≪-separating domains and FS-domains

Achim Jung’s category of FS-domains is one of the two maximal cartesian-closed categories of continuous dcpos [1], and their theory is proposed as Exercises 9.6.25—9.6.32 in the book. Recently, Wei Luan proposed a variation on the notion, and which he calls ≪-separating domains. One of his main theorems is that given an FS-domain X and a ≪-separating domain Y, the space of Scott-continuous maps [X → Y] is a ≪-separating domain. Another one is that the same holds if instead X is assumed to be Noetherian; in passing, he proves an interesting lemma on the way-below relation on spaces of Scott-continuous maps—this will be Lemma C below. Unfortunately, we will see that the category of ≪-separating domains is not Cartesian-closed, but that is life.

RB-domains, FS-domains, ≪-separating domains

Using the denomination used by Wei Luan, FS-domains are defined through the help of something called an approximate identity. (I didn’t define the notion in the book, though, but this is practical.) An approximate identity on a dcpo X is a directed family of Scott-continuous maps (f_i)_{i ∈ I} from X to X such that the pointwise supremum sup_{i ∈ I} f_i is equal to the identity map on X.

An RB-domain is a dcpo with an approximate identity (f_i)_{i ∈ I} consisting of deflations; namely, each f_i (is ≤ id_X and) has finite image. Every RB-domain is a continuous dcpo, is stably compact in its Scott topology (Theorem 9.6.22 in the book), and the category RB of RB-domains is Cartesian-closed (Theorem 9.6.24 in the book). The prefix “RB” means “retract of bifinite”, and this refers to their actual definition: as retracts of bifinite domains, but this will not be important here.

My preferred Cartesian-closed category of continuous dcpos, as far as semantics of programming languages is concerned, is the category BCDom of bc-domains. The prefix “bc” stands for “bounded-complete”, but, following Prakash Panangaden, it is tempting to imagine that it means “bloody convenient”. Unfortunately, bc-domains lack a few occasional closure properties. The Plotkin powerdomain of a bc-domain is not a bc-domain in general, for example. The Plotkin powerdomain of an RB-domain is an RB-domain (I will not attempt to justify this here, and we will slowly forget about the Plotkin powerdomain in the rest of this post), though, and every bc-domain is an RB-domain (Exercise 9.6.20 in the book).

FS-domains form an even larger Cartesian-closed category FS of continuous dcpos, although we still do not know whether it is strictly larger than the category of RB-domains to this day.

A dcpo X is an FS-domain if and only if it has an approximate identity (f_i)_{i ∈ I} of functions that are finitely separated from the identity: namely, for each i ∈ I, there is a finite set M_i of points of X such that for every x ∈ X, there is a point m ∈ M_i that sits between f_i(x) and x (f_i(x) ≤ m ≤ x). In that case, we will say that M_i is a finite separating set for f_i.

In general, if f is a Scott-continuous map from X to X with a finite separating set M, then f ≤ id_X and for every x ∈ X, f(x) ≪ x, where ≪ is the way-below relation on X.

Definition. The ≪-separating domains that Wei Luan introduces are defined as those dcpos X with an approximate identity (f_i)_{i ∈ I} of functions such that for every x ∈ X, f_i(x) ≪ x.

Hence every FS-domain is a ≪-separating domain.

It is also clear that every ≪-separating domain is a continuous dcpo: every point x is the supremum of a directed family of points way-below x, namely (f_i(x))_{i ∈ I}.

As a sanity check, we verify that the notion of ≪-separating domain is not vacuous. First, not every continuous domain is a ≪-separating domain: Plotkin’s tie (see Example 9.6.18 and Figure 9.4 in the book) is an example of a continuous dcpo, in fact even an algebraic dcpo, that is not a ≪-separating domain. I will omit the argument, which can be found at the end of Section 3 of [2].

Second, not every ≪-separating domain is an FS-domain. The example provided by W. Luan [2, Example 3.7] is as follows.

Example 0 (an algebraic ≪-separating domain that is not an FS-domain). Let X_comb be the dcpo pictured above. Its elements are infinitely many, pairwise distinct elements v_n and w_n, where n ranges over N. The ordering is defined by:

• v_m ≤ v_n if and only if m≥n;
• v_m ≤ w_n never;
• w_m ≤ v_n if and only if m≥n;
• w_m ≤ w_n if and only if m=n; in other words, the elements w_n are pairwise incomparable.

X_comb is a dcpo because every directed family must in fact contain a largest element. It follows that every element of X_comb is finite, hence X_comb is algebraic. It also follows that every monotonic function from X_comb to any poset is automatically Scott-continuous. Let I be the one element set {*}. We let f_i : X_comb → X_comb be the identity map for every i ∈ I; in other words, f_* = id. The family (f_*)_{* ∈ I} is trivially an approximate identity, and we have f_*(x) ≪ x for every x ∈ X, since every point of X is finite, namely way-below itself. Therefore X_comb is a ≪-separating domain.

We claim that X_comb is not an FS-domain. Let us assume it is. Since it is algebraic, it must be a bifinite domain, by Exercise 9.6.30 in the book. In other words, there must be an approximate identity consisting of idempotent deflations. But there are no idempotent deflations on X_comb. Indeed, let us imagine there is an idempotent deflation f on X_comb. In other words, f o f = f, f ≤ id, and f has finite image. In particular, there are only finitely many points v_n and finitely many points w_n in the image of f. Let n₀ ∈ N be such that every v_n and every w_n in the image of f is such that n<n₀. Then f(v_n0) ≤ v_n0, so f(v_n0) must be of the form v_n or w_n with n≥n₀… hence not in the image of f, which is absurd. ☐

Since FS is a maximal Cartesian-closed category of continuous dcpos and since the category ≪S of ≪-separating domains is a larger category of continuous dcpos, we have the following alternative: either ≪S is not Cartesian-closed or ≪S=FS. We have just seen that the second term of the alternative is wrong. Therefore:

Fact. The category ≪S of ≪-separating domains is not Cartesian-closed.

Too bad.

The main theorem

The following is the first half of Theorem 4.1 of [2]. We have already announced it. In a sense, this shows how close ≪S is from being Cartesian-closed.

Theorem A. Given an FS-domain X and a ≪-separating domain Y, the space of Scott-continuous maps [X → Y] is a ≪-separating domain.

Proof. Let (f_i)_{i ∈ I} be a finitely separating approximate identity on X, and let M_i be a finite separating set for each i ∈ I. This means that for every x ∈ X, there is an element m of M_i such that f_i(x) ≤ m ≤ x.

Let (g_j)_{j ∈ J} be an approximate identity on Y such that g_j(y) ≪ y for every y ∈ Y.

We claim that:

For every Scott-continuous map h : X → Y, for all i ∈ I and j ∈ J,
g_j o h o f_i is way-below h in [X → Y].

(Please mind that I really mean “g_j o h o f_i is way-below h“, not “(g_j o h o f_i) (x) is way-below h (x) for every x ∈ X: that would be different.)

In order to see this, let (h_k)_{k ∈ K} be a directed family in [X → Y] such that h ≤ sup_{k ∈ K} h_k.

• For every m ∈ M_i, g_j(h(m)) ≪ h(m) ≤ sup_{k ∈ K} h_k(m), so there is a index k_m in K such that g_j(h(m)) ≤ h_km(m). Since M_i is finite and (h_k)_{k ∈ K} is directed, there is a common index k in K such that g_j(h(m)) ≤ h_k(m) for every m ∈ M_i.
• For every x ∈ X, let m ∈ M_i be such that f_i(x) ≤ m ≤ x. Then (g_j o h o f_i) (x) ≤ g_j(h(m)) ≤ h_k(m) ≤ h_k(x). We have just shown that g_j o h o f_i ≤ h_k. Therefore g_j o h o f_i is way-below h in [X → Y].

(Note: the argument in the proof of this claim inside the proof of Theorem 4.1 of [2] is made a bit complicated by the introduction of a set H_y, in my opinion. Additionally, H_y is defined as ↑y ∩ δ⁻¹(↓y), where δ is what I write as f_i; but one should read ↓y ∩ δ⁻¹(↑y) instead.)

Let F_ij : [X → Y] → [X → Y] map every h ∈ [X → Y] to g_j o h o f_i. We have just shown that F_ij(h) ≪ h for every h ∈ [X → Y]. It remains to show that (F_ij)_{i ∈ I, j ∈ J} is an approximate identity.

• The family (F_ij)_{i ∈ I, j ∈ J} is non-empty, and given any two of its elements F_ij and F_i’j’, we can pick i” in I so that f_i, f_i’ ≤ f_i” and j” in J so that g_j, g_j’ ≤ g_j”. Then g_j o h o f_i and g_j’ o h o f_i’ are both below g_j” o h o f_i” for every h ∈ [X → Y], so F_ij and F_i’j’ are both below F_i”j”. Hence (F_ij)_{i ∈ I, j ∈ J} is directed.
• For every h ∈ [X → Y], for every x ∈ X, sup_{i ∈ I, j ∈ J} F_ij (h) maps x to sup_{j ∈ J} sup_{i ∈ I} (g_j o h o f_i) (x) = sup_{j ∈ J} (g_j o h) (sup_{i ∈ I} f_i(x)) (since g_j o h is Scott-continuous) = sup_{j ∈ J} (g_j o h) (x) = h(x), where we have used that (f_i)_{i ∈ I} and (g_j)_{j ∈ J} are approximate identities in the last two steps. Therefore sup_{i ∈ I, j ∈ J} F_ij (h) = h, and hence sup_{i ∈ I, j ∈ J} F_ij is the identity map. ☐

Wei Luan also shows the following. He calls point-open the topology of pointwise convergence on [X → Y], namely the subspace topology induced by inclusion in the space Y^X of all functions from X to Y with the product topology; equivalently, the topology with subbasic open sets [x ∈ V] where x ∈ X and V ranges over the (Scott-)open subsets of Y, where [x ∈ V] denotes the set of continous maps f such that f(x) ∈ V.

Proposition B. Under the assumptions of Theorem A, the compact-open topology, the Isbell topology and the Scott topology all coincide with the topology of pointwise convergence on [X → Y].

Proof. Looking at Figure 5.3 in the book, we have the following inclusions of topologies: pointwise convergence ⊆ compact-open ⊆ Isbell. We also have the inclusion Isbell ⊆ Scott: it suffices to show that every subbasic Isbell-open set [^–1(V) ∈ U] (denoting the space of continuous maps f such that f^–1(V) ∈ U, where V is open in Y and U is Scott-open in the lattice OX of Scott-open subsets of X) is Scott-open, and this is an easy exercise.

Hence it suffices to show that Scott ⊆ pointwise convergence, namely that every Scott-open subset W of [X → Y] is open in the topology of pointwise convergence.

We fix a Scott-open subset W of [X → Y], and we consider an arbitrary element h of W. We wish to find a basic open neighborhood of h in the topology of pointwise convergence, namely a finite intersection of subbasic open sets of the form [x ∈ V] that contains h, and is included in W.

Taking the same notations as in the proof of Theorem A, (F_ij)_{i ∈ I, j ∈ J} is an approximate identity, so sup_{i ∈ I, j ∈ J} (g_j o h o f_i) = h. Since W is Scott-open, g_j o h o f_i is in W for some i ∈ I and some j ∈ J, and we also have g_j o h o f_i ≤ h.

For every m ∈ M_i, (g_j o h) (m) ≪ h(m), because g_j(y) ≪ y for every y ∈ Y. Hence h is in [m ∈ V_m], where V_m is the Scott-open subset ↟(g_j o h) (m) of Y consisting of all the points y such thath (g_j o h) (m) ≪ y. (This is Scott-open by Proposition 5.1.16 in the book, since Y is a ≪-separating domain, hence a continuous dcpo.) This holds for every m ∈ M_i, and M_i is finite, so ⋂_{m ∈ Mi} [m ∈ V_m] is a basic open neighborhood of h in the topology of pointwise convergence.

It remains to see that ⋂_{m ∈ Mi} [m ∈ V_m] ⊆ W. Let h’ be any Scott-continuous map in ⋂_{m ∈ Mi} [m ∈ V_m]. For every x ∈ X, there is an m ∈ M_i such that f_i(x) ≤ m ≤ x. Then h’ is in [m ∈ V_m], so (g_j o h) (m) ≪ h’ (m) by definition. Since m≤x, h’ (m) ≤ h’ (x), and we also have f_i(x) ≤ m hence (g_j o h o f_i) (x) ≤ (g_j o h) (m). Therefore (g_j o h o f_i) (x) ≤ (g_j o h) (m) ≪ h’ (m) ≤ h’ (x), and in particular (g_j o h o f_i) (x) ≤ h’ (x). Since this holds for every x ∈ X, g_j o h o f_i ≤ h’. Now g_j o h o f_i is in W, and W is upwards-closed (since Scott-open), so h’ is in W. ☐

A variant of the main theorem

A Noetherian space is one whose open subspaces are all compact (Section 9.7 in the book). Wei Luan shows that if X is Noetherian and Y is an FS-domain, then [X → Y] is a ≪-separating domain [2, Theorem 4.5]. He also shows that if X is compact in its Skula topology (a property that W. Lua calls strong-compact, an adjective that is part of the title of [2]) and Y is a ≪-separating domain, then [X → Y] is a ≪-separating domain [2, Theorem 4.4]. I will argue that the latter result somehow subsumes the former, and I will give a proof of it, or rather of something that appears more general but is really equivalent.

Let me explain. The Skula topology on a topological space is the topology generated by the open subsets and the downwards-closed subsets with respect to the specialization preordering. This is mentioned in Exercise 9.7.16 of the book, where it is asked to show the following result due to R.-E. Hoffmann: a topological space is sober and Noetherian if and only if it is compact in its Skula topology. For sober spaces, see Chapter 8 of the book. The August 2019 post contains additional material on sober Noetherian spaces, and I have mentioned such spaces in various other posts, too.

We will use the following facts.

• First, the sobrification SX of a topological space X is sober (Corollary 8.2.23 in the book). For an explicit description, SX is the space of irreducible closed subsets of X, with the topology whose open sets are exactly the sets ♢U ≝ {C ∈ SX | C intersects U}, where U ranges over the lattice OX of open subsets of X.
• Second, the map U ↦ ♢U is an order isomorphism of OX onto OSX (Lemma 8.2.26 in the book).
• Third, SX is the free sober space over X (Theorem 8.2.44 in the book), meaning that: (1) there is a continuous map η^S : X → SX, the unit (explicitly, η^S maps every point x to its closure ↓x), and (2) for every sober space Y and for every continuous map f : X → Y, there is a unique continuous map f^! : SX → Y such that f^! o η^S = f. Explicitly, f^! maps every irreducible closed subset C of X to the unique point y such that the closure cl(f [C]) of the image of C under f is equal to ↓y, which exists because Y is sober.
• Fourth, every continuous dcpo is sober in its Scott topology (Proposition 8.2.12 in the book).

Let me also give (a bit less than) half of the proof of R.-E. Hoffman’s theorem: every sober Noetherian space X is compact in its Skula topology. Let X be sober and Noetherian. We use Alexander’s subbase lemma (Theorem 4.4.29 in the book): a set is compact if and only if every cover by subbasic open sets contains a finite subcover. Hence let us assume that X is included in a union ∪_i ∈ I U_i ∪ ∪_j ∈ J C_j where each U_i is open and each C_j is downwards-closed. We wish to show that it is included in a finite union of sets U_i and C_j. Let A_j be the complement of C_j, for each j. Then ⋂_j ∈ J A_j ⊆ ∪_i ∈ I U_i. In a Noetherian space, every subspace is compact (Proposition 9.7.7 in the book), so every subset is compact, too (Exercise 4.9.11 in the book). Therefore ⋂_j ∈ J A_j is compact. We extract a finite cover (U_i)_{i ∈ I’} from its open cover (U_i)_i ∈ I; in particular, I’ is a finite subset of I. We write ⋂_j ∈ J A_j as the filtered intersection of the finite intersections ⋂_{j ∈ J’} A_j where J’ ranges over the finite subsets of J. Because X is sober, it is well-filtered (Proposition 8.3.5 in the book), and each set ⋂_{j ∈ J’} A_j is compact, by the same argument as for ⋂_j ∈ J A_j , and saturated, namely upwards-closed, because each A_j is the complement of the upwards-closed set C_j. Therefore ⋂_{j ∈ J’} A_j ⊆ ∪_{i ∈ I’} U_i for some finite subset J’ of J. It follows that X is included in the finite union ∪_{i ∈ I’} U_i ∪ ∪_j ∈ J C_j’.

The key lemma in W. Luan’s proof is the following [2, Lemma 4.3], slightly reformulated. I have no idea what use his theorems about function spaces from (sober) Noetherian spaces to ≪-separating or FS-domains would have—except for the fact that some natural dcpos are in fact sober Noetherian, such as Plotkin’s tie or the dcpo of Example 0—but I think that the following lemma is pretty remarkable: indeed, the implication from “f (x) ≪ g (x) for every x ∈ X” to “f ≪ g in [X → Y]” is something that we learn pretty quickly not to expect in domain theory.

Lemma C. Let X be a sober Noetherian space and Y be a continuous poset. For all f, g ∈ [X → Y], if f (x) ≪ g (x) for every x ∈ X, then f ≪ g in [X → Y].

Proof. Let me say “Skula-open” for “open in the Skula topology”, and let us assume that f (x) ≪ g (x) for every x ∈ X.

Let (h_k)_{k ∈ K} be a directed family in [X → Y] such that g ≤ sup_{k ∈ K} h_k. For every x ∈ X, g (x) ≤ sup_{k ∈ K} h_k (x). Using interpolation (Proposition 5.1.15 in the book), there is point y ∈ Y such that f (x) ≪ y ≪ g (x), so there is an index k_x in K such that y ≤ h_kx (x), and therefore f (x) ≪ h_kx (x). We rephrase the latter as x ∈ h_kx^–1(↟f (x)), and we note that h_kx^–1(↟f (x)) is open because ↟f (x) is Scott-open (in a continuous poset Y, the sets ↟y of points z such that y≪z are Scott-open, by Proposition 5.1.16 in the book). It is obvious that x ∈ ↓x, and that ↓x is downwards-closed. Hence x is in the Skula-open subset h_kx^–1(↟f (x)) ⋂ ↓x, for every x ∈ X.

We have just shown that the sets h_kx^–1(↟f (x)) ⋂ ↓x form a Skula-open cover of X, where x ranges over the points of X. By R.-E. Hoffmann’s theorem, X is compact in the Skula topology, so finitely many such sets cover X. Let E be a finite set of points such that ∪_{e ∈ E} (h_ke^–1(↟f (e)) ⋂ ↓e) contains X. Since E is finite and (h_k)_{k ∈ K} is directed, there is an index k ∈ K such that every h_ke is below h_k. For every x ∈ X, there is an e ∈ E such that x is in h_ke^–1(↟f (e)) ⋂ ↓e. In other words, f (e) ≪ h_ke (x) and x ≤ e. Then f (x) ≤ f (e) ≪ h_ke (x) ≤ h_k (x). This holds for every x ∈ X, so f ≤ h_k. Therefore f ≪ g. ☐

Theorem D. Given a Noetherian space X and a ≪-separating domain Y, the space of Scott-continuous maps [X → Y] is a ≪-separating domain.

Proof. We first claim that we can assume that X is sober without loss of generality. Explicitly,

• SX is sober Noetherian. Indeed, the sobrification of a Noetherian space is Noetherian by Lemma 9.7.9 in the book, and we have seen that sobrifications are sober.
• [X → Y] and [SX → Y] are order-isomorphic. In one direction, we define our order-isomorphism as the function F : [X → Y] → [SX → Y] that maps every f ∈ [X → Y] to f^!. The function f^! exists because Y is a ≪-separating domain, hence a continuous dcpo, hence is sober in its Scott topology. Conversely, we let G : [SX → Y] → [X → Y] map every g ∈ [SX → Y] to g o η^S. We verify that F and G are mutually inverse and monotonic.
  • For every every f ∈ [X → Y], G (F (f)) = f^! o η^S = f.
  • For every g ∈ [SX → Y], F (G (g)) o η^S = G (g)^! o η^S = G (g) = g o η^S. By the uniqueness of G (g)^!, G (g)^! = g, namely F (G (g)) = g. Hence F and G are mutually inverse.
  • It is clear that G is monotonic.
  • For all f, g ∈ [X → Y], if f≤g (pointwise), then for every irreducible closed subset C of X, f [C] ⊆ ↓g [C] ⊆ cl (g [C]), so cl (f [C]) ⊆ cl (g [C]); the latter is equivalent to ↓f^! (C) ⊆ ↓g^! (C), and this implies f^! (C) ≤ g^! (C). This shows that F is monotonic. Therefore F and G together form an order-isomorphism, which is what we wanted to prove.

Hence, up to the replacement of X by SX, we may assume that X is sober and Noetherian, hence compact in its Skula topology, by R.-E. Hoffmann’s theorem. From then on, we follow W. Luan’s proof of his Theorem 4.4 [2]. (His proof, by the way, is a little bit too complicated. It also relies on a Proposition 2.2 that contains essentially the same thing as Theorem 9.7.12 in the book, once you know that his strong-compact spaces are exactly the sober Noetherian spaces.)

Let (g_j)_{j ∈ J} be an approximate identity on Y with the property that g_j (y) ≪ y for every y ∈ Y. Given any function h ∈ [X → Y], we have that for every x ∈ X, (g_j o h) (x) ≪ h (x). By Lemma C (which applies since X is now assumed to be both sober and Noetherian), g_j o h ≪ h in [X → Y].

Let F_j : [X → Y] → [X → Y] map every h ∈ [X → Y] to g_j o h. We have just shown that F_j (h) ≪ h for every h ∈ [X → Y], and for every j ∈ J. It is easy to see that (F_j)_{j ∈ J} is an approximate identity on [X → Y]. Therefore [X → Y] is a ≪-separating domain. ☐

The following proposition establishes a coincidence of topologies, just as in Proposition B, but one should note that we make no mention of the topology of pointwise convergence, and that Y has to be an FS-domain, not just any ≪-separating domain, contrarily to Theorem D.

Proposition E. Given a Noetherian space X and an FS-separating domain Y, the compact-open topology, the Isbell topology and the Scott topology all coincide on [X → Y].

Proof. As in the proof of Proposition B, we have compact-open ⊆ Isbell ⊆ Scott, so we only have to show that every Scott-open subset W of [X → Y] is open in the compact-open topology. The subbasic open subsets of that topology are the sets [Q ⊆ V] ≝ {h ∈ [X → Y] | h[Q] ⊆ V}, where Q ranges over the compact subsets of X and V ranges over the open subsets of Y.

Let (g_j)_{j ∈ J} be a finitely separating approximate identity on Y, and let M_j be a finite separating set for each j ∈ J. Hence, for every y ∈ Y, there is an element m of M_j such that g_j(y) ≤ m ≤ y. Let us write g_j² for g_j o g_j. By Exercise 9.6.25 in the book, (g_j²)_{i ∈ I} is another finitely separating approximate identity on Y, with the same finitely separating set M_j, and such that for every y ∈ Y, there is an element m of M_j such that g_j²(y) ≤ m ≪ y.

Let us fix a function h in W. We wish to find an open neighborhood of h in the compact-open topology that is included in W. We will one that is a finite intersection of subbasic open subsets [Q ⊆ V].

• Since W is Scott-open and h = sup_{j ∈ J} g_j² o h is in W, g_j² o h is in W for some j ∈ J.
• For every m ∈ M_j, h is in [Q_m ⊆ V_m], where Q_m ≝ h^–1(↟m) and V_m ≝ ↟m. Indeed, for every x ∈ Q_m, we have m ≪ h(x), so h(x) ∈ V_m. V_m is Scott-open (Proposition 5.1.16 in the book), and Q_m is compact since X is Noetherian.
• Hence ∩_{m ∈ Mj} [Q_m ⊆ V_m] is an open neighborhood of h in the compact-open topology. It remains to see that ∩_{m ∈ Mj} [Q_m ⊆ V_m] is included in W. For every h’ ∈ ∩_{m ∈ Mj} [Q_m ⊆ V_m], it suffices to show that g_j² o h ≤ h’: since g_j² o h is in W and W is upwards-closed, it will follow that h’ is in W. For every x ∈ X, there is an m ∈ M_j such that (g_j² o h) (x) ≤ m ≪ h (x). The relation m ≪ h (x) means that x ∈ Q_m; since h’ ∈ [Q_m ⊆ V_m], we infer that h’ (x) is in V_m, namely that m ≪ h‘ (x). Then (g_j² o h) (x) ≤ m ≪ h‘ (x), and in particular (g_j² o h) (x) ≤ h‘ (x). As this holds for every x, g_j² o h ≤ h’, as desired. ☐

Final words

As usual when I review a paper for this blog, I am leaving out a lot of material. The most notable of my omissions from [2], here, is the relationships that there are between ≪-separating domains and what W. Luan calls SAC-domains. A SAC-domain is a dcpo with an approximate identity consisting of idempotent functions whose image does not have infinite ascending chains. (Compare this with bifinite domains, where the image is required to be finite.) He shows that:

• every SAC-domain is an algebraic ≪-separating domain [2, Proposition 3.6];
• the Lawson-compact algebraic ≪-separating domains are exactly the Lawson-compact algebraic FS-domains, which are exactly the bifinite domains [2, Corollary 3.8];
• Noetherian ≪-separating domains and Noetherian FS-domains are the same thing (he says this with “strong-compact” instead of “Noetherian” in [2, Proposition 3.9], but strong-compact means sober Noetherian by R.-E. Hoffmann’s theorem, and ≪-separating domains and FS-domains are continuous dcpos, hence sober anyways).

It would be nice if one could show that ≪-separating domains are retracts of SAC-domains, but that is not in [2], and I would not know how to prove this.

1. Achim Jung. The Classification of Continuous Domains. Pages 35–40 of: Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, 1990. IEEE Computer Society Press.
2. Wei Luan.  ≪-separating domains, strong-compact spaces and FS-domains. Theoretical Computer Science Volume 1075, 15 June 2026, 115935.

— Jean Goubault-Larrecq (May 20th, 2026)