Fuzzy Bases of Fuzzy Domains

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.


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][11][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 Section 2, 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 2 Journal of Applied Mathematics image of a fuzzy domain is also a fuzzy domain. Conclusions are settled in the last section.

Preliminary
A frame will be used as the structures of truth values in this paper. Throughout this paper, unless otherwise stated, always denotes a frame. For more properties about frames, we refer to [3,17,18].
Let be a nonempty set, an -subset on is a mapping from to , and the family of all -subsets on will be denoted by . We denote the constant -subsets on taking the values 0 and 1 by 0 and 1 , respectively. Let , ∈ . The equality of and is defined as the usual equality of mappings; that is, = ⇔ ( ) = ( ) for any ∈ . The inclusion ≤ is also defined pointwisely: ≤ ⇔ ( ) ≤ ( ) for any ∈ .
Furthermore, can be always lifted as̃→ : → , which is defined by for any ∈ , ∈ , 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].
A fuzzy ideal is a fuzzy lower directed subset. We denote the set of all fuzzy directed subsets and all fuzzy ideals on by D ( ) and I ( ), respectively. A fuzzy poset is called a fuzzy dcpo if every fuzzy directed subset has a join. 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.
The crisp Galois connection is defined as follows: ≤ ( ) ⇔ ( ) ≤ for any ∈ , ∈ , and its relative properties can be found in [3]. (1) If is a fuzzy monotone mapping and has a lower adjoint, then for any ∈ such that ⊓ exists, (⊓ ) = ⊓ → ( ).
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.
That is, Proof. Obviously, for any ∈ I ( ), ∈ D ( ). On the one hand, On the other hand, for any ∈ D ( ), it is routine to check that ↓ ∈ I ( ). Then     Journal of Applied Mathematics 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].

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. Obviously, the previous definition is really a generation of the notion of a basis in [3].

Theorem 22. A fuzzy dcpo has a fuzzy basis if and only if it is a fuzzy domain.
Proof. Necessity. Suppose that is a fuzzy basis of , then for any ∈ , ∧ ⇓ ∈ D ( ) with = ⊔( ∧ ⇓ ). It is clear that ∧ ⇓ ≤⇓ . Thus ⇓ ∈ D ( ) with = ⊔ ⇓ follows from Proposition 21. Therefore, ( , ) is a fuzzy domain. Sufficiency. It is easy to check that 1 is a fuzzy basis of .
Since for any ∈ , ↓ is a fuzzy lower set. Then we can deduce the following.

Corollary 26. If 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.

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 (1) If is fuzzy Scott continuous, then preserves fuzzy way-below relation.

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.