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Strongly Semicontinuous Domains and Semi-FS Domains
We are mainly concerned with some special kinds of semicontinuous domains and relationships between them. New concepts of strongly semicontinuous domains, meet semicontinuous domains and semi-FS domains are introduced. It is shown that a dcpo L is strongly semicontinuous if and only if L is semicontinuous and meet semicontinuous. It is proved that semi-FS domains are strongly semicontinuous. Some interpolation properties of semiway-below relations in (strongly) semicontinuous bc-domains are given. In terms of these properties, it is proved that strongly semicontinuous bc-domains, in particular strongly semicontinuous lattices, are all semi-FS domains.
The theory of continuous lattices  and the more general theory of domains  initiated by Scott provide a mathematical foundation for the denotational semantics of programming languages and are closely linked to theoretical computer science, general topology and logics [3–5]. The purpose of the theory of domains is to give models for spaces to define computable functions. The idea is that the semantics of a programming language should be formally specified in terms of a small number of basic mathematical constructions on partial orders of information. Intuitively, we say that a state approximates a state if any computation of yields the information of at some finite stage. A logic-oriented approach to domain theory to formalize the properties of computation is provided in [2, 6].
So far, continuous lattices were generalized to other types of order structures, such as quasicontinuous posets [7, 8], -continuous posets , and -precontinuous posets . Motivated by the concept of semiprime ideals studied by Rav in , Zhao in  first introduced the concept of semicontinuous lattices and showed that semicontinuous lattices have many properties similar to that of continuous lattices. Li and Wu  studied properties of semicontinuous lattices. Bi and Xu  introduced the semi-Scott topology and the semi-Lawson topology on semicontinuous lattices. Jiang and Shi  discussed characterizations of pseudoprimes and studied strong retracts of (stable) semicontinuous lattices. In , Li introduced semiprime sets and generalized semicontinuous lattices to semicontinuous domains.
Note that, in the definition of a semicontinuous lattice , the condition used is that, for every element , . This condition is too weak to guarantee that every element can be approximated by elements which are below and semiway-below it in a semicontinuous lattice. To guarantee the mentioned approximation property in a suitable class of semicontinuous lattices, or even of semicontinuous domains, we in this paper introduce the concept of strongly semicontinuous domains. It turns out that strongly semicontinuous domains have domain-like features that every element can be approximated by elements below and semiway-below it. The concept of meet semicontinuous domains is also introduced. It is obtained that a dcpo is strongly semicontinuous if and only if it is semicontinuous and meet semicontinuous. Moreover, inspired by the definition of FS-domains posed by Jung  in the realm of domains, we introduce the concept of semi-FS domains which are defined by the existence of some approximating identities consisting of finitely separated functions in the realm of dcpos. It is proved that every semi-FS domain is a strongly semicontinuous domain and that every strongly semicontinuous bc-domain, in particular every strongly semicontinuous lattice, is a semi-FS domain. A counterexample is constructed to show that a strongly semicontinuous domain need not be a semi-FS domain.
A subset of a poset is called directed (resp., filtered) if it is nonempty and every finite subset of has an upper (resp., a lower) bound in . A poset in which every directed set has a supremum is called a dcpo. For a subset of a poset , let : and : . We say that is a lower set (resp., an upper set) if (resp., ). A subset of is an ideal (resp., a filter) if and only if it is a directed lower set (resp., filtered upper set). A principal ideal (resp., principal filter) is a set of the form : (resp., : ). The set of all ideals (resp. filters) in is denoted by (resp., ). For an ideal , is said to be prime if or . We denote the family of all prime ideals of by . A poset is called bounded complete, if every subset that is bounded above has a supremum. A bounded complete dcpo is called a bc-dcpo.
Let . We denote and the set of upper bounds and lower bounds of , respectively.
Lemma 1 (see [16, Lemma 3.2.8]). Let be a poset, . If , then we have
Definition 2 (see ). Let be a lattice. An ideal of is said to be semiprime if for any , and imply .
Lemma 3 (see [12, Lemma 1.2]). An ideal of a lattice is semiprime if and only if there exist prime ideals of such that .
Thus every prime ideal is semiprime, and . And by Lemma 3, we can generalize the concept of semiprime ideals to the setting of posets.
Definition 4. Let be a poset and an ideal of . If there is a family of prime ideals such that , then is called a semiprime ideal. The family of all semiprime ideals of is denoted by .
Li in  generalized semiprime ideals to semiprime sets.
Definition 5 (see ). Let be a poset. A set is said to be semiprime if there exists a family of prime ideals of such that .
We denote the family of all semiprime sets of with . Clearly, a semiprime set need not to be directed. If a semiprime set is directed, then is a semiprime ideal. For a dcpo, we have the following relation:
Proposition 6. Let be a bc-dcpo. Then .
Proof. It suffices to show that . Let . Then by Definition 5 there exists a family of prime ideals of such that . For any , we have for all . Since is directed, there exists such that . As is a bc-dcpo, we see that exists and for all . Therefore, . So, is directed and , as desired.
Lemma 7. For a bc-dcpo , let and be the complete lattice obtained from by adjoining a top element . Then we have(i) if and only if ;(ii) if and only if .
Proof. (i) Suppose that . Since , we have and .
Conversely, let . Then and . So, and .
(ii) Suppose that . Then by Definition 4, , where . It follows from (i) above that for all ; thus, .
Conversely, let . By Lemma 3, there exist prime ideals of such that . Since , for some . Let . Then for any , we see that . By (i) above, we have for each . Then , and . By Proposition 6, .
In a poset , we say that is way-below , or approximates , written , and if is directed with existing and , then for some . Equivalently, iff for every ideal of such that whenever exists. We use to denote the set . If is a dcpo and, for every element , the set is directed and , then is called a domain. A domain is called an L-domain if for each , the principal ideal is a complete lattice. A complete lattice which is a domain is called a continuous lattice.
For complete lattices, replacing ideals with semiprime ideals, Zhao in  defined a weak form of the way-below relation.
Definition 8 (see ). Let be a complete lattice. Define the semiway-below relation on as follows: for , if for any semiprime ideal of , implies . For each , we write .
Definition 9 (see ). A complete lattice is said to be semicontinuous, if for any , .
Zhao  showed that the interpolation property holds in semicontinuous lattices.
Theorem 10 (see [12, Theorem 1.8]). If is a semicontinuous lattice, then implies the existence of a such that .
3. Strongly Semicontinuous Domains
In terms of semiprime sets, semicontinuous lattices can be generalized to semicontinuous domains. And then strongly semicontinuous domains will be defined.
Definition 11. Let be a poset. Define the relation on as follows: for any , if for any semiprime set of , implies . An element of is said to be semicompact if . For each , we write and .
Proposition 12. Let be a dcpo; then, So, for each , . If is a bc-dcpo, then .
Proof. Note that for each . So, by Lemma 1, we have that
Let . Then .
Next we show that . Let . For any with , by Definition 5 there exists a family of prime ideals such that . It follows from Lemma 1 that for each . Since , we have that for each and . By Definition 11, . So, , as desired.
In , the semiway-below relation on a poset was defined [16, Definition ] in a different way. For a dcpo and , it is established in [16, Proposition ] that . So, by Proposition 12 above, we see that . This means that for a dcpo and , . Thus, in the setting of dcpos, Definition 11 is equivalent to Definition in .
Note that for a poset and , need not imply .
Remark 13. Let be a dcpo and . Then it is easy to check that(1) does not imply , the typical modular lattice is a counterexample;(2)if , then . If , then ;(3)if exists and , then ;(4)for any , implies .
Definition 14 (see ). A dcpo is called a semicontinuous dcpo if for all , . A semicontinuous dcpo will also be called a semicontinuous domain. A bc-dcpo which is semicontinuous will be called a semicontinuous bc-domain.
By Proposition 12, one can immediately have the following.
Proposition 15. Let be a dcpo. Then is semicontinuous iff for any , is the smallest semiprime set such that .
Proposition 16. Let be a dcpo. If, for any , there exists a subset and , then is semicontinuous.
Strengthening the condition in Definition 14, we give the following.
Definition 17. A dcpo is said to be strongly semicontinuous if for each , A dcpo (bc-dcpo) which is strongly semicontinuous will be called a strongly semicontinuous domain (strongly semicontinuous bc-domain). A complete lattice which is strongly semicontinuous will be called a strongly semicontinuous lattice.
Clearly, every strongly semicontinuous domain is semicontinuous, a semicontinuous domain satisfying the condition is strongly semicontinuous. It is easy to see that, for a dcpo without proper prime ideals, every pair of elements in has the semiway-below relation and is strongly semicontinuous. However, a semicontinuous domain need not be strongly semicontinuous. The following counterexample first appeared in .
Example 18 (see ). Let . The partial order on is defined by , . It is clear that the prime ideals of are and . We observe that, for any , and . Thus, is a semicontinuous domain. However, note that, in this example , which yields that is not strongly semicontinuous.
Proposition 19. Every domain is a strongly semicontinuous domain.
Wu and Li in  introduced the following concept of meet semicontinuous lattices.
Definition 20 (see ). A complete lattice is said to be meet semicontinuous if for any and , .
It is known that a semicontinuous lattice need not be a meet semicontinuous lattice. However, it is proved in  that strongly semicontinuous lattices are all meet semicontinuous. Generalize meet semicontinuous lattices, we have
Definition 21. A dcpo is said to be meet semicontinuous if for any and , .
Proposition 22. Every strongly semicontinuous domain is meet semicontinuous.
Proof. Suppose that is a strongly semicontinuous domain. For any and , by Lemma 1, it is easy to see that . So, it suffices to show that . To this end, let . Then and since is strongly semicontinuous. For any , we have that and . Thus, by Definition 11, and . By the arbitrariness of , we see that . Therefore, . So, , as desired.
Remark 23. Note that, for the complete lattice in Example 18 and the prime ideal , we have , revealing that is not meet semicontinuous.
Theorem 24. A dcpo is strongly semicontinuous iff is semicontinuous and meet semicontinuous.
Proposition 25. For a (strongly) semicontinuous bc-domain , let be the complete lattice obtained from by adjoining a top element . Then is a (strongly) semicontinuous lattice.
Proof. Firstly, we show that for each . Let . Then for any with , if , then . If , then and . By Lemma 7 (ii), . Thus, and . So, and for each .
If itself is a (strongly) semicontinuous lattice, then is isolated in and is trivially a (strongly) semicontinuous lattice.
For a bc-dcpo without the biggest element, we see that . So, if is semicontinuous, then for each . It follows from that . If is strongly semicontinuous, then for all , and .
To sum up, if is a semicontinuous bc-domain, then , and is a semicontinuous lattice; if is a strongly semicontinuous bc-domain, then , , and is a strongly semicontinuous lattice.
Corollary 26. If is semicontinuous bc-domain, then , .
Corollary 27. If is a (strongly) semicontinuous bc-domain, then implies the existence of a such that .
Next we will show that (strongly) semicontinuous bc-domains also exhibit some strong types of interpolation properties.
Proposition 28. In a (strongly) semicontinuous bc-domain ,(i)for all with , one has (ii)for all with , one has
Proof. (i) For any , , and , it follows from Corollary 27 that there exists a such that . Noticing that , by Proposition 6, there exists a such that . Since is a bc-dcpo, exists and . Thus, () holds.
(ii) Let with , and . By Corollary 27 there exists a such that . Since and , there exists such that . Noticing that , we see that exists. Set ; then, . Therefore, () holds.
4. Semi-FS Domains
In this section, we introduce semi-FS domains which are counterparts of FS-domains posed by Jung  in the setting of strongly semicontinuous domains. It is proved that every strongly semicontinuous bc-domain, in particular every strongly semicontinuous lattice, is a semi-FS domain.
Definition 29 (see ). Let be dcpos. A function is said to preserve suprema of prime ideals if it is order-preserving and for any , .
Let be dcpos. We use to denote all order-preserving functions from to , use to denote all the functions preserving suprema of prime ideals from to and use to denote all the Scott-continuous functions from to . All of them are under the pointwise order. It is easy to see that .
Proposition 30. Let be dcpos. Then is a dcpo.
Proof. Let be a directed subset of and for all . Then it is easy to see that is order-preserving. For any prime ideal , we have Thus, , showing that is a dcpo.
Definition 31. Let be a dcpo. If is directed and , then we say that is an approximate identity for .
Proposition 32. Let be a dcpo. If has an approximate identity such that for all and for all , then is a strongly semicontinuous domain.
Definition 33 (see ). Let be a dcpo. A function on is finitely separating if there is a finite set such that, for each , there exists such that .
A dcpo is called a semi-FS domain if there is an approximate identity consisting of finitely separating functions.
For an FS-domain , there is an approximate identity for consisting of finitely separating functions. So, an FS-domain is a semi-FS domain.
Proposition 34. Let be a dcpo. If is finitely separating, then, for all , . Thus a semi-FS domain is a strongly semicontinuous domain.
Proof. Suppose that and with . Since is finitely separating, there exists a finite set such that for each there exists with . Let , a nonempty finite subset of . Then for each , we can get such that . As is a prime ideal, there exists such that for all . Hence for all , and . Therefore, . It follows from Proposition 12 that .
By Proposition 32 we see that is a strongly semicontinuous domain.
The next example gives a strongly semicontinuous domain which is not a semi-FS domain, showing that the reverse of Proposition 34 is not true.
Example 35. Let be the domain showing in Figure 1, where for and . It is clear that is an -domain but not compact in the Lawson topology. By [20, Corollary 2.2], we immediately see that is not an FS-domain. Note that, in , every directed set is finite. So, an order-preserving function from to preserves directed sups in . And . So, . Since is not an FS-domain, is not a semi-FS domain either.
Proposition 36. Every strongly semicontinuous bc-domain is a semi-FS domain.
Proof. Let be a strongly semicontinuous bc-domain. For each , , define by . If , then , the least element of . So, is well-defined. It is easy to see that is order-preserving with for all . Next we show that preserves suprema of prime ideals. For each , it suffices to show that , where . If , then, by the definition of , we see that . Let , then exists in . By the definition of , . Since for , we have that . By () in Proposition 28, there is a such that . It follows from that . Noticing that and , we have which yields that . So, preserves suprema of prime ideals, and .
Suppose that with . It is easy to see that and is directed. Let . Then is finite by the finiteness of . So, is a finitely separating function. Since is strongly semicontinuous, for each and , and
Therefore, is an approximate identity for consisting of finitely separating functions, and is a semi-FS domain.
Corollary 37. Every strongly semicontinuous lattice is a semi-FS domain.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers for their valuable comments and helpful suggestions. This work is supported by the NSF of China (61103018, 61300153) and Open Fund of the State Key Laboratory of Beijing University (no. SKLSDE-2011KF-08).
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