub(A, ↑ x), Au(x) = sub(A, ↓ x);

  2. e(⊔ A, x) = sub(A, ↓ x), e(x, ⊓ A) = sub(A, ↑ x) withA andA exist;

  3. e(x, y) ∧ sub(↑ x, A) ≤ sub (↑ y, A);

  4. sub (A, B) ≤ e(⊔ A, ⊔ B); sub (A, B) ≤ e(⊓ B, ⊓ A) withA, ⊓ A, ⊔ B andB exist;

  5. If A is a lower L-set, then sub (↓ x, A) = A(x);

  6. If A is an upper L-set, then sub (↑ x, A) = A(x).

Definition 2.11

([21, 22, 26, 27]). Let (X, eX), (Y, eY) be L-posets and f : XY be a map. Then f is said to be L-order-preserving or L-monotone (resp., L-antitone) if eX(x, y) ≤ eY(f(x), f(y)) (resp., eX(x, y) ≤ eY(f(y), f(x))) for all x, yX.

Definition 2.12

([21, 22]). Let (X, eX), (Y, eY) be L-posets, f : XY, g : YX be L-order-preserving maps. Then (f, g) is called a fuzzy Galois connection between X and Y if eY(f(x), y)=eX(x, g(y)) for all xX, yY, where f is called the left adjoint of g and dually g the right adjoint of f.

3 The continuity of L-posets via fuzzy way-below relation

Way-below relation was first imported for investigating the continuity of complete lattices. It was also an effective tool to describe the continuity of dcpos. This observation had inspired several authors to study continuous posets [3032]. Unfortunately, these works on posets were rather restrictive since the definition of the way-below relation only considers certain directed subsets, of which join exists. In view of this deficiency, Erné [33] introduced another way-below relation on posets, and studied the continuous posets via it. However, in the quantitative domain theory, the fuzzy way-below relation introduced on fuzzy dcpos [21, 22] has the same deficiency and is not fit for fuzzy posets either. Thus for a fuzzy poset it is necessary to define a reasonable fuzzy way-below relation and reconsider its continuity.

Definition 3.1

Let (X, e) be an L-poset, yX. DefineyLX by xX,y(x)=IIL(X)Iul(y)I(x). Then (X, e) is said to be continuous ify is directed and ⊔ ⇓ y = y.

Note that each of the sets ⇓ y is a lower L-set (but in general not directed), and ⇓ : X × XL can be regarded as the fuzzy way-below relation on (X, e).

When (X, e) is a fuzzy dcpo, then for every I ∈ 𝓘L(X), Iul(y)=zX(Iu(z)e(y,z))=zX(sub(I,z)e(y,z))=zX(e(I,z)e(y,z))=e(y,I). Thus, the fuzzy way-below relation as above is compatible with the usual one for fuzzy dcpos [21, 22].

Some fundamental properties of the fuzzy way-below relation are listed in the following proposition.

Proposition 3.2

Let (X, e) be an L-poset. Then

  1. xX, I ∈ 𝓘L(X), ⋀yX e(x, y) ≤ I(x).

  2. x, yX, ⋀zX e(x, z) ≤ ⇓ y(x)

  3. xX, ⇓ x ≤ ↓ x.

  4. x, u, v, yX, e(u, x)∧ ⇓ y(x) ∧ e(y, v) ≤ ⇓ v(u).

Proof

  1. Refer to Proposition 5.7 (1) in [21].

  2. y(x) = ⋀I∈𝓘L(X) Iul (y) → I(x) ≥ ⋀I∈ 𝓘L(X) I(x) = ⋀zX e(x, z).

  3. x = ⋀I∈ 𝓘L(X) Iul(x) → I ≤ (↓ x)ul(x)→ ↓ x = ↓ x.

  4. For any I ∈ 𝓘L(X), we have e(u,x)(Iul(y)I(x))e(y,v)Iul(v)e(u,x)(Iul(y)I(x))Iul(y)e(u,x)I(x)I(u). Then e(u, x)∧(Iul(y)→ I(x))∧ e(y, v) ≤ Iul(v)→ I(u).

Thus, e(u, x) ∧ ⇓ y(x) ∧ e(y, v) = e(u, x) ∧⋀I∈ 𝓘L(X) (Iul(y) → I(x))∧ e(y, v) ≤ ⋀I∈ 𝓘L(X) Iul(v)→ I(u) = ⇓ v(u). □

Proposition 3.3

If (X, e) is a continuous L-poset, then for all xX, (⇓ x)u = ↑ x, and so (⇓ x)ul(x) = 1.

Proof

Suppose xX, then for every yX, (x)u(y)=zXx(z)e(z,y)=sub(x,y)=e(x,y)=↑x(y). That is, (⇓ x)u = ↑ x. Thus (⇓ x)ul(x) = ↓ x(x) = 1. □

Theorem 3.4

If (X, e) is a continuous L-poset, theny(x) = ⋁zX (⇓ z(x) ∧ ⇓ y(z)) for all x, yX.

Proof

At first, by Proposition 3.2(3)(4), we have ⋁ zX(⇓ z(x)∧⇓ y(z)) ≤ ⋁zX(e(x, z)∧⇓ y(z)∧ e(y, y)) ≤ ⇓ y(x). Next, define ALX by A(x) = ⋁zX(⇓ z(x)∧⇓ y(z)) for every xX. We only need to show that ⇓ y(x) ≤ A(x).

As shown in the proof of Theorem 5.9 in [22], we can see that A ∈ 𝓘L(X).

Furthermore, Aul(y)=wX(Au(w)e(y,w))=wX(tX(A(t)e(t,w))e(y,w))=wX(tX(zX(z(t)y(z))e(t,w))e(y,w))=wX(tXzX((z(t)y(z))e(t,w))e(y,w))=wX(zX(y(z)tX(z(t)e(t,w)))e(y,w))=wX(zX(y(z)sub(z,w))e(y,w))=wX(zX(y(z)e(z,w))e(y,w))=wX(zX(y(z)e(z,w))e(y,w))=wX(sub(y,w)e(y,w))=wX(e(y,w)e(y,w))=1.

Therefore, ⇓ y(x) = ⋀I∈ 𝓘L(X) Iul(y)→ I(x) ≤ Aul(y)→ A(x) = 1 → A(x) = A(x), as needed. □

Theorem 3.5

Let (X, e) be an L-poset, and IL(X) = {I ∈ 𝓘L(X) : ⊔ I exists}. Then (X, e) is continuous iff (⇓, ⊔) is a fuzzy Galois connection between (X, e) and (IL(X),sub).

Proof

To show the necessity, assume (X, e) is continuous. Clearly both ⇓ and ⊔ are fuzzy order-preserving. ∀ xX, IIL(X) , then sub (⇓ x, I) ≤ e(⊔ ⇓ x, ⊔ I) = e(x, ⊔ I). Furthermore, sub(x,I)=yX(x(y)I(y))yX((Iul(x)I(y))I(y))Iul(x)=yX(Iu(y)e(x,y))=yX(sub(I,y)e(x,y))=yX(e(I,y)e(x,y))=e(x,I). Thus sub (⇓ x, I) = e(x, ⊔ I), and it implies that (⇓, ⊔) forms a fuzzy Galois connection.

To show the sufficiency, suppose that (⇓, ⊔) is a fuzzy Galois connection between (X, e) and (IL(X),sub). Then for every xX, we have ⇓ x is directed, and e(x, ⊔⇓ x) = sub(⇓ x, ⇓ x) = 1. On the other hand, e(⊔⇓ x, x) = sub(⇓ x, ↓ x) = 1. So x = ⊔⇓ x, it implies that (X, e) is continuous. □

4 Fuzzy Scott topology on L-posets

Classical Scott topology on complete lattices and dcpos is studied in [2]. After that, many works have been done to generalize that theory on posets [1, 6]. Recently, fuzzy Scott topology has been investigated on fuzzy ordered sets with the necessary condition that fuzzy joins of all directed fuzzy set exist (i.e. dcpos) [22]. In the absence of any sort of join, the previous result is invalid, so an additional consideration for fuzzy Scott topology on L-posets is needed. This is our motivation for this section.

Definition 4.1

[34, 36, 37] L-filter on X is a map 𝓕 : LXL satisfying that:

(LF1) 𝓕(0) = 0 and 𝓕(1) = 1;

(LF2) ∀ A, BLX, 𝓕(AB) = 𝓕(A) ∧ 𝓕(B).

An L-filter is called stratified if it satisfies the following condition: (SF)aL,F(a)aorF(aB)aF(B).

The set of all stratified L-filters on X will be denoted by FLs(X).

Remark 4.2

The condition (LF2) in Definition 4.1 can be equivalently replaced by A,BLX,F(AB)F(A)F(B).

Moreover, for any 𝓕 ∈ FLs(X), we have sub (A, B) ≤ 𝓕(AB) for all A, BLX.

Example 4.3

][34, 35]]

  1. For any xX, define a map [x] : LXL as [x](A) = A(x) for every ALX. Then [x] is a stratified L-filter, called the principal L-filter of x.

  2. Let (X, τ) be an L-fuzzy topological space and xX. Define Uτx : LXL by ALX,Uτx(A)=BAB(x)τ(B).

    Then Uτx is an L-filter, and it is stratified if τ is enriched.

  3. Let (X, δ) be an L-topological space and xX. Define Uδx : LXL byALX, Uτx(A) = A(x), whereis the L-interior operator of (X, δ). Then Uδx is an L-filter, and if δ is stratified then so is Uδx.

Definition 4.4

([34, 35, 38]). A stratified L-generalized convergence structure on X is a map R : FLs(X) × XL satisfying that

  1. xX, R([x], x) = 1;

  2. xX, ∀ 𝓕, 𝓖 ∈ FLs(X), 𝓕 ≤ 𝓖 ⇒ R(𝓕, x) ≤ R(𝓖, x);

If R is a stratified L-generalized convergence structure on X, xX. Define URx : LXL by ALX,URx(A)=FFLs(X)(R(F,x)F(A)).

Then we have the following theorem.

Theorem 4.5

([34]). Each stratified L-generalized convergence structure R on X induces an enriched L-fuzzy topology τR on X given by ALX,τR(A)=xXA(x)FFLs(X),FURxF(A). and a stratified L-topology δR = {ALX : τR(A) = 1}.

Elicited by the well-known results, we aim to study topologies on an L-poset, then the consideration of a kind of convergence structures on it will be effective. To reach that goal we begin with the discussion of the lower bound of a stratified L-filter.

Let (X, e) be an L-poset and 𝓕 ∈ FLs(X). Define 𝓕lLX by xX,Fl(x)=ALXF(A)Al(x).

Proposition 4.6

([22]). Let (X, e) be an L-poset and 𝓕, 𝓖 ∈ 𝔽L(X). Then

  1. 𝓕l(x) = 𝓕(↑ x);

  2. 𝓕 ≤ 𝓖 ⇒ 𝓕l ≤ 𝓖l;

  3. xX, [x]l = ↓ x;

For a fuzzy ideal on (X, e), define 𝓕I : LXL by ALX,FI(A)=xX(I(x)sub(x,A)).

Then 𝓕I is a stratified L-filter on X, and FIl = I (refer to [22] for detail).

Let (X, e) be an L-poset. Define a map S : 𝔽L(X) × XL by (F,x)FL(X)×X,S(F,x)=IIL(X)(sub(I,Fl)Iul(x)).

It is easily seen that S is a stratified L-generalized convergence structure on X, and S(𝓕, x) can be interpreted as the degree of 𝓕 Scott converges to x. Moreover, we define USx : LXL by ALX,USx(A)=FFLs(X)(S(F,x)F(A)).

Then USx is a stratified L-filter.

By Theorem 4.5, there is an enriched L-fuzzy topology associated with S. We denote it as σLF(X, e)(σLF(X) for short), that is, ALX,σLF(X,e)(A)=xX(A(x)USx(A)).

Furthermore, let σL(X,e)={ALX:σLF(X,e)(A)=1}.

Then σL(X, e) (σL(X) for short) is a stratified L-topology called fuzzy Scott topology on (X, e). We say an L-subset A is fuzzy Scott open if AσL(X).

Proposition 4.7

For ALX, the following are equivalent:

  1. A is fuzzy Scott open;

  2. xX, A(x) ≤ ⋀I ∈ 𝓘L(X) (Iul(x) → 𝓕I(A));

  3. A is an upper L-set, and A(x) ≤ ⋀I ∈ 𝓘L(X) (Iul(x) → ⋁yX(I(y) ∧ A(y))) for all xX.

Proof

(1) ⇒(2). Since A is fuzzy Scott open, so for all xX, we have A(x)FFLs(X)(S(F,x)F(A))IIL(X)(S(FI,x)FI(A))IIL(X)Iul(x)FI(A).

(2) ⇒(3). Above all, A(x) ≤ ⋀I ∈ 𝓘L(X) (Iul(x) → 𝓕I(A)) follows immediately from (2) and the fact that sub (↑ x, A) ≤ A(x). Next, for ∀ x, yX, A(x)e(x,y)e(x,y)IIL(X)Iul(x)FI(A)e(x,y)(x)ul(x)Fx(A)=e(x,y)Fx(A)e(x,y)sub(x,A)A(y).

It implies that A is an upper L-set.

(3) ⇒(1). For any xX and 𝓕 ∈ FLs(X), A(x)S(F,x)=A(x)IIL(X)sub(I,Fl)Iul(x)=IIL(X)A(x)Iul(x)sub(I,Fl)IIL(x)yxA(y)I(y)sub(I,Fl)yXA(y)Fl(y)=yX(A(y)F(y))yXF(A(y)y)F(A).

So A is fuzzy Scott open. □

Theorem 4.8

If (X, e) is a continuous L-poset, and xX, thenx is fuzzy Scott open wherex(y) = ⇓ y(x) for every yX.

Proof

First, ⇑ x is an upper L-set obviously for ∀ xX.

For ∀ yX, ∀ I ∈ 𝓘L(X), we have ⇑ x(y) = ⇓ y(x) = ⋁zX(⇓ z(x) ∧ ⇓ y(z)). Since ⇓ y(z) = ⋀J ∈ 𝓘L(X)(Jul(y) → J(z)) ≤ Iul(y) → I(z), so x(y)zXx(z)Iul(y)I(z)zXIul(y)(x(z)I(z))Iul(y)zX(x(z)I(z)).

As follows by Proposition 4.7(3), ⇑ x is fuzzy Scott open. □

Theorem 4.9

If (X, e) is a continuous L-poset, and ALX, then A is fuzzy Scott open iff A is an upper L-set, and A(x) = ⋁yX(A(y) ∧ ⇓ x(y)) for all xX.

Proof

Necessity: Suppose A is fuzzy Scott open, then by Proposition 4.7, A is an upper L-set and for all xX, we have A(x)(x)ul(x)Fx(A)=Fx(A)=yX(x(y)sub(y,A))yX(x(y)A(y)).

Clearly, A(x) = ⋁yX(A(y) ∧↓ x(y)) ≥ ⋁yX(A(y) ∧ ⇓ x(y)). Therefore, A(x) = ⋁yX(A(y) ∧ ⇓ x(y)).

Sufficiency: By Proposition 4.7, it suffices to show that for all xX, I ∈ 𝓘L(X), A(x) ∧ Iul(x) ≤ ⋁yX(I(y) ∧ A(y)). In fact, by the conditions, A(x)Iul(x)=yXA(y)x(y)Iul(x)yXA(y)Iul(x)(Iul(x)I(y))yX(A(y)I(y)). □

Definition 4.10

Let (X, e) be a L-poset, for any λL and xX, defineλX = λ ∧ ⇑ x.

It is worth noting that ⇑λXσ L(X) in a continuous L-poset (X, e) since σL(X) is stratified.

Theorem 4.11

Let (X, e) be a continuous L-poset, then {⇑λX : λL, xX} is a basis of σL(X).

Proof

If AσL(X), then by the above theorem, for any xX, A(x) = ⋁yX(A(y) ∧ ⇓ x(y)) = ⋁yX(A(y) ∧ ⇑ y(x)) = ⋁yXA(y)y(x). This implies that A = ⋁yXA(y)y. □

Theorem 4.12

Let (X, e) be an L-poset, then for any xX, (↑ x) ≤ ⇑ x and (↑ x) = ⇑ x when (X, e) is continuous, whereis the L-interior operator with respect to σL(X).

Proof

Suppose AσL(X) with A ≤ ↑ x. Then by Proposition 4.7, for all yX, we have A(y)IIL(X)Iul(y)zZ(I(z)A(z))IIL(X)Iul(y)zZ(I(z)e(x,z))IIL(X)Iul(y)I(x)=x(y).

So (↑ x) = ⋁{AσL(X) : A ≤ ↑ x} ≤ ⇑ x. If (X, e) is continuous, then ⇑ xσL(X) and ⇑ x ≤ ↑ x. It implies ⇑ x ≤ (↑ x), and so (↑ x) = ⇑ x. □

5 Scott convergence on L-posets

Usually, convergence theory can not be ignored when considering topology. As shown before, on an L-poset, L-fuzzy Scott topology and fuzzy Scott topology naturally exist. A deeper problem arises in order to be compatible with the convergence under the related topology: how to define a fruitful convergence on an L-poset? This section will give the answer.

Definition 5.1

Let (X, e) be an L-poset, xX and 𝓕 a stratified L-filter on X. Then we say 𝓕 is Scott convergent to x if there exists I ∈ 𝓘L(X) such that I ≤ 𝓕l and Iul(x) = 1. We denote this by 𝓕 →s x.

Proposition 5.2

Let I be a fuzzy ideal on an L-poset (X, e). Then Iul(x) ≤ S(𝓕I, x) for all xX.

Proof

By Proposition 3.4 in [38], we have FIl = I. Thus for every xX, S(𝓕I, x) = ⋁J ∈ 𝓘L(X)(sub(J, FIl) ∧ Jul(x)) ≥ sub(I, FIl) ∧ Iul(x) = sub(I, I) ∧ Iul(x) = Iul(x). □

Corollary 5.3

If (X, e) is a continuous L-poset, then for all xX, S(𝓕 X, x) = 1, i. e., 𝓕xSx.

Proof

Since (X, e) is continuous, then for all xX, ⇓ x ∈ 𝓘L(X) and ⊔ ⇓ x = x. Thus, (x)ul(x)=yX((x)U(y)e(x,y))=yX(sub(x,y)e(x,y))=yX(e(x,y)e(x,y))=1.

By Proposition 5.2, S(𝓕x, x) ≥ (⇓ x)ul(x) = 1. □

Recall that for an L-fuzzy topology τ on X, we call a stratified L-filter 𝓕 is convergent to xX, denoting 𝓕 →τ x, if Uτx ≤ 𝓕. Under an L-topology δ on X, a stratified L-filter 𝓕 convergent to xX, denoting 𝓕 →δ x, if Uδx ≤ 𝓕.

Proposition 5.4

Let (X, e) be an L-poset, xX and 𝓕 a stratified L-filter on X. Then 𝓕 →σLF(X) x implies 𝓕 →σL(X)x.

Proof

By the definition, we only need to show that UσL(X)XUσLF(X)X. In fact, for any ALX, UσL(X)X(A)={B(x):BA,σLF(X)(B)=1}BA(B(x)σLF(X)(B))=UσLF(X)X(A).

So UσL(X)xUσLF(X)x, as needed. □

Proposition 5.5

Let (X, e) be an L-poset, then (UσL(X)x)l(UσLF(X)x)l ≤ ⇓ x for all xX. If (X, e) is continuous, then (UσL(X)x)l=(UσLF(X)x)l = ⇓ x.

Proof

At first, (UσL(X)x)l(UσLF(X)x)l just follows from the fact UσL(X)xUσLF(X)x for all xX, which is shown in the above proposition.

Next, if yX and B ≥ ↑ y, then for ∀ I ∈ 𝓘L(X), we have B(x)σLF(X)(B)Iul(x)=B(x)Iul(F,z)FLs(X)×X(B(z)S(F,z)F(B))B(x)Iul(x)zX(B(z)S(FI,z)FI(y))B(x)Iul(x)zXB(z)Iul(z)FIl(y)B(x)Iul(x)B(x)Iul(x)I(y)I(y).

Hence, B(x) ∧ σLF(X)(B) ≥ ⋀I ∈ 𝓘L(X) (Iul(x) → I(y)) = ⇓x(y). So (UσLF(X)x)l(y)=UσLF(X)x (↑ y) = ⋁B ≤ ↑ y(B(x) ∧ σLF(X)(B)) ≥ ⇓ x(y) for every yX, it implies (UσL(X)x)l(UσLF(X)x)l≤⇓x.

If (X, e) is continuous, then σLF(X)(⇑ y) = 1. Therefore, for every yX, (UσL(X)x)l(y)=UσL(X)x(y)={B(x):By,σLF(X)(B)=1}y(x)=x(y). Thus, (UσL(X)x)l=(UσLF(X)x)l=x. □

Proposition 5.6

Let (X, e) be an L-poset, (𝓕, x) ∈ FLs(X) × X. Then S(𝓕, x) ≥ sub (UσLF(X)x,F) . So 𝓕 →s x implies 𝓕 →σLF(X)x.

Proof

For any (𝓕, x) ∈ FLs(X) × X and any ALX, UσLF(X)x(A)=BAB(x)σLF(X)(B)BAB(x)(B(x)S(F,x)F(B))S(F,x)F(A)

Thus, sub (UσLF(X)x,F)=ALxUσLF(X)x(A)F(A)S(F,x). □

Theorem 5.7

Let (X, e) be an L-poset, then the following are equivalent:

  1. (X, e) is continuous;

  2. ∀ 𝓕 ∈ FLs(X), 𝓕 →sX ⇔ 𝓕 →σLF(X)X ⇔ 𝓕 →σL(X)X;

  3. xX, UσL(X)x is Scott convergent to x.

  4. xX, UσLF(X)x is Scott convergent to x

Proof

(1) ⇒ (2). For ∀ 𝓕 ∈ FLs(X), clearly 𝓕 →Sx ⇒ 𝓕 →σLF(X)X ⇒ 𝓕 →σL(X)X by Proposition 5.4 and Proposition 5.6, we only need to show 𝓕 →σL(X)X ⇒ 𝓕 →sX. If 𝓕 →σL(X)X, then (UσL(X)x)lFl. Since (X, e) is continuous, then ⇓ x ∈ 𝓘L(X) and x=(UσL(X)x)lFl. Thus S(𝓕, x) ≥ sub (⇓ x, 𝓕l) ∧ (⇓ x)ul(x) = 1. That is, 𝓕 →sX.

(2) ⇒(3) is obvious.

(3) ⇒(4) follows immediately from the fact UσL(X)xUσLF(X)x.

(4) ⇒(1). For all xX, since UσLF(X)x is Scott convergent to x, then there exists I ∈ 𝓘L(X) such that I(UσLF(X)x)l≤⇓x and Iul(x) = 1. If a, bX, then x(a)x(b)=JIL(X)(Jul(x)J(a))JIL(X)(Jul(x)J(b))I(a)I(b)cX(I(c)e(a,c)e(b,c))cX(x(c)e(a,c)e(b,c)).

Thus ⇓ x is directed by definition. Furthermore, for any yX, we have zX(x(z)e(z,y))zX(x(z)e(z,y))=e(x,y) and zX(x(z)e(z,y))zX(I(z)e(z,y))=sub(I,y)Iul(x)Iu(y)(Iu(y)e(x,y))e(x,y).

Hence, ⋀zX(⇓ x(z) → e(z, y)) = e(x, y), and it implies that x = ⊔ ⇓ x. So (X, e) is continuous. □

Theorem 5.8

Let (X, e) be a continuous L-poset, xX and 𝓕 a stratified L-filter on X. Then 𝓕 is Scott convergent to xiff ⇓ x ≥ 𝓕l.

Proof

The sufficiency is obvious. To show the necessity, assume that 𝓕 is Scott convergent to x. There exists I ∈ 𝓘L (X) such that I ≥ 𝓕l and Iul(x) = 1. For all yX, ⇓ x(y) ≤ Iul(x) → I(y) ≤ 𝓕l(y). Thus ⇓ x ≥ 𝓕l. □

6 Conclusion

In this paper, we first extend the fuzzy way-below relation on fuzzy dcpos to fuzzy ordered sets without any additional conditions, and based on that, the continuity for L-posets is studied. Later on, we propose a kind of stratified L-generalized convergence structure, and then study fuzzy Scott topology. The Scott convergence theory on L-posets is established finally, and the continuity is well described by Scott convergence. That is, an L-poset is continuous if and only if Scott convergence coincides with convergence under either L-fuzzy Scott topology or fuzzy Scott topology. All the works will promote the development of quantitative domain theory.

Acknowledgement

This work is supported by Doctoral Foundation of Shandong Jianzhu University (NO. 0000601373) and a grant from the National Natural Science Foundation of China (No.61471409).

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About the article

Received: 2016-10-06

Accepted: 2017-04-03

Published Online: 2017-06-22


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 815–827, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0067.

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© 2017 Liu and Chen. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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