Countably QC-Approximating Posets

As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-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 QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σ c(L)op of all σ-Scott-closed subsets of L is weakly generalized countably approximating.


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][3][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 ofcontinuous 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.

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., The Scientific World Journal ⊤ ⊥ Figure 1: A complete lattice with countably directed sets having maximal elements. , ≤ }. A subset is ( ) lower set (resp., upper set) if = ↓ (resp., = ↑ ). The supremum of is denoted by ∨ or sup . 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 sup ≥ , 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 sup ≥ , 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 7. Let be the complete lattice formed by uncountably many incomparable unit intervals [0, 1] 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. ∈ there is an element ∈ with ≤ . We say that a nonempty family F of subsets of is (countably) directed if it is (countably) directed in the Smyth preorder. More precisely, F is directed if for all 1 , 2 ∈ F, there exists ∈ F such that 1 , 2 ≤ ; that is, ⊆↑ 1 ∩ ↑ 2 .
Generalizing the relation ≪ on points of to the nonempty subsets of , one obtains the concept of weakly generalized countably approximating posets. 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 P( ) is defined as follows. ⊲ if and only if whenever is a subset of for which ∨ exists, ∨ ∈↑ implies ∩ ↑ ̸ = 0. A complete lattice is called a generalized completely distributive lattice or shortly a GCD lattice, if and only if for all ∈ , ↑ = ∩{↑ | ∈ P fin ( ) and ⊲ }. (

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 -Scottclosed subsets for a poset, Mao and Xu in [12] introduced the concept of countably -approximating posets.

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 .
The next proposition is basic and the proof is omitted.
Definition 19. A poset is said to be countably quasi-approximating, shortly countably -approximating, if for all 4 The Scientific World Journal ∈ , ↑ = ∩{↑ | ∈ ( )}. A countably approximating poset which is also a complete lattice is called a countably -approximating lattice.
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.
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.
Recall that a poset is called a hypercontinuous poset (see [13]) if for all ∈ , the set { ∈ | ≺ ]( ) } is directed and 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 countablyapproximating.