#### Abstract

As a generalization of countably -approximating posets, the concept of countably *QC*-approximating posets is introduced. With the countably -approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably -approximating and weakly generalized countably approximating; (2) a poset having countably directed joins is generalized countably approximating if and only if the lattice of all -Scott-closed subsets of is weakly generalized countably approximating.

#### 1. Introduction

The notion of continuous lattices as a model for the semantics of programming languages was introduced by Scott in [1]. Later, a more general notion of continuous directed complete partially ordered sets (i.e., continuous dcpos or domains) was introduced and extensively studied (see [2–4]). Lawson in [4] gave a remarkable characterization that a dcpo is continuous if and only if the lattice of all Scott-closed subsets of is completely distributive. Gierz et al. in [5] introduced quasicontinuous domains, the most successful generalizations of continuous domains, and proved that quasicontinuous domains equipped with the Scott topology are precisely the spectra of hypercontinuous distributive lattices. Venugopalan in [6] introduced generalized completely distributive (GCD) lattices and Xu in his Ph.D. thesis [7] proved that GCD lattices are precisely the dual of hypercontinuous lattices. Ho and Zhao in [8] introduced the concept of -continuous lattices. And they showed that for any poset , is a -continuous lattice and that is continuous if and only if is continuous.

On the other hand, Lee in [9] introduced the concept of countably approximating lattices, a generalization of continuous lattices, and showed that this new larger class has many properties in common with continuous lattices. In [10], Han et al. further generalized the concept of countably approximating lattices to the concept of countably approximating posets and characterized countably approximating posets via the -Scott topology. Yang and Liu in [11] introduced the concept of generalized countably approximating posets, a generalization of countably approximating posets. Making use of the ideas of [8, 10], Mao and Xu in [12] introduced the concept of countably -approximating posets and showed that the lattice of all -Scott-closed subsets of a poset is a countably -approximating lattice and that a complete lattice is completely distributive if and only if it is countably approximating and countably -approximating.

In this paper, we generalize the concept of countably -approximating posets to the concept of countably -approximating posets. With the countably -approximating property, we present some characterizations of GCD lattices and generalized countably approximating posets.

#### 2. Preliminaries

We quickly recall some basic notions and results (see, e.g., [3, 8] or [11]). Let (, ) be a poset. Then with the dual order is also a poset and denoted by . A* principal ideal* (resp.,* principal filter*) is a set of the form (resp., ). For , we write , . A subset is * lower set* (resp.,* upper set*) if (resp., ). The supremum of is denoted by or . A subset of is* directed* if every finite subset of has an upper bound in . A subset of is* countably directed* if every countable subset of has an upper bound in . Clearly every (countably) directed set is nonempty, and every countably directed set is directed but not vice versa. A poset is a* directed complete partially ordered set* (dcpo, in short) if every directed subset of has a supremum. A poset is said to have countably* directed joins* if every countably directed subset has a supremum.

*Remark 1. *It is clear that if is countably directed and itself is countable, then has a maximal element. By this observation, we see that every countable poset must have countably directed joins and thus a poset having countably directed joins need not be a dcpo.

The following definitions give various induced relations by the order of a poset.

*Definition 2 (see [3]). *Let be a poset and , . We say that is* way-below * or * approximates *, written if whenever is a directed set that has a supremum , then there is some with . For each , we write . A poset is said to be* continuous* if every element is the directed supremum of elements that approximate it. A continuous poset which is also a complete lattice is called a* continuous lattice*.

*Definition 3 (see [10]). *Let be a poset and , . We say that is* countably way-below *, written if for any countably directed subset of with , there is some with . For each , we write and . A poset having countably directed joins is called a* countably approximating poset* if for each , the set is countably directed and . A countably approximating poset which is also a complete lattice is called a* countably approximating lattice*.

In a poset , it is clear that implies that . Since every countably directed set is directed, we have that implies for all , . In other words, for each . However, the following example shows that the reverse implication need not be true.

*Example 4. *Let be the unit interval . For all , , it is easy to check that and that or .

By Remark 1, it is clear that every countable poset is a countably approximating poset.

Proposition 5. *Let be a poset and a countable subset of such that exists. If for all , then .*

*Proof. *Straightforward.

By Proposition 5, in a complete lattice , the set is automatically countably directed for each . So, a complete lattice is countably approximating if and only if for each , . Thus every continuous lattice is a countably approximating lattice.

Proposition 6. *Let be a poset. If every countably directed subset of has a maximal element, then is a countably approximating poset.*

*Proof. *Straightforward by Definition 3.

*Example 7. *Let be the complete lattice formed by uncountably many incomparable unit intervals with all the 0’s being pasted as a and all the 1’s being pasted as a (See Figure 1). Then it is easy to check that the resulting complete lattice satisfies the condition in Proposition 6 and thus is a countably approximating lattice.

Proposition 8. *Let be a poset. If every countably directed subset of is countable, then is a countably approximating poset.*

*Proof. *It is straightforward by Remark 1 and Proposition 6.

*Example 9. *If with its usual order is augmented with uncountably many incomparable upper bounds, then it is easy to check that the resulting poset satisfies the condition in Proposition 8 and thus is a countably approximating poset.

For a set , we use to denote the power set of and to denote the set of all nonempty finite subsets of . For a poset , define a preorder (sometimes called* Smyth preorder*) on by if and only if for all . That is, if and only if for each there is an element with . We say that a nonempty family of subsets of is (countably)* directed* if it is (countably) directed in the Smyth preorder. More precisely, is directed if for all , , there exists such that , ; that is, .

Generalizing the relation on points of to the nonempty subsets of , one obtains the concept of weakly generalized countably approximating posets.

*Definition 10. *Let be a poset having countably directed joins. A binary relation on is defined as follows. if and only if for any countably directed set , implies . We write for and for . If for each , , where , then is called a* weakly generalized countably approximating poset*. A weakly generalized countably approximating poset which is also a complete lattice is called a* weakly generalized countably approximating lattice.*

A weakly generalized countably approximating poset (lattice) with the condition that for each , is countably directed is called a* generalized countably approximating poset* (*lattice*) in [11].

As a generalization of completely distributive lattice, the following concept of GCD lattices was introduced in [6].

*Definition 11 (see [6]). *Let be a poset. A binary relation on is defined as follows. if and only if whenever is a subset of for which exists, implies . A complete lattice is called a* generalized completely distributive lattice* or shortly a GCD lattice, if and only if for all , .

*Definition 12 (see [3]). *A subset of a poset is* Scott-open* if and for any directed set , implies . All the Scott-open sets of form a topology, called the* Scott topology* and denoted by . The complement of a Scott-open set is called a* Scott-closed set*. The collection of all Scott-closed sets of is denoted by . The topology on generated by as a subbase is called the* upper topology* and denoted by .

Replacing directed sets with countably directed sets in Definition 12, we can get the concept of -Scott-open sets.

*Definition 13 (see [10]). *Let be a poset. A subset of is called -*Scott-open* if and for any countably directed set , implies . All the -Scott-open sets of form a topology, called the -*Scott topology* and denoted by . The complement of a -Scott-open set is called a -*Scott-closed* set. The collection of all -Scott-closed sets of is denoted by .

*Remark 14 (see [10], Remark 2.1). *(1) For a poset , the -Scott topology is closed under countably intersections and the Scott topology is coarser than ; that is, .

(2) A subset of a poset is -Scott-closed if and only if it is a lower set and closed under countably directed joins.

To study the order structure of the lattice of all -Scott-closed subsets for a poset, Mao and Xu in [12] introduced the concept of countably -approximating posets.

*Definition 15 (see [12]). *Let be a poset and , . We say that is -*beneath *, denoted by , if for any nonempty -Scott-closed set for which exists, always implies that . Poset is said to be* countably **-approximating* if for each , , where . A complete lattice which is also countably -approximating is called a* countably **-approximating lattice*.

Lemma 16 (see [12]). *For a poset , the lattice is countably approximating.*

*Proof. *Let be a poset and . It is straightforward to check that . For each , we have that . Suppose with . Then for each , since , there exists such that . Noticing that is a lower set, we have . It follows from being a lower set that . Thus by Definition 15, holds in . Hence, . So, and by the arbitrariness of , we conclude that is countably -approximating.

#### 3. Countably -Approximating Posets

In this section, we introduce the concept of countably -approximating posets. Firstly, we generalize the relation on points of a poset to the nonempty subsets of .

*Definition 17. *For a poset , the -beneath relation on nonempty subsets of is defined as follows: if and only if whenever is a nonempty -Scott-closed subset of for which exists, implies . We write for . Set .

The next proposition is basic and the proof is omitted.

Proposition 18. *Let be a poset. Then*(i)*, ;*(ii)*, ;*(iii)*, ;*(iv)*, .*

With the relation , we have the concept of countably -approximating posets.

*Definition 19. *A poset is said to be* countably quasi*--*approximating*, shortly* countably *-*approximating*, if for all , . A countably -approximating poset which is also a complete lattice is called a* countably **-approximating lattice*.

Proposition 20. *Countably -approximating posets are countably -approximating.*

*Proof. *Let be a countably -approximating poset. Then for all ,
Thus . By Definition 19, is countably -approximating.

By Lemma 16 and Proposition 20, we immediately have the following.

Corollary 21. *For any poset , the lattice is countably -approximating.*

In the sequel, we explore relationships between countably -approximating lattices and GCD lattices.

Proposition 22. *Every GCD lattice is weakly generalized countably approximating.*

*Proof. *Let be a GCD lattice. For all and , implies . Then . So . By Definition 10, is weakly generalized countably approximating.

Proposition 23. *Every GCD lattice is countably -approximating.*

*Proof. *Let be a GCD lattice. For each and , implies . Then . Thus . By Definition 19, is countably approximating.

The following theorem characterizes GCD lattices.

Theorem 24. *Let be a complete lattice. Then the following statements are equivalent:*(1)* is a GCD lattice;*(2)* is countably -approximating and weakly generalized countably approximating.*

*Proof. *: follows from Propositions 22 and 23.

: suppose that is countably -approximating and weakly generalized countably approximating. Then for each , by the weakly generalized countably approximating property of , we have . Now for each , we show that . To this end, it suffices to show that . Suppose and . Then for any , . By the countably -approximating property of , there exists such that and . Let . Then is still finite and . It is clear that , contradicting to that . Thus such that .

Suppose , we will show that . For any with , let is a countable subset of . Then is a countably directed set and . Since , there exists a countable subset such that . By Remark 14 (1), is -Scott-closed. It follows from that . This implies , showing that . Thus, . So, . Therefore, is a GCD lattice.

Recall that a poset is called a* hypercontinuous poset* (see [13]) if for all , the set is directed and , where . A hypercontinuous poset which is also a complete lattice is called a* hypercontinuous lattice*.

Lemma 25 (see [7], Theorem ). *Let be a complete lattice. Then is a GCD lattice if and only if is a hypercontinuous lattice.*

It is easy to see that for a finite lattice , both and are continuous, and . It follows from ([14], Theorem 2.1) that and are hypercontinuous lattices; hence by Lemma 25, and are GCD lattices. By this observation, we see that every finite lattice is a countably -approximating lattice. So, countably -approximating lattices need not be distributive.

It is known from Proposition 4.1 in [12] that any countably -approximating lattice is distributive. So, countably -approximating lattices need not be countably -approximating.

Lemma 26 (see [11], Theorem 3.4). *Let be a poset having countably directed joins. Then is generalized countably approximating if and only if the lattice is hypercontinuous.*

So, in view of Lemma 25, a poset having countably directed joins is generalized countably approximating if and only if the lattice is a GCD lattice. The following theorem gives comprehensive characterizations of generalized countably approximating posets.

Theorem 27. *Let be a poset having countably directed joins. Then the following statements are equivalent:*(i)* is a generalized countably approximating poset;*(ii)* is a hypercontinuous lattice;*(iii)* is a GCD lattice;*(iv)* is a weakly generalized countably approximating lattice.*

*Proof. * by Lemma 26.

by Lemma 25.

follows from Theorem 24 and Corollary 21.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable comments and helpful suggestions. This work is supported by NSF of china (11101212 and 61103018).