A note on the problem when FS-domains coincide with RB-domains

In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilattice is an FS-domain if and only if it is an RB-domain; an L-domain is an FS-domain if and only if it is an RB-domain. These results are expected to provide useful hints to the open problem of whether FS-domains are identical with RB-domains.


Introduction
In [4,5], A. Jung introduced the notion of FS-domains (that is, finitely separating domains) and proved that the category FS of FS-domains is a maximal Cartesian closed full subcategrory of continuous dcpos.Also in [4,5], it had been shown that the category RB of RB-domains (or retracts of algebraic FS-domains) is Cartesian closed, but its maximality is still an open question.
A well-known result is that every RB-domain is an FS-domain.Even though much attention has been paid to the question whether each FSdomain is an RB-domain, it is still an open problem [2,4,5].We only make a brief review for the works which are closely related to this problem.In [6], J.D. Lawson proved that the domain of closed formal balls based on a complete metric space is an FS-domain.Meanwhile, it is still unknown whether this domain is an RB-domain.In [7], J.H. Liang and K. Keimel proved that FS-domains and RB-domains are equivalent for L-domains with least elements.In [3], R. Heckmann obtained some characterizations of FS-domains by power domains.In those characterisations, separation by the elements of a finite set is replaced by separation by a continuous nondeterministic function with finite image.
A basic result about RB-domain is that a dcpo is an RB-domain if and only if it has an approximate identity consisting of deflations [4,5].Towards the open problem whether each FS-domain is an RB-domain, a natural ideal is to find a deflation over every finitely separating function.Inspired by the idea of R. Heckmann [3], a possible approach for us is to construct a deflation based on the relating finite subset F δ over every finitely separating function δ.
In this paper, we introduce the notion of super finitely separating functions which is a special case of finitely separating functions.Here, separation by the elements of a finite set is replaced by an order preserving function with finite image.It is shown that a dcpo is an RB-domain if and only if it has an approximate identity consisting of super finitely separating functions, which can be seen as a characterization of RB-domains.Finally, we show that FS-domains always coincide with RB-domains under some special conditions, such as consistent join-semilattices or L-domains (here, the least element is not necessary).Our result may provide useful hints to the open problem mentioned above.

FS-domains and RB-domains
A function f : S → T between dcpos is said to be Scott continuous if it sends directed subsets to directed subsets, and preserves sups of directed subsets.We denote all the Scott continuous funcitons from S to T by [S → T ].Definition 2.1.[2,4] An approximate identity for a dcpo S is a directed subset D ⊆ [S → S] satisfying sup D = id S , the identity on S. Definition 2.2.[2,4] A Scott continuous function δ : S → S on a dcpo S is finitely separating if there exists a finite set F δ such that for each x ∈ S, there exists y ∈ F δ such that δ(x) ≤ y ≤ x.
(1) A dcpo S is called an FS-domain if there is an approximate identity for S consisting of finitely separating functions.
(2) An algebraic FS-domain is called a bifinite domain.
(3) A dcpo S is called an RB-domain if it is isomorphic to the image of some bifinite domain under a Scott continuous projection.That is, an RB-domain is a continuous retract of some bifinite domain.
is an approximate identity for a dcpo S, then D = {δ 2 = δ • δ : δ ∈ D} is also an approximate identity for S.
(2) If a Scott continuous function δ : S → S on a dcpo S is finitely separating, then δ(x) x for each x ∈ S.

Lemma 2.4. [1]
A dcpo S is an RB-domain if and only if there is an approximate identity for S consisting of deflations, where a deflation f : S → S is a Scott continuous function with finite image and f (x) ≤ x holds for each x ∈ S.
Lemma 2.3 indicates that every bifinite domain is an RB-domain and every RB-domain is an FS-domain.
(3) If a dcpo S has an infinite number of minimal elements, then S is not an FS-domain.
Definition 2.6.[7] A dcpo S is an L-domain if for every element x of S, the principal ideal ↓x = {y ∈ S : y ≤ x} is a complete lattice.In this case, we write sup ↓x for the supremum operation in ↓x.
Lemma 2.7.[7] In any L-domain S, if x ≤ y and φ = A ⊆ ↓x, then sup ↓x A = sup ↓y A.
Corollary 2.8.[7] For each L-domain S with the least element, the following statements are equivalent: (1) S is an FS-domain.
Each RB-domain is an FS-domain.However, we do not know whether every FS-domain is an RB-domain.For a positive answer, we need to find a deflation above every finitely separating function δ.We notice that in [3], R. Heckmann uses the existing finite separating set: F δ to give characterizations of FS domains.Therefore, a possible approach for us is to construct a deflation based on the relating F δ .The first trouble thing is that for each x ∈ S, there may exist more than one element y ∈ F δ such that δ(x) ≤ y ≤ x.Using the Axiom of Choice, we provide the following lemma to give an equivalent description of finitely separating functions.Lemma 2.9.A Scott continuous function δ : S → S on a dcpo S is finitely separating if and only if there exists a function f δ : S → S with finite image such that δ(x) ≤ f δ (x) ≤ x for each x ∈ S.
Proof.Suppose δ : S → S is finitely separating.For each x ∈ S, there exists an element y x ∈ F such that δ(x) ≤ y x ≤ x.According to the Axiom of Choice, we define a function f δ : S → S by f δ (x) = y x for each x ∈ S. Obviously, Im(f δ ) ⊆ F is finite.
Conversely, let F = Im(f δ ).It can be checked that δ : S → S is finitely separating.
Remark 2.10.We remind the reader that the function f δ : S → S, given in Lemma 2.9, is not necessary to be order preserving.A typical instance is given in Example 3.10.

Super finitely separating functions
In this section, we introduce the concept of super finitely separating functions and show that a dcpo S is an RB-domain if and only if S has an approximate identity consisting of super finitely separating functions.Then we show that FS-domains coincide with RB-domains in one of the following cases: (1) consistent join-semilattices; (2) dual of consistent join-semilattices; (3) L-domains.Definition 3.1.A Scott continuous function δ : S → S on a dcpo S is called super finitely separating if there exists an order preserving function f δ : S → S with finite image such that δ(x) ≤ f δ (x) ≤ x for each x ∈ S.
An immediate conclusion is that every deflation is super finitely separating and every super finitely separating function is finitely separating.Lemma 3.2.Let S be a domain and δ : S → S be a super finitely separating function.Then there exists a Scott continuous function θ : S → S with finite image such that δ(x) ≤ θ(x) ≤ x for each x ∈ S.
Proof.From Definition 3.1, there exists an order preserving function f δ : S → S with finite image such that δ(x) ≤ f δ (x) ≤ x for each x ∈ S.
Define θ : S → S by θ(x) = sup{f δ (y) : y x} for each x ∈ S. Since S is a domain and f δ : S → S is order preserving, θ : S → S is well defined.It is easy to see that θ has finite image and it is order preserving.Thus θ : S → S is Scott continuous.
Theorem 3.3.A dcpo S is an RB-domain if and only if there is an approximate identity for S consisting of super finitely separating functions.
Proof.Suppose S is an RB-domain.Since every deflation is a super finitely separating function, there is an approximate identity for S consisting of super finitely separating functions.Suppose that there exists an approximate identity {δ i : i ∈ I} for S, consisting of super finitely separating functions.By Lemma 3.2, for each δ i , there exists a deflation θ i such that δ i (x) ≤ θ i (x) ≤ x for each x ∈ S. Since sup{δ i : i ∈ I} = id S , we have sup{θ i : i ∈ I} = id S .We have proved that, S is an RB-domain.If the dual of P is a consistent join-semilattice, we call it a dual consistent join-semilattice.Remark 3.5.(1) A join-semilattice is always a consistent join-semilattice.
(2) A bounded complete domain D is always a consistent join-semilattice.However, the converse does not hold in general even if D is an FS-domain.In fact, a bounded complete domain must have the least element, which is different from a consistent join-semilattice.Proposition 3.6.If a dcpo S is a consistent join-semilattice (or a dual consistent join-semilattice), then each finitely separating function δ : S → S is super finitely separating.
Proof.Since δ : S → S is a finitely separating function, there exists a function f δ : S → S with finite Im(δ) such that δ(x) ≤ f δ (x) ≤ x for each x ∈ S, where Im(δ) stands for the image of the function δ.
If S is a consistent join-semilattice, we denote f δ (x) = sup{f δ (y) : y ≤ x} for each x ∈ S. Then the nonempty subset {f δ (y) : It is easy to see that f δ (x 1 ) ≤ f δ (x 2 ) for all x 1 , x 2 ∈ S with x 1 ≤ x 2 .Thus δ is a super finitely separating function on S.
In case that S is a dual consistent join-semilattice, just let f δ (x) = inf{f δ (y) : y ≥ x} for each x ∈ S. We can get the conclusion that δ is a super finitely separating function on S. Corollary 3.7.A consistent join-semilattice (or a dual consistent joinsemilattice) is an FS-domain if and only if it is an RB-domain.
Proof.This follows immediately from Lemma 2.4, Theorem 3.3 and Proposition 3.6.
It is clear that a sup semilattice is a consistent join-semilattice and an inf semilattice is a dual consistent join-semilattice.Then by Corollary 3.7, for a sup semilattice or an inf semilattice, it is an FS-domain if and only it is an RB-domain.Proof.Based on the proof of Proposition 3.6, to prove this proposition, we only need to show the existence of inf{f δ (y) : y ≥ x} for each x ∈ S.
Since S is an L-domain, every bounded subset of S has the infimum.In particular, f δ (x) ∧ f δ (y) exists for each pair x, y ∈ S with x ≤ y.This can imply that inf{f δ (x) f δ (y) : x ≤ y} exists for each x ∈ S. Observing the sets {f δ (y) : x ≤ y} and {f δ (x) f δ (y) : x ≤ y} have the same lower bounds, we can conclude that inf{f δ (y) : y ≥ x} exists for each x ∈ S. Corollary 3.9.An L-domain is an FS-domain if and only if it is an RBdomain.
Proof.This follows immediately from Lemma 2.3, Theorem 3.3 and Proposition 3.8.
The following example shows that a finitely separating function is not necessary super finitely separating.Since every directed subset in S has a maximum element, S is a domain and the order preserving function δ is Scott continuous.It is easy to see that δ is finitely separating if the associated F δ is chosen as {a, b, c, 0}.But δ is not super finitely separating.In fact: if a function f δ : S → S with finite image separates δ and id S , then f δ (a i ) = a and f δ (c i ) = c hold eventually, but c ≤ a is not true, that is to say, f δ is not order preserving.

Definition 3 . 4 .
A poset P is said to be a consistent join-semilattice if each bounded pair in S has a least upper bound.Equivalently, for each a, b ∈ S, if there exists c ∈ S such that a ≤ c and b ≤ c, then a ∨ b exists.

Proposition 3 . 8 .
If S is an L-domain, then each finitely separating function δ : S → S is super finitely separating.

Example 3 .
10. Let S be the dcpo as Fig.1.Then, δ : S → S is defined as follows:δ(a i ) = b i , δ(b i ) = d i , δ(c i ) = d i for each i ∈ N; δ(a) = band maps others to the least element 0.