# Countably QC-approximating posets.

1. IntroductionThe 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 L is continuous if and only if the lattice [sigma][(L).sup.op] of all Scott-closed subsets of L is completely distributive. Gierz et al. in [5] introduced quasicontinuous domains, the most successful generalizations of continuous domains, and proved that quasi-continuous 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 C-continuous lattices. And they showed that for any poset L, [sigma][(L).sup.op] is a C-continuous lattice and that L is continuous if and only if [sigma][(L).sup.op] 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 cr-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 C-approximating posets and showed that the lattice of all [sigma]-Scott-closed subsets of aposet is acountably C-approximating lattice and that a complete lattice is completely distributive if and only if it is countably approximating and countably C-approximating.

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

2. Preliminaries

We quickly recall some basic not ions and results (see, e.g., [3, 8] or [11]). Let (L, [less than or equal to]) be a poset. Then L with the dual order is also a poset and denoted by [L.sup.op]. A principal ideal (resp., principal filter) is a set of the form [down arrow] x = {y [member of] L | y [less than or equal to] x} (resp., [up arrow] x = {y [member of] L | x [less than or equal to] y}). For X [subset or equal to] L, we write [down arrow] X = [y [member of] L | [there exists] x [member of] X, y [less than or equal to] x], [up arrow] X = [y [member of] L | [there exists] x [member of] X, x [less than or equal to] y}. A subset X is a(n) lower set (resp., upper set) if X = [down arrow] X (resp., X = [up arrow] X). The supremum of X is denoted by [disjunction] X or sup X. A subset D of L is directed if every finite subset of D has an upper bound in D. A subset D of L is countably directed if every countable subset of D has an upper bound in D. Clearly every (countably) directed set is nonempty, and every countably directed set is directed but not vice versa. Aposet L is a directed complete partially ordered set (dcpo, in short) if every directed subset of L 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 D is countably directed and itself is countable, then D 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 L be a poset and x, y [member of] L. We say that x is way-below y or x approximates y, written x [much less than] y if whenever D is a directed set that has a supremum sup D [greater than or equal to] y, then there is some d [member of] D with x [less than or equal to] d. For each x [member of] L, we write [??] x = {y [member of] L | y [much less than] x}. 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 L be a poset and x, y [member of] L. We say that x is countably way-below y, written x [[much less than].sub.c] y if for any countably directed subset D of L with sup D [greater than or equal to] y, there is some d [member of] D with x [less than or equal to] d. For each x [member of] L, we write [[??].sub.c]x = {y [member of] L | y [[much less than].sub.c] x} and [[??].sub.c] x = {y [member of] L | x [[much less than].sub.c] y}. A poset L having countably directed joins is called a countably approximating poset if for each x [member of] L, the set [[??].sub.c]x is countably directed and x = V[[??].sub.c]x. A countably approximating poset which is also a complete lattice is called a countably approximating lattice.

In a poset L, it is clear that x [[much less than].sub.c] y implies that x [less than or equal to] y. Since every countably directed set is directed, we have that x [much less than] y implies [lambda] [[much less than].sub.c] y for all x, y [member of] L. In other words, [??] y [subset or equal to] [[??].sub.c] y for each y [member of] L. However, the following example shows that the reverse implication need not be true.

Example 4. Let L be the unit interval [0,1]. For all x, y [member of] [0,1], it is easy to check that x [[much less than].sub.c] y c [??] x [less than or equal to] y and that x [much less than] y [??] x = 0 = y or x < y.

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

Proposition 5. Let L be a poset and S acountable subset of L such that VS exists. If s [[much less than].sub.c] x for all s [member of] S, then [disjunction]S [[much less than].sub.c] x.

Proof. Straightforward.

By Proposition 5, in a complete lattice L, the set [[??].sub.c]x is automatically countably directed for each x [member of] L. So, a complete lattice L is countably approximating if and only if for each x [member of] L, x = Vlcx. Thus every continuous lattice is a countably approximating lattice.

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

Proof. Straightforward by Definition 3.

Example 7. Let L be the complete lattice formed by uncountably many incomparable unit intervals [0,1] with all the 0's being pasted as a [perpendicular to] 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 L be a poset. If every countably directed subset of L is countable, then L is a countably approximating poset.

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

Example 9. If N 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 X, we use P(X) to denote the power set of X and [P.sub.fin] (X) to denote the set of all nonempty finite subsets of X. For a poset L, define a preorder [less than or equal to] (sometimes called Smyth preorder) on P(L)\{[phi]} by G [less than or equal to] H if and only if [up arrow] H [subset or equal to] [up arrow] G for all G, H [subset or equal to] L. That is, G [less than or equal to] H if and only if for each y [member of] H there is an element x [member of] G with x [less than or equal to] y. We say that a nonempty family F of subsets of L is (countably) directed if it is (countably) directed in the Smyth preorder. More precisely, F is directed if for all [F.sub.1], [F.sub.2] [member of] F, there exists F [member of] F such that [F.sub.1], [F.sub.2] [less than or equal to] F; that is, F [subset or equal to] [up arrow] [F.sub.1] [intersection] [up arrow] [F.sub.2].

Generalizing the relation [[much less than].sub.c] on points of L to the nonempty subsets of L, one obtains the concept of weakly generalized countably approximating posets.

Definition 10. Let L be a poset having countably directed joins. A binary relation [[much less than].sub.c] on P(L)\{[phi]} is defined as follows. A [[much less than].sub.c] B if and only if for any countably directed set D [subset or equal to] L, [disjunction] D [member of] [up arrow] B implies D [intersection] [up arrow] A [not equal to] [phi]. We write F [[much less than].sub.c] x for F [[much less than].sub.c] {x} and y [[much less than].sub.c] H for {y} [[much less than].sub.c] H. If for each x [member of] L, [up arrow] x = [intersection]{[up arrow] F | F [member of] [omega](x)}, where [omega](x) = {F | F [member of] [P.sub.fin](L) and F [[much less than].sub.c] x}, then L 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) L with the condition that for each x [member of] L, [omega](x) 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 L be a poset. A binary relation [??] on P(L) is defined as follows. A [??] B if and only if whenever S is a subset of L for which [disjunction] S exists, [disjunction] S [member of] [up arrow] B implies S [intersection] [up arrow] A [not equal to] [phi]. A complete lattice L is called a generalized completely distributive lattice or shortly a GCD lattice, if and only if for all x [member of] L, [up arrow] x = [intersection] [up arrow] F | F [member of] [P.sub.fin] (L) and F [??] x}.

Definition 12 (see [3]). A subset U of a poset L is Scott-open if [up arrow] U = U and for any directed set D [subset or equal to] L, sup D [member of] U implies U [intersection] D [not equal to] [phi]. All the Scott-open sets of L form a topology, called the Scott topology and denoted by [sigma](L). The complement of a Scott-open set is called a Scott-closed set. The collection of all Scott-closed sets of L is denoted by [sigma][(L).sup.op]. The topology on L generated by {L\ [down arrow] x | x [member of] L} as a subbase is called the upper topology and denoted by v(L).

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

Definition 13 (see [10]). Let L be a poset. A subset U of L is called [sigma]-Scott-open if [up arrow]U = U and for any countably directed set D [subset or equal to] L, sup D [member of] U implies U [intersection] D [not equal to] 0. All the [sigma]-Scott-open sets of L form a topology, called the [sigma]-Scott topology and denoted by [[sigma].sub.c](L). The complement of a [sigma]-Scott-open set is called a [sigma]-Scott-closed set. The collection of all [sigma]-Scott-closed sets of L is denoted by [[sigma].sub.c][(L).sup.op].

Remark 14 (see [10], Remark 2.1). (1) For a poset L, the [sigma]-Scott topology [[sigma].sub.c](L) is closed under countably intersections and the Scott topology [sigma](L) is coarser than ac (L); that is, [sigma](L) [subset or equal to] [[sigma].sub.c](L).

(2) A subset of a poset is [sigma]-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 [sigma]-Scott-closed subsets for a poset, Mao and Xu in [12] introduced the concept of countably C-approximating posets.

Definition 15 (see [12]). Let L be a poset and x, y [member of] L. We say that x is a-beneath y, denoted by x [<.sub.[sigma]] y, if for any nonempty [sigma]-Scott-closed set FcL for which VF exists, [disjunction] F [greater than or equal to] y always implies that x [member of] F. Poset L is said to be countably C-approximating if for each x [member of] L, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A complete lattice which is also countably C-approximating is called a countably C-approximating lattice.

Lemma 16 (see [12]). For a poset L, the lattice [[sigma].sub.c][(L).sup.op] is countably C-approximating.

Proof. Let L be a poset and C [member of] [[sigma].sub.c][([[sigma].sub.c][(L).sup.op]).sup.op]. It is straightforward to check that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For each F [member of] [[sigma].sub.c][(L).sup.op], we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose C [member of] [[sigma].sub.c][([[sigma].sub.c][(L).sup.op]).sup.op] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for each x [member of] F, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists A [member of] C such that x [member of] A. Noticing that A [member of] [[sigma].sub.c][(L).sup.op] is a lower set, we have [up arrow] x [subset or equal to] A [member of] C. It follows from C [member of] [[sigma].sub.c][([[sigma].sub.c][(L).sup.op]).sup.op] being a lower set that [down arrow] x [member of] C. Thus by Definition 15, [down arrow] x [<.sub.[sigma]] F holds in [[sigma].sub.c][(L).sup.op]. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and by the arbitrariness of F [member of] [[sigma].sub.c][(L).sup.op], we conclude that [[sigma].sub.c][(L).sup.op] is countably C-approximating.

3. Countably QC-Approximating Posets

In this section, we introduce the concept of countably QC-approximating posets. Firstly, we generalize the relation [<.sub.[sigma]] on points of a poset L to the nonempty subsets of L.

Definition 17. For a poset L, the [sigma]-beneath relation [<.sub.[sigma]] on nonempty subsets of L is defined as follows: A[<.sub.[sigma]] B if and only if whenever S is a nonempty [sigma]-Scott-closed subset of L for which [disjunction] S exists, [disjunction]S [member of] [up arrow] B implies S [intersection] [up arrow] A [not equal to] 0. We write F [<.sub.[sigma]] x for F [<.sub.[sigma]] {x}. Set c(x) = {F | F [member of] [P.sub.fin] (L) and F [<.sub.[sigma]] x}.

The next proposition is basic and the proof is omitted.

Proposition 18. Let L be a poset. Then

(i) [for all]G, H [subset or equal to] L, G [<.sub.[sigma]] H [??] G [less than or equal to] H;

(ii) [for all]G, H [subset or equal to] L, G [<.sub.[sigma]] H [??] [for all]h [member of] H, G [<.sub.[sigma]] h;

(iii) [??]E, F, G, H [subset or equal to] L, E [less than or equal to] G [<.sub.[sigma]] H [less than or equal to] F [??] E [<.sub.[sigma]] F;

(iv) [for all]x, y [member of] L, {x} [<.sub.[sigma]] {y} [??] x [<.sub.[sigma]] y.

With the relation [<.sub.[sigma]], we have the concept of countably QC-approximating posets.

Definition 19. A poset L is said to be countably quasi-C-approximating, shortly countably QC-approximating, if for all x [member of] L, [up arrow] x = [intersection] {[up arrow] F | F [member of] c(x)}. A countably QC-approximating poset which is also a complete lattice is called a countably QC-approximating lattice.

Proposition 20. Countably C-approximating posets are countably QC-approximating.

Proof. Let L be a countably C-approximating poset. Then for all x [member of] L,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Thus [intersection]{[up arrow] F | F [member of] c(x)} = [up arrow] x. By Definition 19, L is countably QC-approximating.

By Lemma 16 and Proposition 20, we immediately have 1the following.

Corollary 21. For any poset L, the lattice [[sigma].sub.c][(L).sup.op] is countably QC-approximating.

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

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

Proof. Let L be a GCD lattice. For all x [member of] L and F [member of] [P.sub.fin](L), F [??] x implies F [[much less than].sub.c] x. Then [up arrow] x [subset or equal to] [intersection]{[up arrow] F | F [member of] [omega](x)} [subset or equal to] [intersection]{[up arrow] F | F [member of] [P.sub.fin] (L) and F [??] x} = [up arrow]x. So [up arrow] x = [intersection] {[up arrow] F | F [member of] [omega](x)}.By Definition 10, L is weakly generalized countably approximating.

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

Proof. Let L be a GCD lattice. For each x [member of] L and F [member of] [P.sub.fin](L), F [??] x implies F [<.sub.[sigma]] x. Then [up arrow] x [subset or equal to] [intersection]{[up arrow] F | F [member of] c(x)} [subset or equal to] [intersection] {[up arrow] F | F [member of] [P.sub.fin](L) and F [??] x} =[up arrow] x. Thus [up arrow] x = [intersection]{[up arrow] F | F [member of] c(x)}. By Definition 19, L is countably QC-approximating.

The following theorem characterizes GCD lattices.

Theorem 24. Let L be a complete lattice. Then the following statements are equivalent:

(1) L is a GCD lattice;

(2) L is countably QC-approximating and weakly generalized countably approximating.

Proof. (1) [??] (2):follows from Propositions 22 and 23. (2) [??] (1): suppose that L is countably QC-approximating and weakly generalized countably approximating. Then for each x [member of] L, by the weakly generalized countably approximating property of L, we have [up arrow] x = [intersection]{[up arrow] F | F [member of] [omega](x)}.

Now for each F [member of] [omega](x), we show that [up arrow] F = [intersection]{[up arrow] F' | F' [member of] [P.sub.fin](L) and F' [<.sub.[sigma]] F}. To this end, it suffices to show that [intersection]{[up arrow] F' | F' [member of] [P.sub.fin](L) and F'[<.sub.[sigma]] F} [subset or equal to] [up arrow] F. Suppose t [member of] [intersection] {[up arrow] F' | F' [member of] [P.sub.fin] (L) and F' [<.sub.[sigma]] F} and t [not member of] [up arrow] F. Then for any [y.sub.F] [member of] F, t [not member of] [up arrow] [y.sub.F]. By the countably QC-approximating property of L, there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [bar.F] is still finite and [bar.F] [<.sub.[sigma]] F. It is clear that t [not member of] [up arrow] [bar.F], contradicting to that t [member of] [intersection]{[up arrow] F' | F' [member of] [P.sub.fin](L) and F' [<.sub.[sigma]] F}. Thus [up arrow] x = [intersection]{[up arrow] F | F [member of] w(x)} = [intersection]{[up arrow] F' | F' [member of] [P.sub.fin](L), [there exists]F [member of] [P.sub.fin](L) such that F' [<.sub.[sigma]] F [[much less than].sub.c] x}.

Suppose F' [<.sub.[sigma]] F [[much less than].sub.c] x, we will show that F' [??] x. For any A [subset or equal to] L with [disjunction]A [greater than or equal to] x, let G = {[disjunction]E | E is a countable subset of A}. Then G is a countably directed set and [disjunction]G = [disjunction]A [member of] [up arrow] x. Since F [[much less than].sub.c] x, there exists a countable subset E [subset or equal to] A such that [disjunction] E = [disjunction] [down arrow] E [member of] [up arrow] F. By Remark 14 (1), [down arrow] E is [sigma]-Scott-closed. It follows from F' [<.sub.[sigma]] F that [down arrow] E[intersection] [up arrow]F' [not equal to] [phi]. This implies A[intersection] [up arrow] F' [not equal to] [phi], showing that F' [??] x. Thus, [up arrow] x [subset or equal to] [intersection]{[up arrow] W | W [member of] [P.sub.fin](L), W [??] x} [subset or equal to] [intersection]{[up arrow] F' | F' [member of] [P.sub.fin](L), [there exists]F [member of] [P.sub.fin](L), F' [<.sub.[sigma]] F [[much less than].sub.c] = [up arrow] x. So, [up arrow] x = [intersection]{[up arrow] W | W [member of] [P.sub.fin] (L) and W [??] x}. Therefore, L is a GCD lattice.

Recall that a poset L is called a hypercontinuous poset (see [13]) if for all x [member of] L, the set {y [member of] L\y [<.sub.v(L)] x} is directed and x = sup{y [member of] L \ y [<.sub.v(L)] x}, where y[<.sub.v(L)]x [??] x [member of] [int.sub.v(L)] [up arrow] y. A hypercontinuous poset which is also a complete lattice is called a hypercontinuous lattice.

Lemma 25 (see [7], Theorem 4.1.4). Let L be a complete lattice. Then L is a GCD lattice if and only if [L.sup.op] is a hypercontinuous lattice.

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

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

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

So, in view of Lemma 25, a poset having countably directed joins is generalized countably approximating if and only if the lattice [[sigma].sub.c][(L).sup.op] is a GCD lattice. The following theorem gives comprehensive characterizations of generalized countably approximating posets.

Theorem 27. Let L be a poset having countably directed joins.

Then the following statements are equivalent:

(i) L is a generalized countably approximating poset;

(ii) [[sigma].sub.c](L) is a hypercontinuous lattice;

(iii) [[sigma].sub.c][(L).sup.op] is a GCD lattice;

(iv) [[sigma].sub.c][(L).sup.op] is a weakly generalized countably approximating lattice.

Proof. (i) [??] (ii) by Lemma 26.

(ii) [??] (iii) by Lemma 25.

(iii) [??] (iv) follows from Theorem 24 and Corollary 21.

http://dx.doi.org/10.1155/2014/123762

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).

References

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Xuxin Mao (1) and Luoshan Xu (2)

(1) College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

(2) Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Correspondence should be addressed to Luoshan Xu; luoshanxu@hotmail.com

Received 17 May 2014; Accepted 21 July 2014; Published 5 August 2014

Academic Editor: Jianming Zhan

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Title Annotation: | Research Article |
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Author: | Mao, Xuxin; Xu, Luoshan |

Publication: | The Scientific World Journal |

Article Type: | Report |

Date: | Jan 1, 2014 |

Words: | 4431 |

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