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Fuzzy Sets and Systems 161 (2010) 973–987 www.elsevier.com/locate/fss

Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy ଁ directed complete posets Wei Yao∗

Department of Mathematics, Hebei University of Science and Technology, 050018 Shijiazhuang, PR China

Received 1 November 2007; received in revised form 6 June 2009; accepted 8 June 2009 Available online 21 July 2009

Abstract This paper deals with quantitative domain theory via fuzzy sets. It examines the continuity of fuzzy directed complete posets (dcpos for short) based on complete residuated lattices. First, we show that a fuzzy partial order in the sense of Fan and Zhang and an L-order in the sense of Bˇelohlávek are equivalent to each other. Then we redefine the concepts of fuzzy directed subsets and (continuous) fuzzy dcpos. We also define and study fuzzy Galois connections on fuzzy posets. We investigate some properties of (continuous) fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy-double-downward-arrow-operator has a right adjoint. We define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos. We show that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation. © 2009 Elsevier B.V. All rights reserved.

Keywords: Fuzzy relations; Poset; Galois connection; Quantitative domain theory

1. Introduction

Domain theory, a formal basis for the semantics of programming languages, originated in work by Dana Scott [30,31] in the mid-1960s. Domain models for various types of programming languages, including imperative, functional, nondeterministic and probabilistic languages, have been studied extensively. Quantitative domain theory, which models concurrent systems, forms a new branch of domain theory, and has undergone active research in the past three decades. Rutten’s generalized (ultra)metric spaces [29], Flagg’s continuity spaces [11] and Wagner’s -categories [33] are examples of quantitative domain theory frameworks. 1 Recently, based on complete Heyting algebras, Fan and Zhang [10,39] studied quantitative domains through fuzzy set theory. Their approach first defines a fuzzy partial order, specifically a degree function, on a non-empty set. Then they define and study fuzzy directed subsets and (continuous) fuzzy directed complete posets (dcpos for short). Also in [6,7], in order to study fuzzy relational systems, B˘elohlávek defines and studies an L-order on a set. In fact, we can show that a fuzzy partial order in the sense of Fan–Zhang and an L-order in the sense of B˘elohlávek are equivalent to each other.

ଁ This paper is supported by the Foundation of Hebei University of Science and Technology (XL200821, QD200957). ∗ Tel.: +8631181668514. E-mail address: [email protected]. 1 Cited from [39].

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In [21,23], Lai and Zhang studied directed complete -categories, where is a commutative unital quantale. An -category is a non-empty set A together with an assignment of an element A(a, b) ∈ to every ordered pair of (a, b) ∈ A × A, such that (1) ∀a ∈ A, A(a, a) ≥ I ,whereI is the unit of ; (2) A(a, b) ∗ A(b, c) ≤ A(a, c)foralla, b, c ∈ A,where∗ is the tensor on . Roughly speaking, each -category could be considered a fuzzy preordered set in the sense of [10,39,40]. Many results have been obtained [21,23,39,40], but there are still some unsolved problems: (1) The definition of fuzzy directed subsets in [39,40] (Definition 2.6 in [39] and Definition 2.3 in [40], which is based on the well-below relation 2 ) looks relatively complex. 3 ˜ X Y (2) In [39,40], for two fuzzy posets (X, eX )and(Y, eY ) and a monotone map f : X −→ Y . The lift f : L −→ L of f is defined by X ˜ ∀A ∈ L , y ∈ Y, f (A)(y) = A(x) ∧ eY (y, f (x)). x∈X Then it is shown that for any fuzzy directed subset of X, f˜( ) is also a fuzzy directed subset of Y (Theorem 2.11 in [39], Theorem 2.7 in [40] and earlier, Lemma 12 in [11]). By the criterion of extension from crisp settings to fuzzy settings, for L ={0, 1}, f˜ should be the same as f. Meanwhile, for two crisp posets X and Y, a monotone map f : X −→ Y and any directed subset D of X,theimageofD under f˜ is not equal to f →(D) in general, but to ↓ f →(D). Also in [21,23], the authors showed that the image of a fuzzy ideal is also a fuzzy ideal (cf. Lemma 5.3 in [23]). The “lift” 4 in [23] is the same as that in [39,40] by replacing ∧ with the tensor ∗. For this reason, the lift of a map in [21,23,39,40] is not a good extension. 5 (3) The category of crisp dcpos is cartesian closed. It is natural to ask whether the category of fuzzy dcpos is also cartesian closed. (4) In [39,40], a crisp topology, namely the generalized Scott topology, is defined on a given fuzzy dcpo. Can we naturally construct an L-topology on a fuzzy dcpo just like a crisp topology on a crisp dcpo? (5) Many important and nice results in [39,40], especially the definition and results of the generalized Scott topology, are based on a completely distributive complete lattice with the top element ∨-irreducible and the well-below relation multiplicative 6 (see [39,40] for details). It is generally admitted that this condition is too strong. In fact, we can show that any finite lattice with a multiplicative well-below relation must be a chain. Thus many canonical finite lattices, for example, the simplest nontrivial Boolean algebra M2, cannot be supplied as an evaluating lattice in [39,40].

Theorem 1.1. Let L be a ﬁnite lattice with a multiplicative well-below relation Ó. Then L is a chain.

Proof. First, it is easy to see that for any non-zero element a ∈ L,0Óa and 0Ó/ 0. Suppose that two distinct elements a and b cover0.Thenwehavea ∧ b = 0. This contradicts the multiplicativity of Ó. Thus only one element covers 0. Call it a1. Similarly, we ﬁnd only one element a2 that covers a1. Thus, for every natural number n, L ={0, a1, ..., an} with 0 < a1 < ···< an = 1. ç

We will address the above-mentioned problems in a series of three papers. In Part I, we redefine the definition of fuzzy directed sets and fuzzy dcpos and then study the continuity of fuzzy dcpos. In Part II, we prove that, for complete Heyting algebras, the category of fuzzy dcpos is cartesian closed, where the morphisms are defined using L-valued Zadeh functions (differently from [21,23,39,40]). In Part III, for L a complete Heyting algebras, we define an

2 The well-below relation Ó (also called the wedge-below relation or the totally-below relation by some authors) is first introduced in [25].Ina complete lattice L, xÓy iff for each A ⊆ L, y ≤∨A implies x ≤ a for some a ∈ A. Also see Definition 7.1.2 in [1] and Exercise I-2.25 in [13] for the definition of Ó. 3 In fact, in [21,23] Lai and Zhang have already given a simple definition of fuzzy directed subsets (cf. Definition 5.1 in [23]). 4 This is not called a lift in [23]. The definition can be found in the last paragraph above Proposition 2.11 in [23] or Page 5 in [21]. 5 From a purely mathematical viewpoint, a fuzzy concept should have a crisp concept as a special case. 6 Ó is called multiplicative if aÓb, aÓc always implies aÓb ∧ c (See Definition 7.2.18 in [1] for a multiplicative way-below relation). Author's personal copy

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L-topology, namely the fuzzy Scott topology, on any given fuzzy dcpo. Some nice results are obtained. We also define and study the Scott convergence of L-filters in that part. This paper is Part I, and the results are based on complete residuated lattices. In Section 2, some basic concepts and notions which will be used throughout this paper are listed. In Section 3, we show that a fuzzy partial order in the sense of Fan–Zhang and an L-order in the sense of B˘elohlávek are equivalent to each other. In Section 4, the definition of a fuzzy Galois connection is proposed and studied. In Section 5, we redefine the concept of fuzzy directed subsets as well as that of (continuous) fuzzy dcpos. It is shown that a fuzzy dcpo is continuous if and only if the fuzzy-double- downward-arrow-operator and the fuzzy-joins-operator form a fuzzy Galois connection. In Section 6, a fuzzy auxiliary order is presented. We prove that a fuzzy dcpo is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation.

2. Preliminaries

Residuated lattices, introduced by Ward and Dilworth [34], are important algebraic structures associated with fuzzy logic [16]. A residuated lattice is an algebra (L;∧, ∨, ∗, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (R1) (L;∧, ∨, 0, 1) is a bounded lattice with the least (resp., greatest) element 0 (resp., 1); (R2) (L, ∗, 1) is a commutative monoid with the identity 1 and ∗ is isotone at both arguments (∗ is a t-norm); (R3) x ∗ y ≤ z iff x ≤ y → z 7 holds for all x, y, z ∈ L. A complete residuated lattice is a residuated lattice with a complete underlying lattice. A residuated lattice is called a Heyting algebra if ∗=∧. A complete Heyting algebra is also called a frame. In this part, unless otherwise stated, L always denotes a complete residuated lattice with the t-norm ∗ and the implication →. Let X be a nonempty set. The set L X denotes the set of all L-subsets of X.ThenL X is also a complete residuated lattice, where (A ∗ B)(x) = A(x) ∗ B(x), (A → B)(x) = A(x) → B(x). for all A, B ∈ L X and all x ∈ X. −→ → X −→ Y For each map f : X Y ,wehaveamap fL : L L (called the L-forward powerset operator, cf. [26,27]) deﬁned by → = ∀ ∈ , ∀ ∈ X .8 fL (A)(y) A(x)( y Y A L ) f (x)=y → ← The right adjoint to fL is denoted by fL (called the L-backward powerset operator, cf. [26,27]) and given by ∀ ∈ Y , ← = ◦ . B L fL (B) B f An L-relation E on X is an L-subset of X × X.AnL-relation E on X is called an L-preorder if (Ref) ∀x ∈ X, E(x, x) = 1; (Tran) ∀x, y, z ∈ X, E(x, y) ∗ E(y, z) ≤ E(x, z). An L-preorder E on X is called an L-equivalence if (Sym) ∀x, y ∈ X, E(x, y) = E(y, x). An L-equivalence E is called an L-equality if E(x, y) = 1 implies x = y. Let E be an L-equivalence and R an L-relation on X. R is said to be compatible with E [6,7] if for all x1, x2, y1, y2 ∈ X,

E(x1, y1) ∗ E(x2, y2) ∗ R(x1, x2) ≤ R(y1, y2).

3. Two equivalent deﬁnitions of fuzzy partial orders

Deﬁnition 3.1 (A fuzzy partial order in the sense of Bˇelohlávek [6,7]). A Bˇelohlávek-fuzzy partial order (a B-fpo for short) on a set X with an L-equality ≈ is an L-preorder e : X × X −→ L which is compatible w.r.t. ≈ and satisfying (B) (antisymmetry) ∀x, y ∈ X, e(x, y) ∧ e(y, x) ≤ (x ≈ y).

7 → is called the implication or the residuum with respect to ∗. 8 → fL is just the function resulting from Zadeh’s extension principle from f [42]. Author's personal copy

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Besides B-fpos, fuzzy partial orders are also deﬁned and studied in [8,9,15,32, 41] by using different notions of antisymmetry. By Lemma 4 in [7], we know that if e is a B-fpo on X that is compatible w.r.t. an L-equality ≈,then(x ≈ y) = e(x, y) ∧ e(y, x)forallx, y ∈ X. Hence there is a unique L-equality compatible with a B-fpo.

Deﬁnition 3.2 (A fuzzy partial order in the sense of Fan and Zhang [10,39,37]). A Fan–Zhang-fuzzy partial order (an FZ-fpo for short) on a set X is an L-preorder e : X × X −→ L satisfying (FZ) ∀x, y ∈ X, e(x, y) = e(y, x) = 1 implies x = y.

Remark 3.3. (1) In [10,39], Fan–Zhang’s original deﬁnition of a fuzzy partial order is based on Heyting algebras: that is, residuated lattices with ∗=∧. It is generalized onto residuated lattices in [37]. (2) For L = [0, 1] with ∗=min, an FZ-fpo is a dual to a quasi ultrametric in [29].

Obviously, a B-fpo is always an FZ-fpo, since (B) can imply (FZ). Conversely, let e be an FZ-fpo on X.Deﬁne ≈e: X × X −→ L by

(x ≈e y) = e(x, y) ∧ e(y, x)(∀x, y ∈ X).

It is easy to see that e satisﬁes (Ref), (Tran) and (B).

Proposition 3.4. ≈e is an L-equality on X and e is a B-fpo on X which is compatible w.r.t. ≈e.

Proof. (1) ≈e is an L-equality on X. We only need to show that (x ≈e y) ∗ (y ≈e z) ≤ (x ≈e z)forallx, y, z ∈ X.In fact,

(x ≈e y) ∗ (y ≈e z) = (e(x, y) ∧ e(y, x)) ∗ (e(y, z) ∧ e(z, y)) ≤ (e(x, y) ∗ e(y, z)) ∧ (e(z, y) ∗ e(y, x)) ≤ e(x, z) ∧ e(z, x)

= (x ≈e z).

(2) e is compatible with ≈e. In fact, for all x1, x2, y1, y2 ∈ X,

(x1 ≈e y1) ∗ (x2 ≈e y2) ∗ e(x1, x2) ≤ e(y1, x1) ∗ e(x2, y2) ∗ e(x1, x2) ≤ e(y1, y2). ç

Thus a B-fpo and an FZ-fpo are equivalent concepts. The deﬁnition of an FZ-fpo is formally simpler than that of a B-fpo. From now on, a B-fpo or an FZ-fpo will be just called a “fuzzy partial order”. For a fuzzy partial order e on X, we call the pair (X, e)(orjustX if there is no ambiguity) a fuzzy poset.

Proposition 3.5. e is a fuzzy partial order iff e compatible w.r.t. an L-equality ≈ and satisﬁes (Ref), (Tran) and (B )∀x, y ∈ X, e(x, y) ∗ e(y, x) ≤ (x ≈ y).

Proof. The necessity is obvious since (B) can be implied by (B). The sufficiency is also obvious, since if e satisfies (B), that can easily imply that e satisfies (FZ). ç

Example 3.6. (1) Define eL : L × L −→ L by eL (x, y) = x → y for all x, y ∈ L.TheneL is a fuzzy partial order on L [6]. (2) Let e be a fuzzy partial order on X.Theneop : X × X −→ L defined by eop(x, y) = e(y, x)(∀x, y ∈ X)isalso a partial order on X, called the dual fuzzy partial order of e. (3) Let e be a fuzzy partial order on X.Then≤e={(x, y) ∈ X × X| e(x, y) = 1} is a crisp partial order on X. Let (X, ≤) be a crisp poset and define e≤ : X × X −→ L by e≤(x, y) = 1ifx ≤ y and 0 otherwise. Then e≤ is a fuzzy partial order on X [6]. Author's personal copy

W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 977 (4) Let {(Xi , ei )| i ∈ I } be a nonempty family of fuzzy posets. Put X = Xi and deﬁne eX : X × X −→ L by i∈I ∀x = (xi )i∈I , y = (yi )i∈I ∈ X, eX (x, y) = ei (xi , yi ). i∈I

Then eX is a fuzzy partial order on X, called the product fuzzy partial order of {ei | i ∈ I }. (5) Let (X, e) be a fuzzy poset and Y ⊆ X.Thene|Y is a fuzzy partial order on Y,wheree|Y is the restriction of e to Y × Y . ∀ , ∈ X , = → 8 (6) A B L , the subsethood degree [14] of A in B is deﬁned by subX (A B) x∈X A(x) B(x). Then X X X subX : L × L −→ L a fuzzy partial order on L [6].

The following deﬁnitions and propositions can be found in [6,7,10,37–40].

Proposition 3.7. Let (X, e) be a fuzzy poset. Then for all x, y ∈ X, e(x, y) = e(z, x) → e(z, y) = e(y, z) → e(x, z). z∈X z∈X

Deﬁnition 3.8. Let (X, e) be a fuzzy poset and A ∈ L X . (1) Au ∈ L X is deﬁned by ∀x ∈ X, Au(x) = A(y) → e(y, x). y∈X

(2) Al ∈ L X is deﬁned by ∀x ∈ X, Al (x) = A(y) → e(x, y). y∈X

(3) For all A ∈ L X ,supA ∈ L X (resp., inf A ∈ L X )isdeﬁnedby

sup A = Au ∧ Aul (resp., inf A = Al ∧ Alu).

X Deﬁnition 3.9. Let (X, e) be a fuzzy poset, x0 ∈ X, A ∈ L . The element x0 is called a join (resp., meet) of A,in symbols x0 =A (resp., x0 =A), if ∀ ∈ , ≤ , ≤ , (1) x X A(x) e(x x0) (resp., A(x) e(x0 x)); ∀ ∈ , → , ≤ , → , ≤ , (2) y X x∈X A(x) e(x y) e(x0 y) (resp., x∈X A(x) e(y x) e(y x0)).

X It is easy to verify by (FZ) that if x1, x2 are two joins (or meets) of A,thenx1 = x2. That is, each A ∈ L has at most one join (or meet). = = ∀ ∈ , , = → , Proposition 3.10. (1) x0 Aiff sup A(x0) 1 iff y X e(x0 y) x∈X A(x) e(x y). = = ∀ ∈ , , = → , (2) x0 Aiff inf A(x0) 1 iff y X e(y x0) x∈X A(x) e(y x).

Deﬁnition 3.11. A fuzzy poset (X, e) is called “complete” if for all A ∈ L X , A and A exist. , = ∗ = → For example, (L eL ) is a complete fuzzy poset, where A x∈X A(x) x and A x∈X A(x) x for all A ∈ L X .

8 The symbol subX is denoted by S in [3,12] and subs in [4,14]. Author's personal copy

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Proposition 3.12. Let (X, e) be a fuzzy poset. The following statements are equivalent: (1) (X, e) is complete; (2) For any A ∈ L X , Aexists; (3) For any A ∈ L X , Aexists.

Deﬁnition 3.13. A ∈ L X is called a fuzzy upper set (or a fuzzy lower set) if ∀x, y ∈ X, A(x) ∗ e(x, y) ≤ A(y)(or A(x) ∗ e(y, x) ≤ A(y)). ∈ X ∀ ∈ = , → Proposition 3.14. A L is a fuzzy upper (or lower) set if and only if x X, A(x) y∈X e(x y) A(y)(or = , → A(x) y∈X e(y x) A(y)).

Deﬁnition 3.15. For x ∈ X, ↓ x ∈ L X (or ↑ x ∈ L X )isdeﬁnedas∀y ∈ X, ↓ x(y) = e(y, x)(or↓ x(y) = e(x, y)).

An element x ∈ X is called the maximal (or minimal) element of A ∈ L X , in symbols x = max A (or x = min A), if A(x) = 1andforally ∈ X, A(y) ≤ e(y, x)(orA(y) ≤ e(x, y)). It is easy to see that if A has a maximal (or minimal) element, then it is unique.

Proposition 3.16. (1) x = max AiffA(x) = 1 and x =A. (2) ∀x ∈ X, x = max ↓ x. (1) x = min AiffA(x) = 1 and x =A. (2) ∀x ∈ X, x = min ↑ x.

4. Fuzzy Galois connections

This section is also included in our paper [37]. In this section, in order to establish a foundation for the next section, we propose a deﬁnition of a fuzzy Galois connection on fuzzy posets and study its properties. Note that before [37],a fuzzy Galois connection of an antitone form on fuzzy posets was already given in [5]. Let (X, eX )and(Y, eY ) be two fuzzy posets. We call a map f : X −→ Y “order-preserving” or “(L-fuzzy) monotone” (or antitone) if eX (x, y) ≤ eY ( f (x), f (y)) (or eX (x, y) ≤ eY ( f (y), f (x))) for all ∀x, y ∈ X. Two fuzzy posets (X, eX ) −1 and (Y, eY ) are said to be isomorphic if there exists a bijection f : X −→ Y such that both f and f are order-preserving. The map f is called an isomorphism.

Deﬁnition 4.1. Let (X, eX ), (Y, eY ) be two fuzzy posets and f : X −→ Y , g : Y −→ X two order-preserving maps. The pair ( f, g) is called a fuzzy Galois connection between X and Y if

eY ( f (x), y) = eX (x, g(y)) for all x ∈ X, y ∈ Y ,wheref is called the left adjoint of g and dually g the right adjoint of f.

Proposition 4.2. ( f, g) is a fuzzy Galois connection on (X, eX ) and (Y, eY ) iff both f and g are order-preserving and , 9 , ≤ , ≤ ( f g) is a crisp Galois connection on (X eX ) and (Y eY ). , , ≤ Proof. The necessity is obvious. To show the sufﬁciency, suppose that ( f g) is a Galois connection on (X eX )and , ≤ (Y eY ), that is,

eX (x, gf(x)) = eY ( fg(y), y) = 1, for all x ∈ X, y ∈ Y .Then

eY ( f (x), y) ≤ eX (gf(x), g(y)) = eX (gf(x), g(y)) ∗ eX (x, gf(x)) ≤ eX (x, g(y)).

Similarly, we have eX (x, g(y)) ≤ eY ( f (x), y). ç

9 The deﬁnition and properties of (crisp) Galois connections can be found in [13]. Author's personal copy

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Remark 4.3. (1) Obviously, a fuzzy Galois connection is an extension of a crisp Galois connection. Let f : X −→ Y → ← be a map and fL , fL are the L-forward powerset operator and the L-backward powerset operator of f, respectively. →, ← X , Y , By Proposition 4.2, it easy to see that ( fL fL ) is a fuzzy Galois connection between (L subX )and(L subY ). (2) In [2,4],B˘elohlávek introduced an L-Galois connection between X and Y provided by a pair of maps f : L X − → LY and g : LY −→ L X such that both f and g is fuzzy antitone and A ≤ gf(A), B ≤ fg(B)(∀A ∈ L X , B ∈ LY ). It is easy to see that a B˘elohlávek’s L-Galois connection between X and Y is just a fuzzy Galois connection between X , Y , op (L subX )and(L subY ) in Definition 4.1. (3) In [5],anLK -Galois connection on fuzzy posets is also proposed. For an upper set K ⊆ L (called a ≤-filter of L), an LK -Galois connection between two fuzzy posets (X, eX )and(Y, eY ) is a pair of maps ( f : X −→ Y, g : Y −→ X) that satisfies

(i) ∀x1, x2 ∈ X, eX (x1, x2) ≤ eY ( f (x1), f (x2)), whenever eX (x1, x2) ∈ K ; (ii) ∀y1, y2 ∈ Y , eY (y1, y2) ≤ eX (g(y1), g(y2)), whenever eY (y1, y2) ∈ K ; (iii) ∀x ∈ X, y ∈ Y , eX (x, gf(x)) = eY (y, fg(y)) = 1.

By Proposition 4.2, a fuzzy Galois connection between two fuzzy posets (X, eX )and(Y, eY ) is exactly an LK -Galois , , op = connection between (X eX )and(Y eY )forK L. (4) In [20,23,33],fortwo-categories A and B, a pair of -functors f : A −→ B and g : B −→ A is said to be an -adjunction if

B( f (a), b) = A(a, g(b)), for all a ∈ A, b ∈ B (cf. Deﬁnition 2.9 in [23]). A fuzzy Galois connection in Deﬁnition 4.1 is an L-adjunction in the sense of [20,23,33].

Theorem 4.4. Let f :(X, eX ) −→ (Y, eY ) and g :(Y, eY ) −→ (X, eX ) be two maps. Then the following statements are equivalent. (1) ( f, g) is a fuzzy Galois connection. = ← ↓ ∈ (2) f is order-preserving and g(y) max fL ( y) for all y Y . = ← ↑ ∈ (3) g is order-preserving and f (x) min gL ( x) for all x X.

Proof. We only show (1) ⇔ (2). ⇒ ← ↓ =↓ = , = ∈ ← ↓ (1) (2): First, fL ( y)(g(y)) y( fg(y)) eY ( fg(y) y) 1. Second, for all x X,wehave fL ( =↓ = , = , = ← ↓ y)(x) y( f (x)) eY ( f (x) y) eX (x g(y)). Hence g(y) max fL ( y). ⇒ = ← ↓ ∈ (2) (1): Since g(y) max fL ( y), then for all x X,wehave , =↓ = ← ↓ ≤ , . eY ( f (x) y) y( f (x)) fL ( y)(x) eX (x g(y)) , =↓ = ← ↓ = And since eY ( fg(y) y) y( fg(y)) fL ( y)g(y) 1, we have

eX (x, g(y)) ≤ eY ( f (x), fg(y)) = eY ( f (x), fg(y)) ∗ eY ( fg(y), y) ≤ eY ( f (x), y).

Thus eY ( f (x), y) = eX (x, g(y)). , ∈ , ∗ ← ↓ ≤ ← ↓ Also for all y1 y2 Y , it is easy to verify that eY (y1 y2) fL ( y1) fL ( y2). Then , = , = ← ↓ eX (g(y1) g(y2)) eX ( fg(y1) y2) fL ( y2)(g(y1)) ≥ , ∗ ← ↓ = , ∗ = , eY (y1 y2) fL ( y1)(g(y1)) eY (y1 y2) 1 eY (y1 y2) = ← ↓ , ç since g(y1) max fL ( y1). Thus, g is order-preserving. Hence ( f g) is a fuzzy Galois connection.

Theorem 4.5. Let f :(X, eX ) −→ (Y, eY ) and g :(Y, eY ) −→ (X, eX ) be two order-preserving maps. Then = → ∈ X (1) If X is complete, then f is order-preserving and has a right adjoint if and only if f ( A) fL (A) for all A L . = → ∈ Y (2) If Y is complete, then g is order-preserving and has a left adjoint if and only if g( B) gL (B) for all B L . Author's personal copy

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Proof. We only prove (1). Suppose that f is order-preserving and has a right adjoint g.Forally ∈ Y , → → , fL (A)(z) eY (z y) z∈Y ⎛⎛ ⎞ ⎞ ⎝⎝ ⎠ ⎠ = A(x) → eY (z, y) ∈ = zY f (x) z = A(x) → eY ( f (x), y) ∈ = zY f (x) z = A(x) → eX (x, g(y)) ∈ = zY f (x) z = A(x) → eX (x, g(y)) x∈X = eX (A, g(y)) = eX ( f (A), y). = → Hence f ( A) fL (A). = → ∈ X , ∈ =↓ = ↓ Conversely, suppose that f ( A) fL (A)forallA L .Forallx y X, put A y,wehave f (y) f ( = → ↓ y) fL ( y). Then , ≥ → ↓ = ↓ ≥ , . eY ( f (x) f (y)) fL ( y)( f (x)) y(a) eX (x y) f (a)= f (x) −→ = ← ↓ ∈ Thus f is order-preserving. Put g : Y X by g(y) fL ( y)foranyy Y .Then e (g(y ), g(y )) = g( f ←(↓ y ), f ←(↓ y )) X 1 2 L 1 L 2 = ← ↓ → , ← ↓ fL ( y1)(x) eX (x fL ( y2)) x∈X ≥ ← ↓ → ← ↓ fL ( y1)(x) fL ( y2)(x) . x∈X = eY ( f (x), y1) → eY ( f (x), y2) x∈X ≥ eY (y1, y2). Thus g is also order-preserving. Also, since = ← ↓ = → ← ↓ , fg(y) f ( fL ( y)) fL fL ( y) we have , = → ← ↓ , eY ( fg(y) y) eY ( fL fL ( y) y) = → ← ↓ → , fL fL ( y)(z) eY (z y) z∈Y = eY ( f (x), y) → eY (z, y) z∈Y f (x)=z = eY ( f (x), y) → eY (z, y) z∈Y f (x)=z = eY ( f (x), y) → eY ( f (x), y) x∈X = 1.

Also, = ← ↓ , g( f (x)) fL ( f (x)) Author's personal copy

W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 981 and then

, ≥ ← ↓ =↓ = . eX (x gf(x)) fL ( f (x))(x) f (x)( f (x)) 1

By Proposition 4.2, we conclude that ( f, g) is a fuzzy Galois connection. ç

5. Fuzzy dcpos and their continuity

X Definition 5.1 (Definition 5.1 in Lai and Zhang [23]). D ∈ L is called a fuzzy directed set if (FD1) D(x) = 1; x∈X ∀ , ∈ , ∗ ≤ ∗ , ∗ , (FD2) x y X D(x) D(y) z∈X D(z) e(x z) e(y z). A fuzzy directed set I ∈ L X is called a fuzzy ideal if it is also a fuzzy lower set. The set of all fuzzy directed sets (resp., fuzzy ideals) of (X, e) is denoted by L (X) (resp., L (X)). D I A fuzzy directed subset of a fuzzy poset (X, e)in[39,40] is an L-subset A ∈ L X that satisfies (ZFD1) there exists an x ∈ X such that 0ÓA(x); (ZFD2) for all x1, x2 ∈ X, a1, a2, a ∈ L such that a1ÓA(x1), a2ÓA(x2)andaÓ1, there exists an x ∈ X such that aÓA(x), a1Óe(x1, x), a2Óe(x2, x).

Remark 5.2. (1) Condition (FD1) can be interpreted as revealing that each fuzzy directed subset is nonempty. Condition (FD2) can be interpreted as saying that for each fuzzy directed subset D and any a, b ∈ D, there exists c ∈ D such that a ≤ c, b ≤ c. (2) Generally, a fuzzy directed subset in this paper need not be one in [39,40],evenifL is the simplest nontrivial chain {0, a, 1} (0 < a < 1). Obviously, {0, a, 1} satisfies all the conditions required in [39,40], that is, it is a completely distributive lattice with the top element ∨-irreducible and the well-below relation multiplicative. Furthermore, aÓa, X 1Ó1. Let X ={x1, x2} and e(x, y) = 1ifx = y and 0 otherwise. Then e is a fuzzy partial order on X.GivenA ∈ L by A(x1) = a, A(x2) = 1, it is easy to see that A is fuzzy directed subset in the sense of Definition 5.1. Let a1 = a, a2 = 1. Then a1ÓA(x1), a2ÓA(x2), while for any x ∈ X, either e(x1, x) = 0ore(x2, x) = 0 and it follows that a1Ó/ e(x1, x)ora2Ó/ e(x2, x). (3) If L is a completely distributive lattice with a multiplicative well-below relation, then a fuzzy directed subset in [39,40] always implies one in Definition 5.1. In fact, ∈ ∀ , Ó , ∀ Ó ∈ (FD1) by (ZFD1), there exists some x1 X such that A(x1) 0. a1 a2 A(x1) a 1, there exists x X Ó ≤ Ó = = { ∈ such that a A(x)andthena A(x). By the arbitrariness of a 1, we have x∈X A(x) 1 (note that 1 a L| aÓ1}). (FD2) ∀x1, x2 ∈ X, suppose that A(x1) ∧ A(x2) 0. ∀aÓA(x1) ∧ A(x2), we have aÓA(x1), aÓA(x2), then there exists x ∈ X such that aÓA(x), aÓe(x1, x), aÓe(x2, x). It follows that a ≤ A(x) ∧ e(x1, x) ∧ e(x2, x)bythe multiplication of Ó. By the arbitrariness of a, (FD2) holds.

→ Proposition 5.3. Let f :(X, eX ) −→ (Y, eY ) be an order-preserving map. Then ∀D ∈ L (X), f (D) ∈ L (Y ). D L D → = = = Proof. (1) y∈Y fL (D)(y) y∈Y f (x)=y D(x) x∈X D(x) 1. (2) ∀y1, y2 ∈ Y ,

→ → f (D)(y1) ∗ f (D)(y2) L L = D(x1) ∗ D(x2) f (x1)=y1, f (x2)=y2 ≤ D(x) ∗ eX (x1, x) ∗ eX (x2, x) f (x1)=y1, f (x2)=y2 x∈X Author's personal copy

982 W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 ≤ D(x) ∗ eY (y1, f (x)) ∗ eY (y2, f (x)) = , = ∈ f (x1) y1 f (x2) y2 x X = D(x) ∗ eY (y1, f (x)) ∗ eY (y2, f (x)) x∈X ≤ D(x) ∗ eY (y1, y) ∗ eY (y2, y) ∈ = yY f (x) y = → ∗ , ∗ , . ç fL (D)(y) eY (y1 y) eY (y2 y) y∈Y

Deﬁnition 5.4. (X, e) is called a fuzzy dcpo if every fuzzy directed subset has a join.

X For a complete lattice L,amany-valuedL-topology in the sense of [17] is a subset ⊆ L that satisﬁes (i) 0X , 1X ∈ ; ∀ , ∈ X ∗ ∈ X ∀{ | ∈ }⊆ ∈ (ii) A B L , A B L ; (iii) Ai i I , i Ai . It is easy to see that is also a complete residuated lattice.

Example 5.5. Let be a many-valued L-topology on X.PutLpt() ={p : −→ L| p preserves arbitrary joins and X p(A ∗ B) ≥ p(A) ∗ p(B)forallA, B ∈ L }.DeﬁneeLpt : Lpt() × Lpt() −→ L by ∀ f, g ∈ Lpt(), eLpt( f, g) = f (U) → g(U). U∈

Then (Lpt(), eLpt) is a fuzzy dcpo.

Proof. Suppose that D ∈ L Lpt() is a fuzzy directed set. Deﬁne f : −→ L by ∀A ∈ , f (A) = D(g) ∗ g(A). g∈Lpt()

Step 1: f ∈ Lpt(). (a) Obviously, f (0X ) = 0and f (1X ) = D(g) = 1. g∈Lpt() ∀{ | ∈ }⊆ ∅ = ∗ (b) Ai i I (I ), f ( i Ai ) i f (Ai )since is distributive over arbitrary joins. (c) ∀A, B ∈ ,wehave f (A) ∗ f (B) = D(h1) ∗ h1(A) ∗ D(h2) ∗ h2(B) h1∈Lpt() h2∈Lpt() = h1(A) ∗ h2(B) ∗ D(h1) ∗ D(h2) h ,h ∈Lpt() 1 2 ≤ h1(A) ∗ h2(B) ∗ D(g) ∗ eLpt(h1, g) ∗ eLpt(h2, g) h1,h2∈Lpt() g∈Lpt() = h1(A) ∗ eLpt(h1, g) ∗ h2(B) ∗ eLpt(h2, g) ∗ D(g) g∈Lpt() h1,h2∈Lpt() ≤ g(A) ∗ g(B) ∗ D(g) g∈Lpt() h1,h2∈Lpt() ≤ g(A ∗ B) ∗ D(g) g∈Lpt() = f (A ∗ B). Author's personal copy

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Step 2: f =D. In fact, for all g ∈ Lpt(), D(h) → eLpt(h, g) = D(h) → h(A) → g(A) ∈ ∈ ∈ h Lpt( ) hLpt( ) A = (D(h) ∗ h(A)) → g(A) ∈ A∈ h Lpt( ) = (D(h) ∗ h(A)) → g(A) ∈ ∈ A h Lpt( ) = f (A) → g(A) A∈ = eLpt( f, g). ç

Deﬁnition 5.6. Let (X, e) be a fuzzy dcpo. For any x ∈ X,deﬁne⇓ x ∈ L X by ∀y ∈ X, ⇓ x(y) = e(x, I ) → I (y).

I∈ L (X) I

A fuzzy dcpo is called continuous or a fuzzy domain if ⇓ x ∈ L (X)andx =⇓x for all x ∈ X. I We call ⇓: X −→ L X is called the fuzzy-double-download-arrow-operator and call ⇓: X × X −→ L the fuzzy way-below relation on X. ∀x, y ∈ X, ⇓ x(y) can be interpreted as the degree of y being way-below x. ∀ ∈ ∈ , ≤ Proposition 5.7. (1) x X, I L (X), y∈X e(x y) I (x). ∀ , ∈ , ≤⇓I (2) x y X, z∈X e(x z) y(x). (3) ∀x ∈ X, ⇓ x ≤↓ x. (4) ∀x, u,v,y ∈ X, e(u, x)∗⇓y(x) ∗ e(y,v) ≤⇓ v(u).

Proof. (1) e(x, y) = e(x, y) ∗ I (z) ∈ ∈ ∈ y X y X z X = e(x, y) ∗ I (z) ∈ ∈ zX y X ≤ (e(x, z) ∗ I (z)) z∈X ≤ I (x). (2) ⇓ y(x) = e(y, I ) → I (x) ≥ I (x) ≥ e(x, z). I∈ L (X) I ∈ L (X) z∈X (3) ⇓ x(y) ≤ e(x, I↓x) →↓ x(y) = e(x, x) → Ie(y, x) = e(y, x) =↓ x(y). (4) ∀I ∈ L (X), I e(v,I ) ∗ e(u, x) ∗ e(y,v) ∗ (e(y, I ) → I (x)) ≤ e(y, I ) ∗ (e(y, I ) → I (x)) ∗ e(u, x) ≤ e(u, x) ∗ I (x) ≤ I (u), and

e(u, x) ∗ e(y,v) ∗ (e (y, I ) → I (x)) ≤ e (v,I ) → I (u). Author's personal copy

984 W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987

Then e(u, x)∗⇓y(x) ∗ e(y,v) = e(u, x) ∗ e(y,v) ∗ e (y, I ) → I (x) ∈ I L (X) ≤ e(u, x) ∗ e(y,vI) ∗ (e(y, I ) → I (x)) I∈ L (X) ≤ I e(v,I ) → I (u) I∈ L (X) =⇓Iv(u). ç

Remark 5.8. In Proposition 5.7, (1) can be interpreted as saying that the least element (if it exists) that is contained in any ideal, since each ideal is nonempty; (2) can be interpreted as saying that the least element (if it exists) is way below any element; (3) can be interpreted as saying that the way-below relation always implies the less-than-or-equals relation; (4) can be interpreted as the transitivity of the way-below relation.

Theorem 5.9 (Theorem 4.6 in Lai and Zhang [21], Property of interpolation). If (X, e) is a fuzzy domain, then ⇓ = ⇓ ∗⇓ , ∈ y(x) z∈X z(x) y(z) for all x y X.

Proof. Proposition 5.7(3)(4) implies that the right-hand side of the equation is less than or equal to the left-hand side. On the other hand, deﬁne D ∈ L X by ∀a ∈ X, D(a) = ⇓ z(a)∗⇓y(z). z∈X We only need to show that ⇓ y(x) ≤ D(x). Step 1: D is a fuzzy ideal. (1) D(a) = ⇓ z(a)∗⇓y(z) = ⇓ z(a)∗⇓y(z) a∈X a∈X z∈X z∈X a∈X = ⇓ y(z) ∗ ⇓ z(a) = ⇓ y(z) = 1, z∈X a∈X z∈X

since ∀x ∈ X, ⇓ x ∈ L (X). (2) ∀a, b ∈ X, I D(a) ∗ e(b, a) = ⇓ z(a)∗⇓y(z) ∗ e(b, a) ≤ ⇓ z(b)∗⇓y(z) = D(b). z∈X z∈X Thus D is a fuzzy lower set. (3) ∀a, b ∈ X, by Proposition 5.7(4), D(a) ∗ D(b) = ⇓ a1(a)∗⇓b1(b)∗⇓y(a1)∗⇓y(b1) a1,b1∈X ≤ ⇓ a1(a)∗⇓b1(b)∗⇓y(c) ∗ e(a, c) ∗ e(b, c) a1,b1∈X c∈X = (⇓ a1(a) ∗ e(a, c)) ∗ (⇓ b1(b) ∗ e(b, c))∗⇓y(c) a1,b1∈X c∈X ≤ ⇓ c(a)∗⇓c(b)∗⇓y(c) a1,b1∈X c∈X = ⇓ c(a)∗⇓c(b)∗⇓y(c) c∈X ≤ ⇓ c(d) ∗ e(a, d) ∗ e(b, d)∗⇓y(c) c∈X d∈X = e(a, d) ∗ e(b, d) ∗ ⇓ c(d)∗⇓y(c) d∈X c∈X = e(a, d) ∗ e(b, d) ∗ D(d). d∈X Author's personal copy

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Step 2: y =D. In fact, ∀a ∈ X, D(z) → e(z, a) = (⇓ a1(z)∗⇓y(a1)) → e(z, a) z∈X z∈X a1∈X = ⇓ y(a1) → (⇓ a1(z) → e(z, a)) ∈ ∈ a1X z X = ⇓ y(a1) → e(⇓a1, a) a1∈X = ⇓ y(a1) → e(a1, a) a1∈X = e(y, a). Step 3: ⇓ y(x) = e(y, I ) → I (x) ≤ e(y, D) → D(x) = 1 → D(x) = D(x). ç I∈IL (X)

Theorem 5.10. A fuzzy dcpo (X, e) is continuous iff (⇓, ) is a fuzzy Galois connection between (X, e) and ( L (X), subX). I

Proof. ⇒: Obviously, both ⇓ and are order-preserving. ∀x ∈ X, I ∈ L (X), I subX (⇓ x, I ) = ⇓ x(y) → I (y) ∈ y X ≥ (e(x, I ) → I (y)) → I (y) ≥ e(x, I ). y∈I Also, e (x, I ) = ⇓ x(y) → e (y, I ) ≥ ⇓ x(y) → I (y) = subX(⇓ x, I ). y∈X y∈X

⇐: ∀x ∈ X, ⇓ x ∈ L (L)ande(x, ⇓x) = 1. Then x ≤⇓x ≤↓x = x and x =⇓x. ç I 6. Fuzzy auxiliary relations

Deﬁnition 6.1. Let (X, e) be a fuzzy poset. An L-relation ≺ on X is called a fuzzy auxiliary relation if it satisﬁes the following conditions for all u, x, y,v ∈ X: (FAu1) ≺ (x, y) ≤ e(x, y); (FAu2) e(u, x)∗≺(x, y) ∗ e(y,v) ≤≺ (u,v); , ≤≺ , (FAu3) z∈X e(x z) (x y). The set of all fuzzy auxiliary relations on X is denoted by FAux(X).

By Proposition 5.7, the fuzzy way-below relation on a (resp., continuous) fuzzy dcpo is a (resp., an approximating) fuzzy auxiliary relation. X Let FLow(X) be the set of all fuzzy lower sets of (X, e), equipped with the induced fuzzy order of (L , subX ). Let M be the set of order-preserving maps s :(X, e) −→ (FLow(X), subX ) satisfying s(x) ≤↓ x for all x ∈ X.Deﬁnea fuzzy order eM on M by ∀s, t ∈ M, eM (s, t) = subX (s(x), t(x)). x∈X

Deﬁnition 6.2. A fuzzy auxiliary relation ≺ on a fuzzy dcpo X is called “approximating” iff ∀x ∈ X, s≺(x) is fuzzy directed (hence a fuzzy ideal) and x =s≺(x). The set of all approximating fuzzy auxiliary relations on a dcpo X is denoted by FApp(X).

Proposition 6.3. The assignment

≺ s≺ =≺ (·, x) Author's personal copy

986 W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 is a well-deﬁned isomorphism from (FAux(X), subX×X ) to (M, eM ), whose inverse associates to each map s ∈ M, the L-relation ≺s given by

≺s (x, y) = s(y)(x).

Proof. It is straightforward to verify that both ≺ s≺ : FAux(X) −→ M and s ≺s: M −→ FAux(L) are order ∀ ∈ , ≺∈ ≺ , = =≺ , ≺ =≺ =≺ preserving maps. s M FAux(L), s≺ (x y) s≺(y)(x) (x y). Hence s≺ .Also,s≺s (x)(y) s , = , = ç (y x) s(x y). Hence, s≺s s.

Proposition 6.4. In a fuzzy dcpo (X, e), the fuzzy way-below relation is contained in all approximating fuzzy auxiliary relations.

Proof. ∀x, y ∈ X, ⇓ (x, y) = e (y, I ) → I (x) ≤ e (y, ≺(·, y)) →≺ (·, y)(x) =≺ (x, y). ç

I∈ L (X) I By Proposition 6.4, we have

Theorem 6.5. Let (X, e) be a fuzzy dcpo. Then X is continuous if and only if the fuzzy way-below relation is the smallest approximating fuzzy auxiliary relation on X.

7. A remark and some further work

An additional remark: Waszkiewicz [35] also studied the continuity of -valued posets for is a Girard quantale. In [35], a fuzzy ideal is defined by nets (cf. Definition 2.4). For a map betweenQ two -valued posets,Q its lift,whichis used to define the Scott continuity, is also defined like in the sample in [21,23,39,40]Q(cf. Paragraph 1 in Section 2.6 for lift and Definition 2.6 for Scott continuity in [35]). Some further work: In this paper, we define a continuous fuzzy dcpo by degree of way-below relations. We extend this approach further, for example, 1. We can define and study an algebraic fuzzy dcpo. Let (X, e) be a dcpo. Define a map k : X −→ L X by ∀x ∈ X, ∀y ∈ X, k(x)(y) = e(y, x)∗⇓y(y). If for each x ∈ X, k(x) is fuzzy directed and x =k(x), then we call (X, e) an algebraic fuzzy dcpo. For x, y ∈ X, k(x)(y) can be interpreted as requiring the degree of y to be compact and less than or equal to x. 2. Fuzzy suggests many-valued in some sense. Many-valued completely distributive -categories are studied in [22]. We also can study completely distributive fuzzy posets using the approach in this paper. Let (X, e) be a complete fuzzy poset. Define ¨ : X × X −→ L by ∀x, y ∈ X, ¨(x, y) = e(y, A) → A(a) ∗ e(x, a) . A∈L X a∈X We call (X, e) completely distributive if ∀x ∈ X, x =¨(·, x). Clearly, the L-relation ¨ is a generalization of the well-below relation on complete lattices. 3. Since a crisp frame (locale) can be characterized by Galois connections, we can further define a fuzzy frame (locale) with the fuzzy Galois connections in Section 4, and then study Stone representations [19] under the framework of fuzzy posets (cf. a categorical approach in [24] and a kind of fuzzy approach in [43]).

Acknowledgements

The author would like to thank Prof. S.E. Rodabaugh, the Area Editor, and the anonymous reviewers for their valuable comments and suggestions. The author would also like to thank the anonymous language editor for correcting the language throughout this paper. Author's personal copy

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