I have already mentioned quite a few equivalent characterizations of bc-domains in the book (Section 5.7). A bc-domain (short for bounded-complete domain) is a pointed, continuous dcpo in which every pair of elements x, y with a common upper bound has a least upper bound; or, it is a non-empty continuous dcpo in which every family of elements with a common upper bound has a least upper bound; or, it is a non-empty continuous dcpo in which every non-empty family of elements has a greatest lower bound; etc. Additionally, the category of bc-domains is Cartesian-closed, and the bc-domains are the densely injective topological spaces (Exercise 9.3.12).
There is at least one other important characterization of bc-domains, and it has to do with so-called compact semilattices with small semilattices [1, Section VI-3], leading to the fundamental theorem of compact semilattices [1, Theorem VI-3.4]. This is what I want to explain today. Explicitly, the category of bc-domains (with a specific kind of morphisms, we will see which at the end of this post) is equivalent, and in fact isomorphic to the category of compact semilattices with small semilattices (and I will explain what they are) with continuous semilattice homomorphisms.
The compact semilattices with small semilattices were once called Lawson semilattices, and were indeed studied extensively by Jimmie D. Lawson. The precise history of the discovery of the fundamental theorem of compact semilattices eludes me somewhat. Paul Poncet [3] says that early forms of this theorem are due to Jimmie D. Lawson in 1969 and 1973, and appear in [4, 5], and that the original form of the statement is due to Karl Heinrich Hofmann and Albert Stralka in 1976 [6]; I guess that should be Proposition 3.13 there (for continuous lattices, which are very close to bc-domains). Hofmann and Stralka also mention that Mike Mislove and themselves had dealt with the case of algebraic lattices in 1974 [7]. J. W. Lea, Jr. (once a PhD student of Jimmie D. Lawson) gave a very short, self-contained proof of the theorem (for continuous lattices) in 1976 [8].
From bc-domains to compact semilattices with small semilattices
Let X be a bc-domain. We will see that X is naturally a compact semilattice, with an appropriate topology. By the way, here “compact” means “compact Hausdorff”, so that topology cannot be the Scott topology, which is almost never Hausdorff.
The topology of choice is usually taken to be the Lawson topology, but it will be equivalent to work with the patch topology (of the Scott topology on X), for the most part. It will be easier for me to do so, considering that the patch topology is covered in the book in enough detail to make the required proof arguments easier below. Let me review those two topologies briefly.
- The Lawson topology is the join of (the coarsest topology containing) the Scott and of the lower topology. The lower topology has subbasic closed subsets of the form ↑x, x ∈ X. Hence the subsets ↑E, where E ranges over the finite subsets of X, form a base of closed sets: every closed set in the lower topology is an intersection of sets of the form ↑E. The Lawson topology, being the join of the Scott and of the lower topology, has basic open sets of the form U–↑E, where U ranges over the Scott-open subsets of X and E ranges over the finite subsets of X.
- The patch topology is covered in Chapter 9 of the book, and is the join of the original (Scott) topology and the cocompact topology, which has all compact saturated sets Q as closed sets.
There is little the book about the Lawson topology, and a lot more about the patch topology. The patch topology is finer than the Lawson topology, because all the sets ↑E with E finite are compact saturated, and it is in general strictly finer. But they do coincide on continuous posets, in particular on bc-domains; this is Exercise 9.1.36 in the book.
The reason of this coincidence of topologies is that every compact saturated subset Q of a continuous poset X is the intersection of the collection of sets of the form ↑E (E finite) whose interior contains Q (Theorem 5.2.30 in the book). Therefore Q is closed in the lower topology.
Now, as I said at the beginning of this section, let X be a bc-domain, and let us consider Xλ, obtained by equipping X with its Lawson (=patch) topology. We have the following properties.
Fact A. Xλ is compact Hausdorff.
This is because it is the patch space of a stably compact space (Proposition 9.1.27 in the book); X is stably compact in its Scott topology since every bc-domain is (Fact 9.1.6).
Fact B. The meet (binary infimum) operation ⋀ is (jointly) continuous from Xλ × Xλ to Xλ.
Hence (Xλ, ⋀) is a compact semilattice, short for “compact Hausdorff topological semilattice”. A topological semilattice is a topological space with a semilattice operation that is jointly continuous. Notice that “Hausdorff” and “topological” are left implicit in the term “compact semilattice”.
Proof. In order to see that ⋀ is continuous from Xλ × Xλ to Xλ, it suffices to show that the inverse image of a basic set ↟z or X–↑z is open in the product topology.
For those of the latter kind, this is easy: for all points x and y of X, x ⋀ y is in ↑z if and only if both points are, so the inverse image of X–↑z is the complement of the closed sets ↑z × ↑z.
For those of the first kind, it suffices to show that ⋀ is continuous from X × X to X, where X has the usual Scott topology. This way, the inverse image of ↟z will be open in the product topology of X × X (with X given the Scott topology), hence open in the product topology of Xλ × Xλ, which is finer.
We show that ⋀ is separately continuous. This will be enough: by Ershov’s observation, every separately continuous from a product of two spaces, one of which is a c-space (e.g., a continuous poset in its Scott topology) is jointly continuous.
Separate continuity, in turn, means that ⋀ is monotonic in each of its arguments (which is clear), and commutes with directed suprema, namely that x ⋀ supi ∈ I yi = supi ∈ I (x ⋀ yi) for every point x and every directed family (yi)i ∈ I. (This property is usually called meet-continuity, and can be generalized so as to be valid in any continuous poset, and in fact even larger classes of spaces.) Let me give a proof. We only need to show that x ⋀ supi ∈ I yi ≤ supi ∈ I (x ⋀ yi), since the other inequality follows from monotonicity; and to this end, it suffices to show that every point z ≪ x ⋀ supi ∈ I yi is below some x ⋀ yi, since X is a continuous poset. From z ≪ x ⋀ supi ∈ I yi we obtain that z ≪ x (so z≤x), and also that z ≪ supi ∈ I yi, so z≤yi for some i in I; then z ≤ x ⋀ yi, and we are done. ☐
The tricky property is the following. In fact, it is already pretty hard to simply imagine that (Xλ, ⋀) would satisfy that property at all, even without trying to prove it. But going the other way around, from compact semilattices to bc-domains (as we will do below), would fail without this property, so it had better be true!
Fact C. (Xλ, ⋀) has small semilattices.
We have already encountered the notion here, but let me recall the definition. A topological semilattice (L, +) has small semilattices if and only if for every point x of L, for every open neighborhood U of x in L, there is a small semilattice around x included in U, namely a neighborhood A of x included in U that is closed under + (that is, A is a sub-semilattice of L). That (Xλ, ⋀) has small semilattices is rather non-obvious.
Proof. We pick any point x in X, and an open neighborhood U of x in Xλ. Then U contains an open neighborhood of x of the form ↟y ∩ (X–↑E), for some finite set E. But neither ↟y nor X–↑E is closed under ⋀, or at least there is no reason why they ought to be, so how do we conclude?
Well, first, we call V that open neighborhood ↟y ∩ (X–↑E). We will only need to remember three properties of V, and we will use the shape of V to build a further point z (see below), and then the desired neighborhood sub-semilattice A. The three properties are:
- (a) x ∈ V,
- (b) V ⊆ U,
- (c) V is order-convex: any point lying between two points of V is also in V.
Property (c) is due to the fact that ↟y is upwards-closed, hence order-convex, that X–↑E is downwards-closed, hence order-convex, and that any intersection of order-convex subsets is order-convex.
Next,
- (d) there is a point z in V such that z ≪ x.
Indeed, by interpolation (Proposition 5.1.15 in the book), since we already know that y ≪ x, there is a point z such that y ≪ z ≪ x. We have z ∈ ↟y, and z must be in X–↑E, since otherwise we would have e≤z for some e in E, hence e≤z≤x, which is impossible since x ∈ ↟y ∩ (X–↑E); therefore z is in V.
We define the desired neighborhood A of x as V ∩ ↑z. This is a neighborhood of x, because it contains the open set V ∩ ↟z, of which x is an element by (a) and (d). For all points a, b of A, we have z ≤ a⋀b, so a⋀b is in ↑z. Also, since z is in V by (d), since a is in V (because A ⊆ V), and since V is order-convex by (c), the inequalities z ≤ a⋀b ≤ a entail that a⋀b is in V. Therefore a⋀b is both in in ↑z and in V, and thus it is in A. This shows that A is closed under ⋀. ☐
We summarize what we have found in Facts A, B, and C.
Proposition D. For every bc-domain X, (Xλ, ⋀) is a compact semilattice (i.e., a compact Hausdorff topological semilattice) with small semilattices.
From compact semilattices with small semilattices to bc-domains
We examine the converse. We start with a compact semilattice (L, +) with small semilattices, and we will show that it is of the form (Xλ, ⋀) for some bc-domain X.
Clearly, the points of X must be exactly those of L. We keep the two letters X and L distinct: X, as a bc-domain, will be given the Scott topology, while L has a (compact) Hausdorff topology.
For the ordering ≤ on X, we do not have any choice either. We want the semilattice operation + to coincide with the meet operation ⋀ on X, so we need to define x≤y if and only if x=x+y. We will equip X with the Scott topology of ≤.
Fact E. ≤ is a partial ordering on X, which turns X into an inf-semilattice, whose binary infimum operation ⋀ is simply +.
Proof. This is very standard, and works for any semilattice: no compactness, no topology even is needed. We have x=x+x since + is idempotent, so x≤x. If x≤y≤z, then x=x+y and y=y+z, so x=x+(y+z)=(x+y)+z=x+z, so x≤z. If x≤y and y≤x, then x=x+y and y=y+x, so x=y. Therefore ≤ is a partial ordering. We have x+y≤x since x+y=(x+y)+x, and similarly x+y=(x+y)+y yields x+y≤y, so x+y is a lower bound of x and y. If z is any other lower bound, namely if z≤x, y, then by definition z=z+x and z=z+y, so z=z+y=(z+x)+y=z+(x+y), showing that z≤x+y. Therefore x+y is the greatest lower bound (the infimum) of x and y with respect to ≤. ☐
Fact F. (L, ≤) is a compact pospace.
Proof. This means that L is compact (we assumed so), and that the graph (≤) of the ordering ≤ is closed in the topological product L × L. We check this as follows. Let Δ be the diagonal {(x, x) | x ∈ L}. Since L is Hausdorff, Δ is closed in the product topology. Since + is (jointly) continuous, there is a continuous map f from L × L to L × L that maps every pair (x,y) to (x,x+y). Then f–1 (Δ) is closed, too, and we conclude since f–1 (Δ) = (≤). ☐
By the general theory of compact pospaces (Section 9.1 in the book), retopologizing L (equivalently, X) with the upward topology of (L, ≤) yields a stably compact space, which I will temporarily write as Lup. (We will eventually see that Lup=X, namely that the topology of Lup is exactly the topology of X, namely the Scott topology of ≤.) Additionally, the specialization ordering of Lup is ≤, and the patch space (Lup)patch is L. Since Lup is sober, it is a monotone convergence space (Proposition 8.2.34 in the book), meaning that X is a dcpo (in the ordering ≤), and that the topology of Lup is coarser than the Scott topology of ≤, namely the topology that we have chosen on X.
The same can be said of L with the downward topology of (L, ≤)—equivalently, the upward topology of (L, ≥)—which I will write as Ldown: Ldown is a stably compact space, whose specialization ordering is ≥, the opposite of ≤. Additionally, Lup and Ldown are de Groot duals: the compact saturated subsets of each one are the closed subsets of the other.
Lemma G. Every non-empty subset A of L has a greatest lower bound inf A in L with respect to ≤; inf A is the infimum of the filtered family of elements inf B, where B ranges over the non-empty finite subsets of A.
Proof. Every non-empty finite subset B of A has a greatest lower bound inf B, obtained by summing the elements of B. (By Fact E, non-empty finite infima are non-empty finite sums.) The set A↓ of all lower bounds of A is equal to the set of elements x that are smaller than or equal to inf B for every non-empty finite subset B of A, so A↓ = ∩B ↓inf B, where B runs over the non-empty finite subsets of A. Also, this intersection is filtered, or equivalently, the family of elements inf B is filtered; all this with respect to ≤.
With respect to the opposite ordering ≥, the family of elements inf B, where B ranges over the non-empty finite subsets of A, is directed. Since Ldown is stably compact, hence sober, hence a monotone convergence space, hence a dcpo in its specialization ordering ≥, that family has a supremum y with respect to ≥, namely an infimum with respect to ≤. Then ∩B ↓inf B = ↓y. We have obtained that A↓ = ↓y. Therefore y is the largest element of A↓, namely the greatest lower bound of A. ☐
Lemma H and Lemma I below are where we start to see why having small semilattices is important.
Lemma H. For every compact sub-semilattice K of (L, +), y ≝ inf K is in ↑K, and therefore ↑K = ↑y.
Proof. Looking at the proof of Lemma G, y ≝ inf K is obtained as the supremum (with respect to the opposite ordering ≥, which is also the specialization ordering of Ldown) of the directed family of elements inf B, where B ranges over the non-empty finite subsets of K. For each such B, inf B is the sum of all elements of B, which must therefore lie in K, since B ⊆ K and K is a sub-semilattice of (L, +).
Taking the same notations as in the proof of Lemma G (except for the use of K for A), K↓ = ↓y = ∩B ↓inf B, where B ranges over the non-empty finite subsets of K. Each set ↓inf B is closed in L: it is the image by the first projection map π1 : (x,y) ↦ x of (≤) ∩ (L × {inf B}); the latter is the intersection of the closed set (≤) (by Fact E) with the compact set L × {inf B}, so ↓inf B is compact, and therefore closed, since L is Hausdorff (by Fact E again). Also, ↓inf B intersects K (at inf B). Since K is compact, the filtered intersection ∩B ↓inf B must also intersect K. In other words, ↓y intersects K, or equivalently, y is in ↑K.
As a consequence, ↑y ⊆ ↑K. The reverse inclusion follows from the fact that y ≝ inf K. ☐
Lemma I. L has small compact semilattices (not just small semilattices): for every point x of L, for every open neighborhood U of x in L, there is a small compact semilattice around x included in U, namely a compact neighborhood K of x included in U that is closed under +.
Proof. We know that L has small semilattices, and we want to show that it has small compact semilattices. In order to show this, we first observe that for every sub-semilattice A of L, the closure cl(A) in L is also a sub-semilattice of L. Indeed, let us imagine that we could find points x, y in cl(A) whose sum x+y were not in cl(A). Since the complement of cl(A) is open, and since + is jointly continuous, there would be open neighborhoods U of x and V of y such that the sum of any element of U with any element of V would fall outside of cl(A). Since x ∈ U, U intersects cl(A), and therefore also A, say at x’. Similarly, V intersects A at some point y’. Then, as we have just seen, x’+y’ lies outside cl(A). But this is impossible, since x’ and y’ are in A, which is closed under +.
This being proved, let x be any point of L, and U be an open neighborhood of x in L. Since L is compact Hausdorff hence locally compact, x has a compact neighborhood Q included in U. Since L has small semilattices, the interior int(Q) contains a sub-semilattice neighborhood A of x. We have just seen that cl(A) is also a sub-semilattice of L. Since A ⊆ cl(A), cl(A) is also a neighborhood of x; since A ⊆ Q and Q is compact, hence closed (because L is Hausdorff), cl(A) is also included in Q, hence in U; and finally, cl(A) is a closed subset of the compact set Q, so cl(A) is itself compact. The desired K is therefore cl(A). ☐
Lemma J. Lup is a c-space, and therefore a continuous dcpo, and its topology is the Scott topology of ≤.
See Section 5.1.2 in the book for the notion of c-spaces: those are topological spaces in which every point has a neighborhood base consisting of upward closures of points ↑y. The sober c-spaces are exactly the continuous dcpos, and their topologies are the Scott topologies of their specialization orderings (Proposition 8.3.36).
Proof. For every point x of L, for every open subset U of Lup that contains x, we will show that there is a point y in U and an open neighborhood V of x in Lup that is included in ↑y. This will show that Lup is a c-space. Since Lup is stably compact (by Fact F), it is sober, hence it will be a continuous dcpo and have the Scott topology of its specialization ordering, which is ≤.
Since U is open in Lup, it is also open in L: by definition the open subsets of Lup are the open subsets of L that are upwards-closed with respect to ≤. By Lemma I, there is a small compact semilattice K around x included in U. By Lemma H, ↑K = ↑y for some point y. Since K is included in U and since U is upwards-closed with respect to ≤, ↑K is also included in U, so y is in U.
It remains to show that ↑y is a neighborhood of x in Lup, namely that ↑K (=↑y) contains an open subset V of Lup that contains x. That open subset will be int(↑y), the interior of ↑y in L (not Lup yet—that is the catch). Since K is a neighborhood of x in L, ↑K = ↑y is a neighborhood of x, too, so int(↑y) contains x. Naturally, int(↑y) is included in ↑y, hence in U. It remains to show that int(↑y) is open in Lup, and to this end, it remains to show that int(↑y) is upwards-closed with respect to ≤.
Let u be any element of int(↑y), and v be such that u≤v. By definition, u=u+v, so v is in (u+_)–1 (int(↑y)). The set (u+_)–1 (int(↑y)) is open in L, since + is continuous, and we claim that it is included in ↑y. This will complete the proof, since that will imply (u+_)–1 (int(↑y)) ⊆ int(↑y), hence v ∈ int(↑y).
For every z in (u+_)–1 (int(↑y)), u+z is in int(↑y), hence in ↑y. Therefore y≤u+z≤z (since + is the infimum operation of ≤, by Fact E), and thus z is in ↑y. We are done. ☐
All right. So, to sum up, by Lemma J, Lup is a continuous dcpo, and its topology is the Scott topology of ≤, namely the topology of X. In other words, X=Lup, and X is a continuous dcpo. By Lemma G, every non-empty family in X has a greatest lower bound, so every bounded family has a least upper bound; equivalently, X is bounded-complete (see Exercise 5.7.5 in the book for details). We have obtained the following.
Proposition K. For every compact semilattice (L, +) with small semilattices, the poset X whose points are those of L and whose ordering is defined by x≤y if and only if x=x+y, is a bc-domain. Additionally, (L, +) = (Xpatch, ⋀) = (Xλ, ⋀).
Conversely, by Proposition D, for every bc-domain X, (Xλ, ⋀) (= (Xpatch, ⋀)) is a compact semilattice (i.e., a compact Hausdorff topological semilattice) with small semilattices, and the relation defined ≤’ defined by x≤’y if and only if x=x⋀y is simply ≤. Therefore the two constructions are inverse of each other.
An isomorphism of categories
This extends to an isomorphism (not just an equivalence!) of categories as follows.
Let Bcd/inf be the category of bc-domains and non-empty-inf-preserving Scott-continuous maps. I am using “non-empty-inf-preserving” to mean that such maps f should send the greatest lower bound of any non-empty subset A to the greatest lower bound of its image f[A]. Any such map f : X → Y has the property that f–1(↑y) = ↑inf f–1(↑y) for every y ∈ Y such that f–1(↑y) is non-empty (for every x in the domain of f, if x ∈ f–1(↑y) then x≥inf f–1(↑y), and conversely, if x≥inf f–1(↑y) then f(x)≥f(inf f–1(↑y))=inf f[f–1(↑y)]≥y, since every element of f[f–1(↑y)] is larger than or equal to y); therefore f is continuous with respect to the lower topologies on X and on Y. Since it is also Scott-continuous, it is continuous from Xλ to Yλ. Every non-empty-inf-preserving map also preserves binary infima, so f is a continuous semilattice homomorphism from (Xλ, ⋀) to (Yλ, ⋀).
Let CSLattSmall be the category of compact semilattices with small semilattices, with continuous semilattice homomorphisms as morphisms. We have just defined a functor from Bcd/inf to CSLattSmall: on objects, it maps every bc-domain X to (Xλ, ⋀), and every morphism (every non-empty-inf-preserving Scott-continuous map) f : X → Y to f itself, seen as a continuous semilattice homomorphism from (Xλ, ⋀) to (Yλ, ⋀).
In the reverse direction, there is a morphism from CSLattSmall to Bcd/inf which sends every compact semilattice with small semilattices (L, +) to the bc-domain X≝Lup, and every morphism (every continuous semilattice homomorphism) f : (L,+) → (L’,+’) to f itself. It suffices to verify that f is Scott-continuous from the bc-domain X≝Lup to the bc-domain Y≝L’up. Since + is binary infimum in X and +’ is binary infimum in Y, and f maps +-sums to +’-sums, it is in particular monotonic with respect to the specialization orderings ≤ and ≤’ of X and Y (respectively). Every continuous monotonic map from L to L’ is in particular continuous from Lup to L’up, hence Scott-continuous from X to Y. It is also continuous from Ldown to L’down, and we will use a lemma that I have already mentioned there: any continuous map from a quasi-monotone convergence space Z to a (T0) topological space Z’ is Scott-continuous with respect to the underlying specialization orderings. Since the specialization orderings of Ldown and of L’down are the opposite orderings of X and Y respectively, f preserves all filtered infima. It also preserves non-empty finite infima (non-empty finite sums) since f is a semilattice homomorphism. As we can write the infimum of any non-empty family F as the filtered infimum of the finite infima inf B where B ranges over the non-empty finite subsets of F, f preserves all non-empty infima. In other words, f is a morphism in Bcd/inf.
And that’s it! We have obtained:
Theorem (the fundamental theorem of compact semilattices). There is an isomorphism of categories between Bcd/inf and CSLattSmall, which is the identity on morphisms.
A variant is between the subcategories of bc-domains with a top element (a.k.a. continuous complete lattices) and compact unital semilattices (L, +, 0) with small semilattices; unital means that 0 is a unit for the semilattice operation. Morphisms in continuous complete lattices must now also preserve the top element (and therefore all infima, not just non-empty infima), and the morphisms in the category of compact unital semilattices must also preserve the unit 0.
There is more information on all this in Section VI-3 of [1] (additional characterizations of the morphisms in Bcd/inf, the related question whether continuous semilattice homomorphisms from (L, +) to ([0,1], min) separate points, the free continuous lattice over a compact Hausdorff space, the case of totally disconnected compact semilattices and algebraic complete lattices, etc.), and my only purpose here was to give a different proof of the fundamental theorem of compact semilattices that relies on the theory of compact pospaces, with not too much extra effort—but some effort is still required.
Oh, a final word. One may wonder whether there are compact semilattices that do not have small semilattices. This was solved by Jimmie D. Lawson in 1970 [2], and I have given his example of a compact semilattice (L’, ⋀) without small semilattices there. From what I understand, this is one of two examples of compact semilattices without small semilattices, and no other example is known.
- Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. Continuous Lattices and Domains. Number 93 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003.
- Jimmie D. Lawson. Lattices with no Interval Homomorphisms. Pacific Journal of Mathematics 32(2):459–465, 1970
- Paul Poncet. Domain theory and mirror properties in inverse semigroups. arXiv report arXiv:1301.5718v1 [math.RA], January 24th, 2013.
- Jimmie D. Lawson. Topological semilattices with small semilattices. J. London Math. Soc. (2), 1:719–724, 1969.
- Jimmie D. Lawson. Intrinsic topologies in topological lattices and semilattices. Pacific J. Math., 44:593–602, 1973.
- Karl Heinrich Hofmann and Albert Stralka. The algebraic theory of compact Lawson semilattices Applications of Galois connections to compact semilattices. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1976.
- Karl Heinrich Hofmann, Mike Mislove and Albert Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Lecture Notes in Math. 396, Springer-Verlag, Heidelberg, 1974.
- Jim Wesley Lea, Jr. Continuous lattices and compact Lawson semilattices. Semigroup Forum 13, 387–388 (1976).
— Jean Goubault-Larrecq (August 20th, 2023)