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).