Abstract
We introduce a new construction—-domain—and prove that the category with -domains as objects and Scott continuous functions as morphisms is a Cartesian closed category. We obtain that the Plotkin powerdomain over an -domain is an -domain.
1. Introduction
Powerdomains are very important structures in Domain theory, which play an important role in modeling the semantics of nondeterministic programming languages. Three classical powerdomains are the Hoare or lower powerdomain [1], the Smyth or upper powerdomain [2], and the Plotkin or convex powerdomain [3]. They are all free dcpo-algebras over (continuous) dcpos with special binary operators satisfying some equations and inequalities (see [4–12]).
In [13], Huth et al. concluded that the Hoare powerdomain over a pointed domain is a distributive -lattice. In [14], Meng and Kou obtained that the Smyth powerdomain of a Lawson compact domain is an -domain. Then we have a problem whether the Plotkin powerdomain can be characterized by some special -domain. In this paper, we will introduce a new domain construction called the -domain which is a +-semilattice and there exists a directed family of finitely separated Scott continuous and +-semilattice homomorphisms which can approximate , where the operation + is Scott continuous which it satisfied the commutative, associative, and idempotency laws. And the category with -domains as objects and Scott continuous functions as morphisms is a Cartesian closed category. We will show that the Plotkin powerdomain over an -domain is an -domain, where the Plotkin powerdomain is the free dcpo-semilattice over a continuous dcpo.
Next, we collect some basic notions needed in this paper. The reader can also consult [4, 5, 15, 16]. A poset is called a directed complete poset (a dcpo, for short) if any nonempty directed subset of has a sup in . For , is way below (denoted by ) if and only if, for all directed subsets for which exists, the relation implies the existence of a with . A dcpo is called a continuous domain if, for all , ; that is, the set is directed and . For a subset of , let , . We use (resp., ) instead of (resp., ) when . is called an upper (resp., a lower) set if (resp., ). If is a dcpo, we define the Scott topology, denoted by , which has as its topology of closed sets all directed complete lower subsets, that is, lower sets closed under directed sups. A function from a dcpo into a dcpo is continuous with respect to the Scott topologies if preserves suprema of directed subsets.
Recall the definition of -domain: a dcpo is called an -domain if is approximated directly by a family of finitely separated Scott continuous functions. A Scott continuous function is called finitely separated if there exists a finite set such that, for each , there exists such that .
2. -Domains
2.1. Categories of -Domains
For dcpos and , the function space of all Scott continuous functions from to with the pointwise order is a dcpo. Then for dcpo +-semilattices and , we conclude that the function space of all the Scott continuous and +-semilattice homomorphisms from to with the pointwise order is a dcpo +-semilattice from the following theorem, where the operation + satisfies the commutative, associative, and idempotency laws.
Theorem 1. Let and be dcpo +-semilattices; then is a dcpo +-semilattice.
Proof. For any directed family and , set . It is obvious that is Scott continuous. Then
So is also a Scott continuous and +-semilattice homomorphism. Hence is a dcpo.
For any , , we define . For a directed set , we have
Then is Scott continuous.
For a pair of points in ,
That is, is a +-semilattice homomorphism. So is a +-semilattice.
Finally, by the Scott continuity of the operation +, we obtain the following conclusion. For the sup of the directed set and , if , then
So is Scott continuous.
We have obtained that is a dcpo +-semilattice.
With respect to these special Scott continuous functions, we will introduce some new order structures.
Definition 2. A dcpo is called an -domain if it is a +-semilattice and there exists a directed family of finitely separated Scott continuous and +-semilattice homomorphisms which can approximate .
For example, an -domain is a continuous dcpo -semilattice where is approximated by a directed family of finitely separated Scott continuous functions preserving finite infs.
We know that an -domain is an -domain.
Theorem 3. Let and be -domains; then and are -domains.
Proof. Suppose that and are approximate identities for and , respectively. Then we claim that the family
defined by
for is an approximate identity for where is finitely separated. The proof is similar to the case of -domains.
It suffices to show that . Firstly, it is obvious that is Scott continuous. Secondly, for a pair of points , we have for any
So we conclude that is a +-semilattice homomorphism. Then is an -domain. Similarly, is also an -domain.
Theorem 4. The category with -domains as objects and Scott continuous functions as morphisms is a Cartesian closed category.
Note that the category with -domains as objects and Scott continuous and +-semilattice homomorphisms as morphisms is not a Cartesian closed category generally, because the evaluation maps do not preserve the finite +-operation.
2.2. Classify the Powerdomains
Definition 5 (see [5]). Let be a dcpo-algebra equipped with a Scott continuous binary operation + that satisfies the following equations: for any (1) (idempotency law);(2) (commutative law);(3) (associative law).Then the dcpo-algebra is a commutative idempotent semigroup, called a dcpo-semilattice. The free dcpo-semilattice over a dcpo is called the convex or Plotkin powerdomain of and it is denoted by .
If the binary operation + satisfies the inequality , then we obtain the upper or Smyth powerdomain, and it is denoted by , where .
Similarly, if the binary operation + satisfies , then it is called the lower or Hoare powerdomain, denoted by , where .
Proposition 6 (see [5]). For subsets and of a preordered set one has (1);(2) iff ;(3) iff there exists a finite subset such that ;(4);(5) iff ;(6) iff iff ;(7), where iff and ;(8) iff and ;(9) iff and .
Next, we draw the conclusion that some special -domain categories concerning the operation + can be used to classify the powerdomains.
Theorem 7. If is an -domain, then the convex powerdomain is an -domain.
Proof. Suppose that is an -domain; then is a Lawson compact domain. Thus, is also a domain. Assume that is the approximate identity for , where is a family of finitely separated Scott continuous functions; that is, for any , there exists a finite set such that, for any , there exists some such that . We claim that is the approximate identity for . It suffices to consider four steps as follows.(1). For , define . By Proposition 6, . For any , let ; then implies and . Hence .(2). For any , it is obvious that . Suppose . There is such that and . By and , there is some finite set such that . But for any finite set , we have , where iff and . Then . This is a contradiction. Then we conclude that .(3) is Scott continuous and finitely separated. For a directed family in , we have Then is Scott continuous. For any , is a finite set. By and , it follows that . Let . Since and is finite, it follows that is a finite family of . And we have that, for any , there exists such that ; that is, is finitely separated.(4) is a +-semilattice homomorphism. For , since is a +-semilattice, : Then we conclude that is the approximate identity for . Thus the convex powerdomain is an -domain.
Combined with the work of Huth et al. [13] and Meng and Kou [14], we conclude the following theorem.
Theorem 8. Let be a domain. Then the following statements hold:(1)if is Lawson compact, then the Smyth powerdomain is an -domain in [14];(2)if has a least point, then the Hoare powerdomain is a distributive -lattice in [13];(3)if is an -domain, then the Plotkin powerdomain is an -domain.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the Foundation of the Education Department of Henan Province (13A110552), the Foundation of the Science and Technology Department of Henan Province (142300410165), and the Foundation of Henan Normal University (2013PL03).