# Strongly Semicontinuous Domains and Semi-FS Domains

Authors: Qingyu He; Luoshan Xu
Journal: The Scientific World Journal
 (2014)

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## Abstract

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 dcpoL 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.


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## Body

## 1. Introduction

The theory of continuous lattices [1] and the more general theory of domains [2] 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 x approximates a state y if any computation of y yields the information of x 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], Z-continuous posets [9], and Z-precontinuous posets [10]. Motivated by the concept of semiprime ideals studied by Rav in [11], Zhao in [12] first introduced the concept of semicontinuous lattices and showed that semicontinuous lattices have many properties similar to that of continuous lattices. Li and Wu [13] studied properties of semicontinuous lattices. Bi and Xu [14] introduced the semi-Scott topology and the semi-Lawson topology on semicontinuous lattices. Jiang and Shi [15] discussed characterizations of pseudoprimes and studied strong retracts of (stable) semicontinuous lattices. In [16], Li introduced semiprime sets and generalized semicontinuous lattices to semicontinuous domains.Note that, in the definition of a semicontinuous latticeL, the condition used is that, for every element x
∈
L, ∨
⇓
x
≥
x. 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 [17] 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.We organize the paper as follows: Section2 gives preliminaries; Section 3 investigates strongly semicontinuous domains; Section 4 introduces semi-FS domains and discusses their properties.


## 2. Preliminaries

We give some basic concepts and results which will be used in the sequel. Most of them come from [4, 12]. For other unstated concepts please refer to [16].A subsetD of a poset L is calleddirected (resp.,filtered) if it is nonempty and every finite subset of D has an upper (resp., a lower) bound in D. A poset in which every directed set has a supremum is called adcpo. For a subset A of a poset L, let ↓
A
=
{
y
∈
L: ∃
x
∈
A
,
y
≤
x
} and ↑
A
=
{
y
∈
L: ∃
x
∈
A
,
y
≥
x
}. We say that A is alower set (resp., anupper set) if ↓
A
=
A (resp., ↑
A
=
A). A subset of L is anideal (resp., afilter) if and only if it is a directed lower set (resp., filtered upper set). Aprincipal ideal (resp.,principal filter) is a set of the form ↓
x
=
{
y
∈
L: y
≤
x
} (resp., ↑
x
=
{
y
∈
L: x
≤
y
}). The set of all ideals (resp. filters) in L is denoted by I
d
(
L
) (resp., Filt
(
L
)). For an ideal P
∈
I
d
(
L
), P is said to beprime if L
∖
P
=
∅ or L
∖
P
∈
Filt
(
L
). We denote the family of all prime ideals of L by P
I
(
L
). A poset is calledbounded complete, if every subset that is bounded above has a supremum. A bounded complete dcpo is called abc-dcpo.LetS
⊆
L. We denote S
↓
=
{
y
∈
L
:
∀
x
∈
S
,
y
≤
x
} and S
↑
=
{
y
∈
L
:
∀
x
∈
S
,
y
≥
x
} the set of upper bounds and lower bounds of S, respectively.Lemma 1 (see [16, Lemma  3.2.8]).
LetL be a poset, A
,
B
⊆
L. If A
⊆
B, then we have
(1)
A
↑
↓
⊆
B
↑
↓
,
A
↓
↑
⊆
B
↓
↑
.Definition 2 (see [11]).
LetL be a lattice. An ideal I of L is said to besemiprime if for any x
,
y
,
z
∈
L, x
∧
y
∈
I and x
∧
z
∈
I imply x
∧
(
y
∨
z
)
∈
I.Lemma 3 (see [12, Lemma 1.2]).
An idealP of a lattice L is semiprime if and only if there exist prime ideals P
i
(
i
∈
I
) of L such that P
=
⋂
i
∈
I
‍
P
i.Thus every prime ideal is semiprime, andP
I
(
L
)
⊆
R
d
(
L
). And by Lemma 3, we can generalize the concept of semiprime ideals to the setting of posets.Definition 4.
LetL be a poset and S
⊆
L an ideal of L. If there is a family of prime ideals {
P
i
}
i
∈
I such that S
=
⋂
i
∈
I
P
i, then S is called asemiprime ideal. The family of all semiprime ideals of L is denoted by R
d
(
L
).Li in [16] generalized semiprime ideals to semiprime sets.Definition 5 (see [16]).
LetL be a poset. A set S
⊆
L is said to be semiprime if there exists a family of prime ideals P
i
(
i
∈
I
) of L such that S
=
⋂
i
∈
I
P
i.We denote the family of all semiprime sets ofL with S
P
(
L
). Clearly, a semiprime set need not to be directed. If a semiprime set S is directed, then S is a semiprime ideal. For a dcpo, we have the following relation:
(2)
P
I
(
L
)
⊆
R
d
(
L
)
=
S
P
(
L
)
∩
I
d
l
(
L
)
⊆
S
P
(
L
)
.Proposition 6.
LetL be a bc-dcpo. Then S
P
(
L
)
=
R
d
(
L
).Proof.
It suffices to show thatS
P
(
L
)
⊆
R
d
(
L
). Let A
∈
S
P
(
L
). Then by Definition 5 there exists a family of prime ideals P
i
(
i
∈
I
) of L such that A
=
⋂
i
∈
I
P
i. For any y
1
,
y
2
∈
A, we have y
1
,
y
2
∈
P
i for all i
∈
I. Since P
i is directed, there exists y
∈
P
i such that y
1
,
y
2
≤
y. As L is a bc-dcpo, we see that y
1
∨
y
2 exists and y
1
∨
y
2
∈
P
i for all i
∈
I. Therefore, y
1
∨
y
2
∈
A. So, A is directed and A
∈
R
d
(
L
), as desired.Lemma 7.
For a bc-dcpoL, let M
⊆
L and L
⊤
=
L
∪
{
⊤
} be the complete lattice obtained from L by adjoining a top element ⊤. Then we have(i)
M
∈
P
I
(
L
) if and only if M
∈
P
I
(
L
⊤
);
(ii)
M
∈
R
d
(
L
) if and only if M
∈
R
d
(
L
⊤
).Proof.
(i) Suppose thatM
∈
P
I
(
L
⊤
). Since M
⊆
L, we have L
∖
M
=
L
⊤
∖
M
∈
Filt
(
L
) and M
∈
P
I
(
L
).
Conversely, letM
∈
P
I
(
L
). Then M
∈
I
d
(
L
) and L
∖
M
∈
Filt
(
L
). So, L
⊤
∖
M
=
(
L
∖
M
)
∪
{
⊤
}
∈
Filt
(
L
⊤
) and M
∈
P
I
(
L
⊤
).
(ii) Suppose thatM
∈
R
d
(
L
). Then by Definition 4, M
=
⋂
i
∈
I
P
i, where P
i
∈
P
I
(
L
)
(
i
∈
I
). It follows from (i) above that P
i
∈
P
I
(
L
⊤
) for all i
∈
I; thus, M
∈
R
d
(
L
⊤
).
Conversely, letM
∈
R
d
(
L
⊤
). By Lemma 3, there exist prime ideals P
i
(
i
∈
I
) of L
⊤ such that M
=
⋂
i
∈
I
P
i. Since ⊤
∉
M, ⊤
∉
P
i
0 for some i
0
∈
I. Let J
=
{
j
∈
I
:
⊤
∉
P
j
}. Then for any i
∈
I
∖
J, we see that ⊤
∈
P
i
=
L
⊤. By (i) above, we have P
j
∈
P
I
(
L
) for each j
∈
J. Then M
=
⋂
j
∈
J
P
j, and M
∈
S
P
(
L
). By Proposition 6, M
∈
R
d
(
L
).In a posetL, we say that x isway-below
y, or x
approximates
y, written x
≪
y, and if D is directed with existing sup
⁡
D and sup
⁡
D
≥
y, then x
≤
d for some d
∈
D. Equivalently, x
≪
y iff x
∈
I for every ideal I of L such that y
≤
sup
⁡
I whenever sup
⁡
I exists. We use ↡
x to denote the set {
a
∈
L
:
a
≪
x
}. If L is a dcpo and, for every element x
∈
L, the set ↡
x is directed and sup
⁡
↡
x
=
x, then L is called adomain. A domain L is called anL-domain if for each x
∈
L, the principal ideal ↓
x is a complete lattice. A complete lattice which is a domain is called acontinuous lattice.For complete lattices, replacing ideals with semiprime ideals, Zhao in [12] defined a weak form of the way-below relation.Definition 8 (see [12]).
LetL be a complete lattice. Define the semiway-below relation ⇐ on L as follows: for x
,
y
∈
L, x
⇐
y if for any semiprime ideal I of L, y
≤
∨
I implies x
∈
I. For each x
∈
L, we write ⇓
x
=
{
y
∈
L
:
y
⇐
x
}
,
⇑
x
=
{
y
∈
L
:
x
⇐
y
}.Definition 9 (see [12]).
A complete latticeL is said to besemicontinuous, if for any x
∈
L, x
≤
∨
(
⇓
x
).Zhao [12] showed that the interpolation property holds in semicontinuous lattices.Theorem 10 (see [12, Theorem 1.8]).
IfL is a semicontinuous lattice, then x
⇐
y implies the existence of a z
∈
L such that x
⇐
z
⇐
y.


## 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.
LetL be a poset. Define the relation ⇐ on L as follows: for any x
,
y
∈
L, y
⇐
x if for any semiprime set S of L, x
∈
S
↑
↓ implies y
∈
S. An element k of L is said to besemicompact if k
⇐
k. For each x
∈
L, we write ⇓
x
=
{
y
∈
L
:
y
⇐
x
} and ⇑
x
=
{
y
∈
L
:
x
⇐
y
}.Proposition 12.
LetL be a dcpo; then,
(3)
⇓
x
=
∩
{
S
∈
S
P
(
L
)
:
x
∈
S
↑
↓
}
=
∩
{
S
∈
R
d
(
L
)
:
x
≤
∨
S
}
=
∩
{
P
∈
P
I
(
L
)
:
x
≤
∨
P
}
.
So, for each x
∈
L, ⇓
x
∈
S
P
(
L
). If L is a bc-dcpo, then ⇓
x
∈
R
d
(
L
).Proof.
Note thatJ
↑
↓
=
↓
(
∨
J
) for each J
∈
I
d
(
L
). So, by Lemma 1, we have that
(4)
⇓
x
=
∩
{
S
∈
S
P
(
L
)
:
x
∈
S
↑
↓
}
⊆
∩
{
S
∈
R
d
(
L
)
:
x
∈
↓
(
∨
S
)
}
⊆
∩
{
P
∈
P
I
(
L
)
:
x
∈
↓
(
∨
P
)
}
.
LetA
=
∩
{
P
∈
P
I
(
L
)
:
x
∈
↓
(
∨
P
)
}. Then A
∈
S
P
(
L
).
Next we show thatA
⊆
⇓
x. Let y
∈
A. For any M
∈
S
P
(
L
) with x
∈
M
↑
↓, by Definition 5 there exists a family of prime ideals {
P
i
}
i
∈
I such that M
=
⋂
i
∈
I
P
i. It follows from Lemma 1 that x
∈
M
↑
↓
⊆
P
i
↑
↓
=
↓
(
∨
P
i
) for each i
∈
I. Since y
∈
A
=
∩
{
P
∈
P
I
(
L
)
:
x
∈
↓
(
∨
P
)
}, we have that y
∈
P
i for each i
∈
I and y
∈
⋂
i
∈
I
P
i
=
M. By Definition 11, y
⇐
x. So, A
⊆
⇓
x, as desired.In [16], the semiway-below relation ⇐
s on a poset was defined [16, Definition 3.2
.
2] in a different way. For a dcpo L and x
∈
L, it is established in [16, Proposition 3.2
.
5] that ⇓
s
x
=
∩
{
P
∈
P
I
(
L
)
:
x
≤
∨
P
}. So, by Proposition 12 above, we see that ⇓
x
=
⇓
s
x. This means that for a dcpo L and x
,
y
∈
L, y
⇐
s
x
⇔
y
⇐
x. Thus, in the setting of dcpos, Definition 11 is equivalent to Definition 3.2
.
2 in [16].Note that for a posetL and x
,
y
∈
L, y
⇐
s
x need not imply y
⇐
x.Remark 13.
LetL be a dcpo and a
,
b
,
c
,
d
∈
L. Then it is easy to check that(1)
a
⇐
b does not imply a
≤
b, the typical modular lattice M
5 is a counterexample;
(2)
ifa
≤
b
⇐
c
≤
d, then a
⇐
d. If a
⇐
b
⇐
c, then a
⇐
c;
(3)
ifa
∨
b exists and a
,
b
⇐
c, then a
∨
b
⇐
c;
(4)
for anya
,
b
∈
L, a
≪
b implies a
⇐
b.Definition 14 (see [16]).
A dcpoL is called asemicontinuous dcpo if for all x
∈
L, x
∈
(
⇓
x
)
↑
↓. A semicontinuous dcpo will also be called asemicontinuous domain. A bc-dcpo which is semicontinuous will be called asemicontinuous bc-domain.By Proposition12, one can immediately have the following.Proposition 15.
LetL be a dcpo. Then L is semicontinuous iff for any x
∈
L, ⇓
x is the smallest semiprime set S such that x
∈
S
↑
↓.Proposition 16.
LetL be a dcpo. If, for any x
∈
L, there exists a subset A
⊆
⇓
x and x
∈
A
↑
↓, then L is semicontinuous.Proof.
It follows from Lemma1 and Definition 14.Strengthening the condition in Definition14, we give the following.Definition 17.
A dcpoL is said to bestrongly semicontinuous if for each x
∈
L,
(5)
x
∈
(
⇓
x
∩
↓
x
)
↑
↓
.
A dcpo (bc-dcpo) which is strongly semicontinuous will be called astrongly semicontinuous domain (strongly semicontinuous bc-domain). A complete lattice which is strongly semicontinuous will be called astrongly semicontinuous lattice.Clearly, every strongly semicontinuous domain is semicontinuous, a semicontinuous domainL satisfying the condition x
∈
(
⇓
x
∩
↓
x
)
↑
↓
(
∀
x
∈
L
) is strongly semicontinuous. It is easy to see that, for a dcpo L without proper prime ideals, every pair of elements in L has the semiway-below relation ⇐ and L is strongly semicontinuous. However, a semicontinuous domain need not be strongly semicontinuous. The following counterexample first appeared in [18].Example 18 (see [18]).
LetL
=
{
⊥
,
a
,
b
,
x
,
⊤
}
∪
{
x
n
:
n
=
0,1
,
…
}. The partial order on L is defined by ⊥
≤
a
,
b
≤
x
0
≤
⋯
≤
x
n
⋯
≤
⊤, ⊥
≤
x
≤
⊤. It is clear that the prime ideals of L are L
∖
↑
x and L. We observe that, for any t
∈
L, ⇓
t
=
L
∖
↑
x and t
≤
∨
⇓
t. Thus, L is a semicontinuous domain. However, note that, in this example x
∉
(
⇓
x
∩
↓
x
)
↑
↓, which yields that L is not strongly semicontinuous.Proposition 19.
Every domain is a strongly semicontinuous domain.Proof.
It follows from Remark13 (4) that ∀
x
∈
L, ↡
x
⊆
⇓
x
∩
↓
x. By Lemma 1, x
∈
↓
x
=
↓
(
∨
↡
x
)
=
(
↡
x
)
↑
↓
⊆
(
⇓
x
∩
↓
x
)
↑
↓. Thus, L is a strongly semicontinuous.Wu and Li in [19] introduced the following concept of meet semicontinuous lattices.Definition 20 (see [19]).
A complete latticeL is said to bemeet semicontinuous if for any x
∈
L and I
∈
R
d
(
L
), x
∧
(
∨
I
)
=
∨
(
x
∧
I
).It is known that a semicontinuous lattice need not be a meet semicontinuous lattice. However, it is proved in [18] that strongly semicontinuous lattices are all meet semicontinuous. Generalize meet semicontinuous lattices, we haveDefinition 21.
A dcpoL is said to bemeet semicontinuous if for any x
∈
L and S
∈
S
P
(
L
), (
↓
x
∩
S
)
↑
↓
=
↓
x
∩
S
↑
↓.It is easy to check that, for complete lattices, the meet semicontinuity in Definitions20 and 21 are equivalent.Proposition 22.
Every strongly semicontinuous domain is meet semicontinuous.Proof.
Suppose thatL is a strongly semicontinuous domain. For any x
∈
L and S
∈
S
P
(
L
), by Lemma 1, it is easy to see that ↓
x
∩
S
↑
↓
⊇
(
↓
x
∩
S
)
↑
↓. So, it suffices to show that ↓
x
∩
S
↑
↓
⊆
(
↓
x
∩
S
)
↑
↓. To this end, let t
∈
↓
x
∩
S
↑
↓. Then t
∈
S
↑
↓ and t
∈
(
⇓
t
∩
↓
t
)
↑
↓ since L is strongly semicontinuous. For any r
∈
⇓
t
∩
↓
t, we have that r
⇐
t
∈
S
↑
↓ and r
≤
t
≤
x. Thus, by Definition 11, r
∈
S and r
∈
↓
x
∩
S. By the arbitrariness of r
∈
⇓
t
∩
↓
t, we see that ⇓
t
∩
↓
t
⊆
↓
x
∩
S. Therefore, t
∈
(
⇓
t
∩
↓
t
)
↑
↓
⊆
(
↓
x
∩
S
)
↑
↓. So, ↓
x
∩
S
↑
↓
⊆
(
↓
x
∩
S
)
↑
↓, as desired.Remark 23.
Note that, for the complete lattice in Example18 and the prime ideal L
∖
↑
x, we have ↓
x
∩
(
L
∖
↑
x
)
↑
↓
≠
(
↓
x
∩
(
L
∖
↑
x
)
)
↑
↓, revealing that L is not meet semicontinuous.Theorem 24.
A dcpoL is strongly semicontinuous iff L is semicontinuous and meet semicontinuous.Proof.
⇒: By Proposition 22.
⇐: For each x
∈
L, by the semicontinuity of L we have that x
∈
(
⇓
x
)
↑
↓. It follows from meet semicontinuity of L that x
∈
↓
x
∩
(
⇓
x
)
↑
↓
=
(
↓
x
∩
⇓
x
)
↑
↓. So, by Definition 17, L is strongly semicontinuous.Proposition 25.
For a (strongly) semicontinuous bc-domainL, let L
⊤
=
L
∪
{
⊤
} be the complete lattice obtained from L by adjoining a top element ⊤. Then L
⊤ is a (strongly) semicontinuous lattice.Proof.
Firstly, we show that⇓
L
⊤
x
⊇
⇓
L
x for each x
∈
L. Let y
⇐
L
x. Then for any P
∈
R
d
(
L
⊤
) with ∨
L
⊤
P
≥
x, if P
=
L
⊤, then y
∈
P. If P
≠
L
⊤, then ⊤
∉
P and P
⊆
L. By Lemma 7 (ii), P
∈
R
d
(
L
). Thus, ∨
L
P
=
∨
L
⊤
P
≥
x and y
∈
P. So, y
⇐
L
⊤
x and ⇓
L
x
⊆
⇓
L
⊤
x for each x
∈
L.
IfL itself is a (strongly) semicontinuous lattice, then ⊤ is isolated in L
⊤ and L
⊤ is trivially a (strongly) semicontinuous lattice.
For a bc-dcpoL without the biggest element, we see that ∨
L
⊤
L
=
⊤. So, if L is semicontinuous, then ∨
L
⊤
⇓
L
⊤
x
≥
∨
L
⊤
⇓
L
x
≥
x for each x
∈
L. It follows from
(6)
⇓
L
⊤
⊤
⊇
∪
x
∈
L
⇓
L
⊤
x
⊇
∪
x
∈
L
⇓
L
x
that ∨
L
⊤
(
⇓
L
⊤
⊤
∩
↓
⊤
)
=
∨
L
⊤
⇓
L
⊤
⊤
≥
∨
L
⊤
∪
x
∈
L
⇓
L
x
≥
∨
L
⊤
L
=
⊤. If L is strongly semicontinuous, then for all x
∈
L, x
≥
∨
L
⊤
(
⇓
L
⊤
x
∩
↓
⁡
x
)
≥
∨
L
⊤
(
⇓
L
x
∩
↓
x
)
≥
x and ∨
L
⊤
(
⇓
L
⊤
x
∩
↓
x
)
=
x.
To sum up, ifL is a semicontinuous bc-domain, then ∀
y
∈
L
⊤, y
≤
∨
L
⊤
⇓
L
⊤
y and L
⊤ is a semicontinuous lattice; if L is a strongly semicontinuous bc-domain, then ∀
y
∈
L
⊤, ∨
L
⊤
(
⇓
L
⊤
y
∩
↓
y
)
=
y, and L
⊤ is a strongly semicontinuous lattice.Corollary 26.
IfL is semicontinuous bc-domain, then ∀
x
∈
L, ⇓
L
⊤
x
=
⇓
L
x.Proof.
By the proof of Proposition25, we have that ⇓
L
⊤
x
⊇
⇓
L
x and L
⊤ is a semicontinuous lattice. Then it follows from Propositions 12 and 15 and Lemma 7 (ii) that ⇓
L
⊤
x
⊆
⇓
L
x. So, ⇓
L
⊤
x
=
⇓
L
x.Whether the interpolation property holds or not in semicontinuous domains is still unknown. So, by Theorem10 and Corollary 26, we immediately have the following.Corollary 27.
IfL is a (strongly) semicontinuous bc-domain, then x
⇐
y implies the existence of a z
∈
L such that x
⇐
z
⇐
y.Next we will show that (strongly) semicontinuous bc-domains also exhibit some strong types of interpolation properties.Proposition 28.
In a (strongly) semicontinuous bc-domainL,(i)
for allx
,
y
∈
L with x
≤
y, one has
(7)
(
S
I
≤
)
x
⇐
y
i
m
p
l
i
e
s
(
∃
z
∈
L
)
(
x
⇐
z
⇐
y
,
x
≤
z
)
;
(ii)
for allx
,
y
∈
L with x
<
y, one has
(8)
(
S
I
<
)
x
⇐
y
i
m
p
l
i
e
s
(
∃
z
∈
L
)
(
x
⇐
z
⇐
y
,
x
<
z
)Proof.
(i) For anyx
,
y
∈
L, x
≤
y, and x
⇐
y, it follows from Corollary 27 that there exists a z
0
∈
L such that x
⇐
z
0
⇐
y. Noticing that ⇓
y
∈
R
d
(
L
), by Proposition 6, there exists a w
∈
R
d
(
L
) such that x
,
z
0
≤
w. Since L is a bc-dcpo, x
∨
z
0 exists and x
⇐
z
0
≤
x
∨
z
0
⇐
y. Thus, (S
I
≤) holds.
(ii) Letx
,
y
∈
L with x
<
y, and x
⇐
y. By Corollary 27 there exists a z
0
∈
L such that x
⇐
z
0
⇐
y. Since ⇓
y
∈
R
d
(
L
) and y
≤
∨
⇓
y, there exists z
1
∈
⇓
y such that z
1
≰
x. Noticing that z
0
,
z
1
∈
⇓
y
∈
R
d
(
L
), we see that z
0
∨
z
1 exists. Set z
=
z
0
∨
z
1; then, x
<
z. Therefore, (S
I
<) holds.


## 4. Semi-FS Domains

In this section, we introduce semi-FS domains which are counterparts of FS-domains posed by Jung [17] 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 [16]).
LetL
,
M be dcpos. A function f
:
L
→
M is said topreserve suprema of prime ideals if it is order-preserving and for any P
∈
P
I
(
L
), f
(
⋁
P
)
=
⋁
f
(
P
).LetL
,
M be dcpos. We use [
L
,
M
] to denote all order-preserving functions from L to M, use [
L
↪
M
] to denote all the functions preserving suprema of prime ideals from L to M and use [
L
→
M
] to denote all the Scott-continuous functions from L to M. All of them are under the pointwise order. It is easy to see that [
L
→
M
]
⊆
[
L
↪
M
]
⊆
[
L
,
M
].Proposition 30.
LetL
,
M be dcpos. Then [
L
↪
M
] is a dcpo.Proof.
LetG be a directed subset of [
L
↪
M
] and f
(
x
)
=
∨
g
∈
G
g
(
x
) for all x
∈
L. Then it is easy to see that f is order-preserving. For any prime ideal P
∈
P
I
(
L
), we have
(9)
∨
f
(
P
)
=
∨
x
∈
P
f
(
x
)
=
∨
x
∈
P
∨
g
∈
G
g
(
x
)
=
∨
g
∈
G
∨
g
(
P
)
=
∨
g
∈
G
g
(
∨
P
)
=
f
(
∨
P
)
.
Thus, f
∈
[
L
↪
M
], showing that [
L
↪
M
] is a dcpo.Definition 31.
LetL be a dcpo. If D
⊆
[
L
,
L
] is directed and sup
⁡
δ
∈
D
δ
=
I
d
L, then we say that D is anapproximate identity for L.Proposition 32.
LetL be a dcpo. If L has an approximate identity D
⊆
[
L
↪
L
] such that δ
(
x
)
⇐
x for all δ
∈
D and for all x
∈
L, then L is a strongly semicontinuous domain.Proof.
For anyx
∈
L, let A
=
{
δ
(
x
)
:
δ
∈
D
}. Then A
⊆
⇓
x
∩
↓
x. Note that D is directed and x
∈
↓
(
∨
A
)
=
A
↑
↓. By the Proposition 16 and Definition 17, it follows that L is a strongly semicontinuous domain.Definition 33 (see [17]).
LetL be a dcpo. A function δ
:
L
→
L on L is finitely separating if there is a finite set F
δ such that, for each x
∈
L, there exists y
∈
F
δ such that δ
(
x
)
≤
y
≤
x.
A dcpoL is calleda semi-FS domain if there is an approximate identity D
⊆
[
L
↪
L
] consisting of finitely separating functions.For an FS-domainL, there is an approximate identity D
⊆
[
L
→
L
]
⊆
[
L
↪
L
] for L consisting of finitely separating functions. So, an FS-domain is a semi-FS domain.Proposition 34.
LetL be a dcpo. If δ
∈
[
L
↪
L
] is finitely separating, then, for all x
∈
L, δ
(
x
)
⇐
x. Thus a semi-FS domain is a strongly semicontinuous domain.Proof.
Suppose thatx
∈
L and P
∈
P
I
(
L
) with x
≤
∨
P. Since δ is finitely separating, there exists a finite set F
δ such that for each d
∈
P there exists y
d
∈
F
δ with δ
(
d
)
≤
y
d
≤
d. Let F
δ
′
=
{
y
d
∈
F
δ
:
d
∈
P
}, a nonempty finite subset of F
δ. Then for each y
∈
F
δ
′, we can get d
y
∈
P such that δ
(
d
y
)
≤
y
≤
d
y. As P is a prime ideal, there exists d
0
∈
P such that y
≤
d
y
≤
d
0 for all y
∈
F
δ
′. Hence for all d
∈
P, δ
(
d
)
≤
y
d
≤
d
0 and δ
(
x
)
≤
δ
(
∨
P
)
=
∨
d
∈
P
δ
(
d
)
≤
d
0. Therefore, δ
(
x
)
∈
P. It follows from Proposition 12 that δ
(
x
)
⇐
x.
By Proposition32 we see that L 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 Proposition34 is not true.Example 35.
LetL
=
{
0
,
b
1
,
b
2
}
∪
{
a
j
:
j
=
1,2
,
…
} be the domain showing in Figure 1, where 0
<
b
i
<
a
j for i
=
1,2 and j
=
1,2
,
…. It is clear that L is an L-domain but not compact in the Lawson topology. By [20, Corollary 2.2], we immediately see that L is not an FS-domain. Note that, in L, every directed set is finite. So, an order-preserving function from L to L preserves directed sups in L. And [
L
→
L
]
⊆
[
L
↪
L
]
⊆
[
L
,
L
]
=
[
L
→
L
]. So, [
L
→
L
]
=
[
L
↪
L
]. Since L is not an FS-domain, L is not a semi-FS domain either.Figure 1
Lis strongly semicontinuous but not a semi-FS domain.Proposition 36.
Every strongly semicontinuous bc-domain is a semi-FS domain.Proof.
LetL be a strongly semicontinuous bc-domain. For each x
∈
L, S
∈
P
fin
(
L
), define δ
S
:
L
→
L by δ
S
(
x
)
=
∨
{
y
∈
S
∩
↓
x
:
y
⇐
x
}. If {
y
∈
S
∩
↓
x
:
y
⇐
x
}
=
∅, then δ
S
(
x
)
=
⊥, the least element of L. So, δ
S
(
x
) is well-defined. It is easy to see that δ
S is order-preserving with δ
S
(
x
)
≤
x for all x
∈
L. Next we show that δ
S preserves suprema of prime ideals. For each P
∈
P
I
(
L
), it suffices to show that δ
S
(
∨
P
)
=
δ
S
(
k
)
≤
∨
p
∈
P
δ
S
(
p
), where k
=
∨
P. If {
y
∈
S
∩
↓
k
:
y
⇐
k
}
=
∅, then, by the definition of δ
S, we see that δ
S
(
k
)
=
⊥
≤
∨
p
∈
P
δ
S
(
p
). Let {
y
1
,
…
,
y
l
}
=
{
y
∈
S
∩
↓
k
:
y
⇐
k
}
≠
∅, then m
=
∨
i
=
1
l
y
i exists in L. By the definition of δ
S, δ
S
(
k
)
=
m
≤
k. Since y
i
⇐
k for i
=
1,2
,
…
,
l, we have that m
⇐
k. By (SI
≤) in Proposition 28, there is a m
*
≥
m such that m
⇐
m
*
⇐
k. It follows from m
*
⇐
k
=
∨
P that m
,
m
*
∈
P. Noticing that m
≤
m
* and y
i
≤
m
⇐
m
*
∈
P, we have {
y
1
,
…
,
y
l
}
⊆
{
y
∈
S
∩
↓
m
*
:
y
⇐
m
*
} which yields that ∨
p
∈
P
δ
S
(
p
)
≥
δ
S
(
m
*
)
≥
δ
S
(
k
). So, δ
S preserves suprema of prime ideals, and δ
∈
[
L
↪
L
].
Suppose thatS
,
T
∈
P
fin
(
L
) with S
⊆
T. It is easy to see that δ
S
≤
δ
T and D
=
{
δ
S
}
S
∈
P
fin
(
L
) is directed. Let F
δ
S
=
{
δ
S
(
x
)
:
x
∈
L
}. Then F
δ
S is finite by the finiteness of S. So, δ
S is a finitely separating function. Since L is strongly semicontinuous, for each x
∈
L and δ
S
∈
D, δ
S
(
x
)
=
∨
{
y
∈
S
∩
↓
x
:
y
⇐
x
}
≤
x and
(10)
∨
δ
S
∈
D
δ
S
(
x
)
=
∨
δ
S
∈
D
(
∨
{
y
∈
S
∩
↓
x
:
y
⇐
x
}
)
≥
∨
z
⇐
x
(
∨
{
y
∈
{
z
}
∩
↓
x
:
y
⇐
x
}
)
=
∨
(
⇓
x
∩
↓
x
)
=
x
.
Therefore,D
=
{
δ
S
}
S
∈
P
fin
(
L
)
⊆
[
L
↪
L
] is an approximate identity for L consisting of finitely separating functions, and L is a semi-FS domain.Corollary 37.
Every strongly semicontinuous lattice is a semi-FS domain.



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*Source: 262648-2014-08-17.xml*