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Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy 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]. 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.06.018 Author's personal copy 974 W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 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 W. Yao / Fuzzy Sets and Systems 161 (2010) 973–987 975 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.