# Fuzzy Bases of Fuzzy Domains

Authors: Sanping Rao; Qingguo Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is an attempt to develop quantitative domain theory over frames. Firstly, we propose the notion of a fuzzy basis, and several equivalent characterizations of fuzzy bases are obtained. Furthermore, the concept of a fuzzy algebraic domain is introduced, and a relationship between fuzzy algebraic domains and fuzzy domains is discussed from the viewpoint of fuzzy basis. We finally give an application of fuzzy bases, where the image of a fuzzy domain can be preserved under some special kinds of fuzzy Galois connections.


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## Body

## 1. Introduction

Since the pioneering work of Scott [1, 2], domain and its generalization have attracted more and more attention. Domain provides models for various types of programming languages that include imperative, functional, nondeterministic, and probabilistic languages. When domains appear in theoretical computer science, one typically wants them to be objects suitable for computation. In particular, one is motivated to find a suitable notion of a recursive or recursively enumerable domain. This leads to the notion of a basis (cf. [3]).Quantitative domain theory has been developed to supply models for concurrent systems. Now it forms a new focus on domain theory and has undergone active research. Rutten's generalized (ultra)metric spaces [4], Flagg's continuity spaces [5], and Wagner's Ω-categories [6] are good examples, which consist of basic frameworks of quantitative domain theory (cf. [7]).Recently, based on complete residuated lattices, Yao and Shi [8, 9] investigated quantitative domains via fuzzy set theory. They defined a fuzzy way-below relation via fuzzy ideals to examine the continuity of fuzzy domains and later discussed fuzzy Scott topology over fuzzy dcpos. Zhang and Fan [7] studied quantitative domains over frames. From the very beginning, they defined a fuzzy partial order which is really a degree function on a nonempty set. After that, they defined and studied fuzzy dcpos and fuzzy domains. Roughly speaking, the definition of a fuzzy directed subset in [7] which is based on a kind of special relations looks relatively complex. Furthermore, from the viewpoint of category, Hofmann and Waszkiewicz [10–12], Lai and Zhang [13, 14], and Stubbe [15, 16] studied quantitative domain theory.It is well known that the notion of a basis plays an important role in domain theory. The results not only are handy in establishing certain equivalent characterizations for domains but also are critical to study some properties of domains. Then, how can we describe a fuzzy basis in a fuzzy dcpo? And what is the role of it in fuzzy ordered set theory? For this purpose, we are motivated to introduce the notion of a fuzzy basis as a new approach to study fuzzy domains. From the viewpoint of fuzzy basis, we try to build a relationship between fuzzy domains and fuzzy algebraic domains. Moreover, we investigate some applications of fuzzy bases to examine the relationships of the definitions.The contents of this paper are organized as follows. In Section2, some preliminary concepts and properties are recalled. In Section 3, the concept of a fuzzy basis is proposed, and an equivalent characterization of fuzzy bases is obtained. Furthermore, the notion of a fuzzy algebraic domain is proposed; it is proved that a fuzzy dcpo is a fuzzy algebraic if and only if it is a fuzzy domain and the fuzzy basis satisfies some special interpolation property. In Section 4, an application of fuzzy bases is given, where we investigate some special kinds of fuzzy Galois connections, under which the image of a fuzzy domain is also a fuzzy domain. Conclusions are settled in the last section.


## 2. Preliminary

A frame will be used as the structures of truth values in this paper. Throughout this paper, unless otherwise stated,L always denotes a frame. For more properties about frames, we refer to [3, 17, 18].LetX be a nonempty set, an L-subset on X is a mapping from X to L, and the family of all L-subsets on X will be denoted by LX. We denote the constant L-subsets on X taking the values 0 and 1 by 0X and 1X, respectively. Let A,B∈LX. The equality of A and B is defined as the usual equality of mappings; that is, A=B⇔A(x)=B(x) for any x∈X. The inclusion A≤B is also defined pointwisely: A≤B⇔A(x)≤B(x) for any x∈X.The following definitions and propositions can be found in [7–9, 14, 19–24].Definition 1.
A fuzzy poset is a pair(X,e) such that X is a nonempty set, and e:X×X→L is a mapping, called a fuzzy order, that satisfies for any x,y,z∈X, (1)
e(x,x)=1,(2)
e(x,y)∧e(y,z)≤e(x,z),(3)
e(x,y)=e(y,x)=1 implies x=y.To study fuzzy relational systems, Bělohlávek [19] defined and studied an L-order over complete residuated lattices. It is shown in [25] that the previous notion is equivalent to Bělohlávek's one.Definition 2.
Let(X,e) be a fuzzy poset. An element x0∈X is called a join (or meet) of a fuzzy subset A, in symbols x0=⊔A (or x0=⊓A) if (1)
for anyx∈X,A(x)≤e(x,x0)(orA(x)≤e(x0,x)),(2)
for anyy∈X,⋀x∈X(A(x)→e(x,y))≤e(x0,y)(or ⋀x∈X(A(x)→e(y,x))≤e(y,x0)).It is easy to check ifx1,x2 are two joins (or meets) of A, then x1=x2. This means if A∈LX has a join (or meet), then it is unique.Proposition 3.
Let(X,e) be a fuzzy poset. Then (1)
x0=⊔A if and only if for any y∈X,e(x0,y)=⋀z∈X(A(z)→e(z,y));(2)
x0=⊓A if and only if for any y∈X,e(y,x0)=⋀z∈X(A(z)→e(y,z)).Example 4.
Given a nonempty setX, the subsethood degree mapping sub(-,-):LX×LX→L is defined by for each pair (A,B)∈LX×LX, sub(A,B)=⋀x∈X(A(x)→B(x)). Then sub(-,-) is an L-partial order on LX. Moreover, if A≤B, then sub(A,B)=⋀x∈X(A(x)→B(x))=1.Definition 5.
Let(X,e) be a fuzzy poset. A∈LX is called a fuzzy upper set (or a fuzzy lower set) if for any x,y∈X,A(x)∧e(x,y)≤A(y)(orA(x)∧e(y,x)≤A(y)).Definition 6.
Let(X,e) be a fuzzy poset. For x∈X,↓x∈LX(or ↑x∈LX) is defined as for any y∈X,↓x(y)=e(y,x)(or↑x(y)=e(x,y)). And ↓D is defined by for any x∈X,↓D(x)=⋁y∈XD(y)∧e(x,y).
Note thatx=⊔↓x. When A=↓x, by Proposition 3, we have
(1)e(x,y)=⋀z∈X(e(z,x)⟶e(z,y)).Definition 7.
Let(X,eX) and (Y,eY) be two fuzzy posets. A mapping f:(X,eX)→(Y,eY) is called a fuzzy monotone mapping if for any x,y∈X, eX(x,y)≤eY(f(x),f(y)).Definition 8.
LetX,Y be two nonempty sets. For each mapping f:X→Y, the L-forward powerset operator fL→:LX→LY is defined by
(2)foranyy∈Y,A∈LX,fL→(A)(y)=⋁f(x)=yA(x).
The L-backward powerset operator fL←:LY→LX is defined by
(3)foranyB∈LY,fL←(B)=B∘f.
Furthermore, f can be always lifted as f~→:LX→LY, which is defined by
(4)foranyy∈Y,A∈LX,f~→(A)(y)=⋁x∈XA(x)∧e(y,f(x)).In the literature one can find several different fuzzy versions of directed subsets. We will focus on one of them, which is introduced in [8, 14].Definition 9.
Let(X,e) be a fuzzy poset. D∈LX is called a fuzzy directed subset if (1)
⋁x∈XD(x)=1,(2)
for anya,b∈X, D(a)∧D(b)≤⋁d∈XD(d)∧e(a,d)∧e(b,d).A fuzzy ideal is a fuzzy lower directed subset. We denote the set of all fuzzy directed subsets and all fuzzy ideals onX by 𝒟L(X) and ℐL(X), respectively. A fuzzy poset is called a fuzzy dcpo if every fuzzy directed subset has a join.Definition 10.
Let(X,eX) and (Y,eY) be two fuzzy dcpos. A fuzzy monotone mapping f:(X,eX)→(Y,eY) is said to be fuzzy Scott continuous if for any D∈𝒟L(X), f(⊔D)=⊔f~→(D).We now introduce one of the most efficient tools in dealing with fuzzy poset, which were extensively studied in [8, 13, 14, 19, 20, 25]. One reason for this great efficiency is that the pairs of mappings of the kind we are about to single out exist in great profusion.Definition 11.
Let(X,eX) and (Y,eY) be two fuzzy posets, f:(X,eX)→(Y,eY) and g:(Y,eY)→(X,eX) two fuzzy monotone mappings. The pair (f,g) is called a fuzzy Galois connection between (X,eX) and (Y,eY) provided that
(5)foranyx∈X,y∈Y,eY(y,f(x))=eX(g(y),x),
where f is called the upper adjoint of g and dually g is called the lower adjoint of f.Proposition 12.
Let(X,eX) and (Y,eY) be two fuzzy posets. (f,g) is a fuzzy Galois connection on (X,eX) and (Y,eY) if and only if both f and g are fuzzy monotone mappings, and (f,g) is a crisp Galois connection on (X,≤eX) and (Y,≤eY), where ≤eX is defined as follows: eX(x1,x2)=1⇔x1≤eXx2.The crisp Galois connection is defined as follows:y≤f(x)⇔g(y)≤x for any x∈X,y∈Y, and its relative properties can be found in [3].Proposition 13.
Let(X,eX) and (Y,eY) be two fuzzy posets, f:(X,eX)→(Y,eY) and g:(Y,eY)→(X,eX) two mappings. (1)
Iff is a fuzzy monotone mapping and has a lower adjoint, then for any S∈LX such that ⊓S exists, f(⊓S)=⊓fL→(S).(2)
Ifg is a fuzzy monotone mapping and has an upper adjoint, then for any D∈LY such that ⊔D exists, g(⊔D)=⊔gL→(D).The fuzzy visions of way-below relations were extensively studied in [7, 8, 10–14]. Hofmann and Waszkiewicz [11] presented a systematic investigation of such relation in quantale-enriched categories.Definition 14.
Let(X,e) be a fuzzy dcpo. For any x,y∈X, define ⇓x∈LX by
(6)⇓x(y)=⋀I∈ℐL(X)(e(x,⊔I)⟶I(y)).It is a fact that in the crisp setting, the way-below relation can be defined by ideals and directed subsets, respectively. And in this case, the two way-below relations are equivalent. Then, does the equivalence of such relations also hold? Here we present a proof to confirm it.Lemma 15.
Let(X,eX) and (Y,eY) be two fuzzy posets, f:(X,eX)→(Y,eY) a fuzzy monotone mapping. Then for any D∈𝒟L(X),f~→(D)∈ℐL(Y).Proposition 16.
Let(X,e) be a fuzzy dcpo. Then for any x,y∈X,
(7)⋀D∈𝒟L(X)(e(x,⊔D)⟶(⋁d∈XD(d)∧e(y,d)))=⋀I∈ℐL(X)(e(x,⊔I)⟶I(y)).
That is, ⇓x(y)=⋀D∈𝒟L(X)(e(x,⊔D)→(⋁d∈XD(d)∧e(y,d))).Proof.
Obviously, for anyI∈ℐL(X), I∈𝒟L(X). On the one hand,
(8)⋀D∈𝒟L(X)(e(x,⊔D)⟶(⋁d∈XD(d)∧e(y,d)))≤⋀I∈ℐL(X)(e(x,⊔I)⟶(⋁d∈XI(d)∧e(y,d)))≤⋀I∈ℐL(X)(e(x,⊔I)⟶⋁d∈XI(y))=⋀I∈ℐL(X)(e(x,⊔I)⟶I(y)).
On the other hand, for anyD∈𝒟L(X), it is routine to check that ↓D∈ℐL(X). Then
(9)⋀I∈ℐL(X)(e(x,⊔I)⟶I(y))≤⋀D∈𝒟L(X)(e(x,⊔↓D)⟶↓D(y))=⋀D∈𝒟L(X)(e(x,⊔D)⟶(⋁d∈XD(d)∧e(y,d))).By Proposition16, for all statements, it is valid for the fuzzy way-below relation over fuzzy directed subsets if and only if it holds for the one over fuzzy ideals.Some basic properties of the fuzzy relation are listed in the following proposition.Proposition 17.
Let(X,e) be a fuzzy dcpo. For any x,y,u,v∈X, then (1)
⇓x≤↓x,(2)
e(u,x)∧⇓y(x)∧e(y,v)≤⇓v(u).Definition 18.
A fuzzy dcpo(X,e) is called a fuzzy domain or continuous if for any x∈X, ⇓x∈𝒟L(X)(or ⇓x∈ℐL(X)) and x=⊔⇓x.The following theorem exhibits an important property of the fuzzy way-below relation on fuzzy domains, the interpolation property. It has been widely discussed in [7, 8, 11, 13].Theorem 19.
If(X,e) is a fuzzy domain, then for any x,y∈X, ⇓y(x)=⋁z∈X⇓y(z)∧⇓z(x).


## 3. Fuzzy Bases and Fuzzy Algebraic Domains

In this section, we define a fuzzy basis in a fuzzy dcpo, and we obtain some equivalent characterizations of fuzzy bases. Moreover, we also study fuzzy algebraic domains from the viewpoint of fuzzy basis.Definition 20.
Let(X,e) be a fuzzy dcpo. B∈LX is called a fuzzy basis of X if (1)
for anyx∈X, B∧⇓x is a fuzzy directed subset of X, and(2)
for anyx∈X, x=⊔(B∧⇓x).Obviously, the previous definition is really a generation of the notion of a basis in [3].Proposition 21.
Let(X,e) be a fuzzy dcpo. For any x∈X, if there exists a fuzzy directed subset A such that x=⊔A and A≤⇓x, then ⇓x is a fuzzy directed subset with x=⊔⇓x.Proof.
For anyy∈X, we firstly show that ⇓x(y)≤⋁d∈XA(d)∧e(y,d). Indeed,
(10)⇓x(y)=⋀D∈𝒟L(X)(e(x,⊔D)⟶(⋁d∈XD(d)∧e(y,d)))≤e(x,⊔A)⟶(⋁d∈XA(d)∧e(y,d))=1⟶(⋁d∈XA(d)∧e(y,d))=⋁d∈XA(d)∧e(y,d).
Then for any a,b∈X, we have
(11)⇓x(a)∧⇓x(b)≤⋁d1,d2∈XA(d1)∧A(d2)∧e(a,d1)∧e(b,d2)≤⋁d1,d2∈X⋁d∈XA(d)∧e(d1,d)∧e(d2,d)∧e(a,d1)∧e(b,d2)≤⋁d1,d2∈X⋁d∈XA(d)∧e(a,d)∧e(b,d)=⋁d∈XA(d)∧e(a,d)∧e(b,d)≤⋁d∈X⇓x(d)∧e(a,d)∧e(b,d).
Moreover, ⋁y∈X⇓x(y)=1 follows from 1=⋁y∈XA(y)≤⋁y∈X⇓x(y). Hence ⇓x is fuzzy directed.
It is easy to verify that⊔ is fuzzy monotone. Since A≤⇓x, then 1=sub(A,⇓x)≤e(⊔A,⊔⇓x)=e(x,⊔⇓x). Meanwhile, 1=sub(⇓x,↓x)≤e(⊔⇓x,⊔↓x)=e(⊔⇓x,x). Therefore, x=⊔⇓x.Theorem 22.
A fuzzy dcpo has a fuzzy basis if and only if it is a fuzzy domain.Proof.
Necessity. Suppose that B is a fuzzy basis of X, then for any x∈X, B∧⇓x∈𝒟L(X) with x=⊔(B∧⇓x). It is clear that B∧⇓x≤⇓x. Thus ⇓x∈𝒟L(X) with x=⊔⇓x follows from Proposition 21. Therefore, (X,e) is a fuzzy domain.
Sufficiency. It is easy to check that 1X is a fuzzy basis of X.Proposition 23.
Let(X,e) be a fuzzy domain. For any z∈X, if z=⊔D for some D∈𝒟L(X), then ⇓z=⋁d∈XD(d)∧⇓d.Proof.
For anyx∈X, denote that A∈LX as A(x)=⋁d∈XD(d)∧⇓d(x).
(a)A is a fuzzy directed subset as follows:
(12)⋁x∈XA(x)=⋁x∈X⋁d∈XD(d)∧⇓d(x)=⋁d∈XD(d)∧(⋁x∈X⇓d(x))=1.
Furthermore, for any a,b∈X,
(13)A(a)∧A(b)=⋁d1,d2∈XD(d1)∧⇓d1(a)∧D(d2)∧⇓d2(b)≤⋁d1,d2∈X⋁d∈XD(d)∧e(d1,d)∧e(d2,d)∧⇓d1(a)∧⇓d2(b)≤⋁d1,d2∈X⋁d∈XD(d)∧⇓d(a)∧⇓d(b)=⋁d∈XD(d)∧⇓d(a)∧⇓d(b)≤⋁d∈XD(d)∧(⋁c∈X⇓d(c)∧e(a,c)∧e(b,c))=⋁c∈Xe(a,c)∧e(b,c)∧(⋁d∈XD(d)∧⇓d(c))=⋁c∈XA(c)∧e(a,c)∧e(b,c).
(b)⊔A=z. For any y∈X, we have
(14)⋀x∈X(A(x)⟶e(x,y))=⋀x∈X((⋁d∈XD(d)∧⇓d(x))⟶e(x,y))=⋀x∈X⋀d∈X(D(d)⟶(⇓d(x)⟶e(x,y)))=⋀d∈X(D(d)⟶⋀x∈X(⇓d(x)⟶e(x,y)))=⋀d∈X(D(d)⟶e(⊔⇓d,y))=⋀d∈X(D(d)⟶e(d,y))=e(⊔D,y).
Hence ⊔D=⊔A=z.
(c)⇓z=⋁d∈XD(d)∧⇓d. On one hand, for any x∈X,
(15)⋁d∈XD(d)∧⇓d(x)≤⋁d∈Xe(d,⊔D)∧⇓d(x)≤⋁d∈X⇓z(x)=⇓z(x).
On the other hand,
(16)⇓z(x)=⋀S∈𝒟L(X)(e(z,⊔S)⟶(⋁b∈XS(b)∧e(x,b)))≤e(z,⊔(⋁d∈XD(d)∧⇓d))⟶(⋁b∈X⋁d∈XD(d)∧⇓d(b)∧e(x,b))≤e(z,⊔A)⟶(⋁b∈X⋁d∈XD(d)∧⇓d(x))=⋁d∈XD(d)∧⇓d(x).
Therefore, ⇓z=⋁z∈XD(d)∧⇓d.Theorem 24.
Let(X,e) be a fuzzy domain. For B∈LX, the following are equivalent: (1)
B is a fuzzy basis of X;(2)
for anyx∈X, there exists a fuzzy directed subset D≤B∧⇓x such that x=⊔D;(3)
for anyx,y∈X, ⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧⇓b(x);(4)
for anyx,y∈X, ⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧↓b(x).Proof.
(
1
) implies (2). It is  evident from Definition 20.
(
2
) implies (3). Since (X,e) is a fuzzy domain, by Proposition 23, for any z∈X, if z=⊔D for some D∈𝒟L(X), then ⇓z=⋁b∈XD(b)∧⇓b. Indeed, for any x,y∈X,
(17)⇓y(x)=⋁z∈X⇓y(z)∧⇓z(x)=⋁z∈X⇓y(z)∧(⋁b∈XD(b)∧⇓b(x))≤⋁z∈X⇓y(z)∧(⋁b∈X(B∧⇓z)(b)∧⇓b(x))=⋁z∈X⋁b∈XB(b)∧⇓y(z)∧⇓z(b)∧⇓b(x)≤⋁z∈X⋁b∈XB(b)∧⇓y(b)∧⇓b(x)=⋁b∈XB(b)∧⇓y(b)∧⇓b(x).
(
3
) implies (4). For any x,y∈X, we have
(18)⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧⇓b(x)≤⋁b∈XB(b)∧⇓y(b)∧↓b(x)≤⋁b∈XB(b)∧⇓y(x)≤⇓y(x).
Hence ⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧↓b(x).
(
4
) implies (1). Assuming (4), we next show that B is a fuzzy basis of X. In fact, for any y∈X,
(a)y=⊔(B∧⇓y). Since y=⊔⇓y, then for any u∈X,
(19)e(y,u)=⋀x∈X(⇓y(x)⟶e(x,u))=⋀x∈X((⋁b∈XB(b)∧⇓y(b)∧↓b(x))⟶e(x,u))=⋀x∈X⋀b∈X((B(b)∧⇓y(b)∧↓b(x))⟶e(x,u))=⋀x∈X⋀b∈X((B(b)∧⇓y(b))⟶(e(x,b)⟶e(x,u)))=⋀b∈X((B(b)∧⇓y(b))⟶⋀x∈X(e(x,b)⟶e(x,u)))=⋀b∈X((B∧⇓y)(b)⟶e(b,u))=e(⊔(B∧⇓y),u).
Hencey=⊔(B∧⇓y).
(b)B∧⇓y is a fuzzy directed subset. Firstly, for any a,b∈X,
(20)(B∧⇓y)(a)∧(B∧⇓y)(b)≤⇓y(a)∧⇓y(b)≤⋁c∈X⇓y(c)∧e(a,c)∧e(b,c)=⋁c∈X⋁d∈XB(d)∧⇓y(d)∧↓d(c)∧e(a,c)∧e(b,c)≤⋁c∈X⋁d∈XB(d)∧⇓y(d)∧e(a,d)∧e(b,d)=⋁d∈X(B∧⇓y)(d)∧e(a,d)∧e(b,d).
Moreover, note that ⋁x∈X⇓y(x)=1 and ⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧↓b(x). Then ⋁x∈X⇓y(x)=⋁b∈XB(b)∧⇓y(b)∧⋁x∈X↓b(x). Hence ⋁b∈X(B∧⇓y)(b)=1. By Definition 20, (1) holds.Proposition 25.
IfB is a fuzzy basis of X, then so is ↓B.Proof.
In fact, for anyx∈X,
(a)x=⊔(↓B∧⇓x). It is clear that B∧⇓x≤↓B∧⇓x. Then
(21)1=sub(B∧⇓x,↓B∧⇓x)≤e(⊔(B∧⇓x),⊔(↓B∧⇓x))=e(x,⊔(↓B∧⇓x)).
Note that↓B∧⇓x≤⇓x≤↓x. Then
(22)1=sub(↓B∧⇓x,↓x)≤e(⊔(↓B∧⇓x),⊔↓x)=e(⊔(↓B∧⇓x),x).
Hence x=⊔(↓B∧⇓x).
(b)↓B∧⇓x is a fuzzy directed subset. Since B is a fuzzy basis of X, by Theorems 24, for any c∈X, we have ⇓x(c)=⋁d∈XB(d)∧⇓x(d)∧↓d(c). Then for any a,b∈X,
(23)(↓B∧⇓x)(a)∧(↓B∧⇓x)(b)≤⇓x(a)∧⇓x(b)≤⋁c∈X⇓x(c)∧e(a,c)∧e(b,c)=⋁c∈X⋁d∈XB(d)∧⇓x(d)∧↓d(c)∧e(a,c)∧e(b,c)≤⋁c∈X⋁d∈XB(d)∧⇓x(d)∧e(a,d)∧e(b,d)=⋁d∈X(B∧⇓x)(d)∧e(a,d)∧e(b,d)≤⋁d∈X(↓B∧⇓x)(d)∧e(a,d)∧e(b,d).
Furthermore, ⋁d∈X(↓B∧⇓x)(d)=1 follows from 1=⋁d∈X(B∧⇓x)(d)≤⋁d∈X(↓B∧⇓x)(d). Therefore, ↓B is a fuzzy basis of X.Since for anyD∈LX, ↓D is a fuzzy lower set. Then we can deduce the following.Corollary 26.
IfX has a fuzzy basis, then there exists a fuzzy lower one.Although the definition of fuzzy algebraic domain was introduced by compact elements in [8], we next introduce the notion of a fuzzy algebraic domain and discuss the relationships between fuzzy algebraic domains and fuzzy domains from the viewpoint of fuzzy basis.Definition 27.
A fuzzy dcpo(X,e) is called a fuzzy algebraic domain if (1)
for anyx∈X, K∧↓x is a fuzzy directed subset of X, and(2)
for anyx∈X, x=⊔(K∧↓x),
whereK∈LX is defined as follows: for any y∈X, K(y)=⇓y(y). If no confusion arises, K always denotes the previous definition in the sequel.Theorem 28.
Let(X,e) be a fuzzy dcpo. (X,e) is a fuzzy algebraic domain if and only if K is precisely a fuzzy basis of X.Proof.
By Definitions20 and 27, it suffices to show that for any x∈X, K∧⇓x=K∧↓x.
It is clear thatK∧⇓x≤K∧↓x. Conversely, for any y∈X, obviously, (K∧↓x)(y)≤K(y). Meanwhile, (K∧↓x)(y)=⇓y(y)∧↓x(y)≤⇓x(y). Thus (K∧↓x)(y)≤K(y)∧⇓x(y)=(K∧⇓x)(y). By the arbitrariness of y, K∧↓x≤K∧⇓x. Therefore, K∧⇓x=K∧↓x.Theorem 29.
Let(X,e) be a fuzzy dcpo. Then (X,e) is a fuzzy algebraic domain if and only if (1)
(X,e) is a fuzzy domain, and(2)
for anyx,y∈X,⇓y(x)=⋁z∈XK(z)∧↓y(z)∧↓z(x).Proof.
Necessity. Suppose that (X,e) is a fuzzy algebraic domain, by Theorem 28, and K is a fuzzy basis of X. Thus (X,e) is a fuzzy domain follows from Theorem 22. It remains to show that for any x,y∈X,⇓y(x)=⋁z∈XK(z)∧↓y(z)∧↓z(x). On one hand,
(24)⋁z∈XK(z)∧↓y(z)∧↓z(x)=⋁z∈X⇓z(z)∧↓y(z)∧↓z(x)≤⋁z∈X⇓y(z)∧↓z(x)≤⇓y(x).
On the other hand,
(25)⇓y(x)=⋀D∈𝒟L(X)(e(y,⊔D)⟶(⋁z∈XD(z)∧e(x,z)))≤e(y,⊔(K∧↓y))⟶(⋁z∈X(K∧↓y)(z)∧e(x,z))=⋁z∈XK(z)∧↓y(z)∧↓z(x).
Therefore, ⇓y(x)=⋁z∈XK(z)∧↓y(z)∧↓z(x).
Sufficiency. In fact, for any y∈X,
(a)y=⊔(K∧↓y). Since y=⊔⇓y, then for any u∈X,
(26)e(y,u)=⋀x∈X(⇓y(x)⟶e(x,u))=⋀x∈X((⋁z∈XK(z)∧↓y(z)∧↓z(x))⟶e(x,u))=⋀x∈X⋀z∈X((K(z)∧↓y(z))⟶(e(x,z)⟶e(x,u)))=⋀z∈X((K(z)∧↓y(z))⟶⋀x∈X(e(x,z)⟶e(x,u)))=⋀z∈X((K∧↓y)(z)⟶e(z,u))=e(⊔(K∧↓y),u).
Hence y=⊔(K∧↓y).
(b)K∧↓y is fuzzy directed. For any x∈X, (K∧↓y)(x)≤⇓y(x) and ⇓y(x)=⋁d∈XK(d)∧↓y(d)∧↓d(x). Then for any a,b∈X,
(27)(K∧↓y)(a)∧(K∧↓y)(b)≤⇓y(a)∧⇓y(b)≤⋁c∈X⇓y(c)∧e(a,c)∧e(b,c)=⋁c∈X⋁d∈XK(d)∧↓y(d)∧↓d(c)∧e(a,c)∧e(b,c)≤⋁c∈X⋁d∈XK(d)∧↓y(d)∧e(a,d)∧e(b,d)=⋁d∈X(K∧↓y)(d)∧e(a,d)∧e(b,d).
Furthermore, since ⋁x∈X⇓y(x)=1 and ⇓y(x)=⋁d∈XK(d)∧↓y(d)∧↓d(x), then
(28)⋁x∈X⇓y(x)=⋁d∈XK(d)∧↓y(d)∧⋁x∈X↓d(x).
Thus ⋁d∈X(K∧↓y)(d)=1.
Therefore,(X,e) is a fuzzy algebraic domain.Remark 30.
The main results of Theorems24, 28, and 29 indicate that the definitions of the fuzzy basis and the fuzzy algebraic domain are reasonable.


## 4. An Application of Fuzzy Bases

This section is mainly devoted to giving an application of fuzzy bases. Our aim is to investigate some special kinds of fuzzy Galois connections, under which the image of a fuzzy domain can be preserved.Definition 31.
Let(X,eX) and (Y,eY) be two fuzzy dcpos. A fuzzy monotone mapping f:(X,eX)→(Y,eY) is said to preserve fuzzy way-below relation if for any x,y∈X, ⇓x(y)≤⇓f(x)(f(y)).Proposition 32.
Let(X,eX) and (Y,eY) be two fuzzy dcpos, (f,g) a fuzzy Galois connection from (X,eX) to (Y,eY). (1)
Iff is fuzzy Scott continuous, then g preserves fuzzy way-below relation.(2)
Iff is surjective, then for any x,y∈Y, ⇓g(x)(g(y))≤⇓x(y).Proof.
(1) For anyD∈𝒟L(X), by Lemma 15, f~→(D)∈ℐL(Y). Then for any x,y∈Y,
(29)⇓x(y)=⋀S∈𝒟L(Y)(eY(x,⊔S)⟶(⋁d∈YS(d)∧eY(y,d)))≤⋀D∈𝒟L(X)((⋁d∈Yf~→(D)(d)∧eY(y,d))eY(x,⊔f~→(D))⟶(⋁d∈Yf~→(D)(d)∧eY(y,d)))=⋀D∈𝒟L(X)(eY(x,f(⊔D))⟶f~→(D)(y))=⋀D∈𝒟L(X)((⋁b∈XD(b)∧eY(y,f(b)))eX(g(x),⊔D)⟶(⋁b∈XD(b)∧eY(y,f(b))))=⋀D∈𝒟L(X)((⋁b∈XD(b)∧eY(y,f(b)))eX(g(x),⊔D)⟶(⋁b∈XD(b)∧eX(g(y),b)))=⇓g(x)(g(y)).
(2) Sincef is surjective, fg=idY. Note that for any S∈𝒟L(Y), g~→(S)∈ℐL(X), and g is a lower adjoint of f. Then for any x,y∈Y, by Proposition 13, we have
(30)⇓g(x)(g(y))=⋀D∈𝒟L(X)((⋁b∈XD(b)∧eX(g(y),b))eX(g(x),⊔D)⟶(⋁b∈XD(b)∧eX(g(y),b)))≤⋀S∈𝒟L(Y)((⋁b∈Xg~→(S)(b)∧eX(g(y),b))b∈XeX(g(x),⊔g~→(S))⟶(⋁b∈Xg~→(S)(b)∧eX(g(y),b)))=⋀S∈𝒟L(Y)(eX(g(x),g(⊔S))⟶g~→(S)(g(y)))=⋀S∈𝒟L(Y)((⋁d∈YS(d)∧eX(g(y),g(d)))eY(x,fg(⊔S))⟶(⋁d∈YS(d)∧eX(g(y),g(d))))=⋀S∈𝒟L(Y)((⋁d∈YS(d)∧eX(g(y),g(d)))eY(x,⊔S)⟶(⋁d∈YS(d)∧eX(g(y),g(d))))≤⋀S∈𝒟L(Y)((⋁d∈YS(d)∧eX(g(y),g(d)))eY(x,⊔S)⟶(⋁d∈YS(d)∧eY(fg(y),fg(d))))=⋀S∈𝒟L(Y)(eY(x,⊔S)⟶(⋁d∈YS(d)∧eY(y,d)))=⇓x(y).Proposition 33.
Let(X,eX) and (Y,eY) be two fuzzy dcpos, (f,g) a fuzzy Galois connection from (X,eX) to (Y,eY). If (Y,eY) is a fuzzy domain and g preserves fuzzy way-below relation, then f is fuzzy Scott continuous.Proof.
For anyD∈𝒟L(X), we need to show that f(⊔D)=⊔f~→(D). Indeed,
(31)eY(⊔f~→(D),f(⊔D))=⋀y∈Y(f~→(D)(y)⟶eY(y,f(⊔D)))=⋀y∈Y((⋁x∈XD(x)∧eY(y,f(x)))⟶eY(y,f(⊔D)))=⋀y∈Y⋀x∈X(D(x)⟶(eY(y,f(x))⟶eY(y,f(⊔D))))=⋀x∈X(D(x)⟶⋀y∈Y(eY(y,f(x))⟶eY(y,f(⊔D))))=⋀x∈X(D(x)⟶eY(f(x),f(⊔D)))≥⋀x∈X(D(x)⟶eX(x,⊔D))≥⋀x∈X(D(x)⟶D(x))=1.
For the converse, letu=⊔D. Since (Y,eY) is a fuzzy domain, f(⊔D)=f(u)=⊔⇓f(u). Note that eX(gf(u),u)=1. Then for any v∈Y,
(32)⇓f(u)(v)≤⇓gf(u)(g(v))=⇓gf(u)(g(v))∧eX(gf(u),u)≤⋀S∈𝒟L(X)(eX(u,⊔S)⟶(⋁d∈XS(d)∧eX(g(v),d)))≤eX(u,⊔D)⟶(⋁d∈XD(d)∧eX(g(v),d))=1⟶(⋁d∈XD(d)∧eX(g(v),d))=⋁d∈XD(d)∧eX(g(v),d)=⋁d∈XD(d)∧eY(v,f(d))=f~→(D)(v).
Thus 1=sub(⇓f(u),f~→(D))≤eY(⊔⇓f(u),⊔f~→(D))=eY(f(⊔D),⊔f~→(D)). Therefore, f(⊔D)=⊔f~→(D).Definition 34.
Let(X,eX) and (Y,eY) be two fuzzy dcpos. f:(X,eX)→(Y,eY) is called a fuzzy morphism if f is a fuzzy Scott continuous upper adjoint.Theorem 35.
Let(X,eX) be a fuzzy domain, and let (Y,eY) be a fuzzy dcpo. If f:(X,eX)→(Y,eY) is a surjective fuzzy morphism, then (Y,eY) is a fuzzy domain.Proof.
By Theorem22 and Corollary 26,  there exists a fuzzy lower basis B of X. Now we show that f~→(B) is a fuzzy basis of Y.
Sincef is a surjective morphism, then for any y∈Y, there exists a lower adjoint g of f and an x in X such that y=f(x). For any u∈Y, by Proposition 32 (2), we have
(33)f~→(⇓gf(x))(u)=⋁v∈X⇓gf(x)(v)∧eY(u,f(v))=⋁v∈X⇓gf(x)(v)∧eX(g(u),v)≤⋁v∈X⇓gf(x)(g(u))≤⇓f(x)(u).
Obviously,f~→(B∧⇓gf(x))≤f~→(B).
Hence(34)f~→(B∧⇓gf(x))≤f~→(B)∧f~→(⇓gf(x))≤f~→(B)∧⇓f(x).
Furthermore, since B is a fuzzy lower set, then by Proposition 32 (1),
(35)(f~→(B)∧⇓f(x))(u)=(⋁v∈XB(v)∧eY(u,f(v)))∧⇓f(x)(u)≤(⋁v∈XB(v)∧eX(g(u),v))∧⇓gf(x)(g(u))=B(g(u))∧⇓gf(x)(g(u))≤⋁v∈XB(v)∧⇓gf(x)(v)∧eX(g(u),v)=⋁v∈XB(v)∧⇓gf(x)(v)∧eY(u,f(v))=f~→(B∧⇓gf(x))(u).
Therefore, f~→(B∧⇓gf(x))=f~→(B)∧⇓f(x). Note that f is fuzzy Scott continuous and B∧⇓gf(x)∈𝒟L(X), then
(36)⊔(f~→(B)∧⇓y)=⊔(f~→(B)∧⇓f(x))=⊔f~→(B∧⇓gf(x))=f(⊔(B∧⇓gf(x)))=f(gf(x))=f(x)=y.
It follows from Lemma 15 that f~→(B)∧⇓y=f~→(B)∧⇓f(x)=f~→(B∧⇓gf(x))∈𝒟L(Y). Thus f~→(B) is a fuzzy basis of Y. Therefore, by Theorem 22, (Y,eY) is a fuzzy domain.


## 5. Conclusion

In this paper, we propose the notion of a fuzzy basis in a fuzzy dcpo, which generalizes the concept of an ordinary basis. It provides a new approach to explore fuzzy domains. We can extend this approach further; for example, we can define a fuzzy complete basis on a fuzzy complete lattice [24] to investigate fuzzy completely distributive lattices introduced in [8, 13]. Moreover, in crisp setting, the definition of a wight is in close touch with the notion of a basis, and fuzzy Scott topology on fuzzy directed complete posets was given in [9]. As a followup of this paper, we can further give a fuzzy vision of a weight on fuzzy Scott topology and study its relative properties.



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