Topological semilattices with small semilattices

My purpose today is to explain a counterexample due to Jimmie Lawson [3], or rather a slight variant of it, pertaining to the theory of topological semilattices (which I will not assume to be Hausdorff, contrarily to [3]) and to a property that crops up naturally, namely having small semilattices. Before I can do this, I will have to spend some time explaining what topological semilattices are, and how small semilattices arise naturally.

Semilattices

A semilattice is a pair (L, +) where L is a set and + is an associative, commutative, and idempotent operator +. By idempotence, I mean that x+x=x for every x in L. Examples abound. Notably, there are sup-semilattices, which are posets in which every pair of elements x, y has a supremum x ⋁ y, in which case we can define + as ⋁; dually, there are inf-semilattices, in which every pair of elements x, y has an infimum x ⋀ y, in which case we define + as ⋀.

Every semilattice (L, +) has an associated ordering ⊑, defined by x⊑y if and only if x=x+y. Notice that, if we start from an inf-semilattice, we recover the original ordering this way. Conversely, this definition of ⊑ turns L into a poset, in which x+y is the infimum of x and y, for any two points x and y. Therefore semilattices are merely another presentation of inf-semilattices. (Or also of sup-semilattices, if you decide to take the opposite of ⊑ as your preferred ordering!)

Semilattices occur in a variety of other situations. Instead of making a comprehensive list, let me concentrate on one particular case where they arise naturally: the question of characterizing the algebras of the finitary Smyth hyperspace monad, as studied by Andrea Schalk [1]. That as well is something I need to introduce.

The finitary Smyth hyperspace Q_fin(X)

For every topological space X, let the finitary Smyth hyperspace Q_fin(X) be the set of non-empty finitary compact subsets of X, by which I mean the subsets of X of the form ↑{x₁, …, x_n}, where n≥1, and ↑ denotes upward closure with respect to the specialization preordering of X.

It is easy to see that (Q_fin(X), ∪) is a semilattice. Its associated ordering ⊑ is defined by: Q is below Q’ if and only if Q = Q ∪ Q’, if and only if Q’ is included in Q. In other words, its associated ordering ⊑ is the familiar reverse inclusing ordering that we are accustomed to when handling Smyth powerdomains and their variants.

What other properties does it have?

We equip Q_fin(X) with the upper Vietoris topology, whose basic open subsets are the sets ☐U ≝ {Q ∈ Q_fin(X) | Q ⊆ U}, where U ranges over the open subsets of X. This is the subspace topology induced by the topology with the same name on the Smyth hyperspace Q(X); Q(X) is the set of all non-empty compact saturated subsets of X, whose basic open subsets I usually also write as ☐U, and are instead defined as {Q ∈ Q(X) | Q ⊆ U}. I am only considering Q_fin(X) for now instead of Q(X) for simplicity.

With that topology, (Q_fin(X), ∪) is even a topological semilattice. A topological semilattice is a semilattice (L, +) where L is given a topology such that + is jointly continuous. In the case of Q_fin(X), we check that binary union is jointly continuous by verifying that ∪^–1(☐U) is simply ☐U × ☐U.

Rather easily, Q_fin(X) is T₀, too: its specialization preordering is reverse inclusion.

Finally, for all elements Q and Q’ of Q_fin(X), we always have Q ⊆ Q ∪ Q’, namely Q ∪ Q’ is below Q in the specialization ordering. The topological semilattices (L, +) in which x+y ≤ x for all points x and y, where ≤ is the specialization preordering of L (not the canonical ordering ⊑ defined from +) are called deflationary. You may compare this with the notion of inflationary semilattices, which we have looked at here.

Algebras of the Q_fin monad

This raises the question whether (Q_fin(X), ∪) would be the free T₀ deflationary topological semilattice over the space X. For that question to make sense, we need to define a category of T₀ deflationary topological semilattices, with continuous, addition-preserving maps as morphisms, and we ask whether the forgetful functor from that category to the category Top₀ of T₀ topological spaces has a left adjoint.

Let us approach the problem in a different way. If (Q_fin(X), ∪) were the free T₀ deflationary topological semilattice over X, then the required adjunction would produce a monad (T, η, μ) on Top₀ such that TX=Q_fin(X) for every space X, at least up to natural isomorphism. We already know of such a monad: the unit η_X : X → Q_fin(X) maps every point x of X to ↑x, and the multiplication μ_X : Q_fin(Q_fin(X)) → Q_fin(X) maps every element Q of Q_fin(Q_fin(X)) to ∪Q, the union of all the elements of Q; explicitly, μ_X(↑{↑E₁, …, ↑E_n}) = ↑(E₁ ∪ … ∪ E_n). (I will let you check that this is indeed a monad by yourself.)

Then we characterize the algebras of that monad. The category Top₀^Q_fin of Q_fin-algebras, also known as the Eilenberg-Moore category of the monad, is automatically equipped with an adjunction, which turns Q_fin(X) into the free Q_fin-algebra over the T₀ space X. Our question then becomes: are the Q_fin-algebras on Top₀ exactly the T₀ deflationary topological semilattices?

This was studied, among other things, by Andrea Schalk [1]. One can check that every Q_fin-algebra has a canonical structure of deflationary topological semilattice, but not all deflationary topological semilattices give rise to a Q_fin-algebra. The reasons are the following, and I will explain them below.

• Every Q_fin-algebra is a deflationary topological semilattice with small semilattices. (Ah, at last! We finally meet the small semilattices. In fact, the Q_fin-algebras are exactly the deflationary topological semilattices with small semilattices.)
• There is a deflationary topological semilattice that does not have small semilattices, as shown by Jimmie Lawson [3] (see also [2, Section VI-4]). This is actually the main thing I would like to explain here; the rest is mostly motivation for this, if you remember what I said in the introduction.

Of course, I haven’t said yet what having small semilattices means. Here we go.

A subsemilattice of a semilattice (L, +) is just a subset A of L that is closed under addition: for all x, y in A, x+y is in A. A topological semilattice (L, +) has small semilattices if and only if every point has a neighborhood base consisting of semilattices, namely if and only if:

• for every x ∈ L, for every open neighborhood U of x, there is a subsemilattice A of L such that x ∈ int(A) ⊆ A ⊆ U.

Every Q_fin-algebra is canonically a T₀ deflationary topological semilattice with small semilattices

Let us check the first of our claims: every Q_fin-algebra (on Top₀) is a T₀ deflationary topological semilattice with small semilattices. For that, we need to remember that a Q_fin-algebra (on Top₀) is a T₀ topological space L together with a so-called structure map α : Q_fin(L) → L such that:

• α o η_L = id_L, namely α(↑x)=x for every x in L;
• α o μ_L = α o Q_fin(α). The action of the functor Q_fin on morphisms f is defined by Q_fin(f) : ↑{x₁, …, x_n} ↦ ↑{f(x₁), …, f(x_n)}, so we require that α(↑(E₁ ∪ … ∪ E_n)) = α(↑{α(↑E₁), …, α(↑E_n)}) for every n≥1 and all finite non-empty sets E₁, …, E_n.

There is a canonical way of defining a semilattice structure on such a Q_fin-algebra L: we let x+y ≝ α(↑{x, y}). Then x+x = α(↑x) = x, so + is idempotent. It is commutative by definition. In order to show that + is associative, we apply the law α(↑(E₁ ∪ … ∪ E_n)) = α(↑{α(↑E₁), …, α(↑E_n)}) to the case n≝2, E₁≝{x, y}, E₂≝{z}, yielding α(↑{x,y,z}) = (x+y)+z, and to the case n≝2, E₁≝{x}, E₂≝{y, z}, yielding α(↑{x,y,z}) = x+(y+z).

This is a topological semilattice, because + is the composition of η_L × η_L, of the binary union operation ∪ (which, as we have seen already, is jointly continuous on Q_fin(L)), and of α.

Note that the law α(↑(E₁ ∪ … ∪ E_n)) = α(↑{α(↑E₁), …, α(↑E_n)}) can now be interpreted, when n=2, as saying that α maps unions to sums, namely α(↑E₁ ∪ ↑E₂) = α(↑E₁) + α(↑E₂).

Finally, L has small semilattices. Let x be any point of L, and U be any open neighborhood of x in L. We have α(↑x)=x, so ↑x is in the open subset α^–1(U) of Q_fin(L). By definition of the upper Vietoris topology, there is an open subset V of X such that ↑x is in ☐V, and ☐V is included in α^–1(U). Now we notice that:

• x is in V, since ↑x is in ☐V;
• ☐V is a sub-semilattice of Q_fin(L): the union of any two elements of ☐V is again in ☐V;
• the direct image A ≝ α[☐V] of ☐V is a sub-semilattice of L: indeed, ☐V is closed under binary unions, and α maps unions to sums;
• V is included in A: for every point y of V, ↑y is in ☐V, so y=α(↑y) is in A, by definition of A;
• A is included in U: in other words, α[☐V] ⊆ U, and that is equivalent to ☐V ⊆ α^–1(U), which is by construction.

In short, x ∈ V ⊆ A ⊆ U. Therefore L has small semilattices, as promised.

Q_fin-algebras = T₀ deflationary topological semilattices with small semilattices

We have proved one half of the equivalence between Q_fin-algebras and T₀ deflationary topological semilattices with small semilattices. In the other direction, let (L, +) be a T₀ deflationary topological semilattice with small semilattices. We show that there is a unique Q_fin-algebra structure on L, namely a unique structure map α : Q_fin(L) → L that is compatible with +, in the sense that α(↑{x, y}) = x+y for all points x, y of L.

Well, we do not have much choice: we must define α(↑{x₁, …, x_n}) as the sum x₁+ …+x_n. Showing that this makes sense makes heavy use of the fact that + is associative, commutative, idempotent, and deflationary.

There only remains to show that this map α is continuous, and this is where we require (L, +) to be topological, and to have small semilattices. Let ↑{x₁, …, x_n} be mapped by α to a point (necessarily x₁+ …+x_n) in some open subset U of L. We wish to show that there is an open neighborhood of ↑{x₁, …, x_n} in Q_fin(L)—which we shall take of the form ☐V, with x₁, …, x_n all in the open set V—that is mapped by α into U.

Here is how we might think of proceeding, hoping to use only joint continuity, not small semilattices. Since + is jointly continuous, there is a product U₁ × … × U_n of open neighborhoods of x₁, …, x_n respectively, such that every sum of n elements, one from U₁, …, one from U_n, lies in U. We might think of defining V as some kind of intersection, or of union, or of sum of the open sets U₁, …, U_n, but none of that works, really. Notably the sum U₁ + … + U_n, which we define as the set of sums of n elements, one from U₁, …, one from U_n, is not open in general—at least there is no reason why it should be.

Instead, we first use the fact that L has small semilattices: there is a sub-semilattice A of U such that x₁+ …+x_n is in the interior int(A) of A. Now (and only now) we use the fact that + is jointly continuous, so there is a product U₁ × … × U_n of open neighborhoods of x₁, …, x_n such that the sum U₁ + … + U_n is included in int(A). We define V as int(A).

• It is easy to see that ↑{x₁, …, x_n} is in ☐V, namely that every x_i is in int(A): this is because + is deflationary, so that x_i ≥ x₁+ …+x_n, and because x₁+ …+x_n is in the open, hence upwards-closed, set int(A).
• For every non-empty finitary compact set ↑{y₁, …, y_m} in ☐V, all the points y_j are in V, hence in A, and therefore their sum y₁+ …+y_m is also in A, because A is a sub-semilattice. Since A is included in U, this means that α maps ☐V into U, and this finishes to show that α is continuous.

A T₀ deflationary topological semilattice that does not have small semilattices

At this point, you should be convinced that the notion of having small semilattices arises naturally. But is that a useful notion? Namely, if all T₀ deflationary topological semilattices had small semilattices, that would be a useless notion.

Be reassured: there is a T₀ deflationary topological semilattice that does not have small semilattices. Such a counterexample was found by Jimmie Lawson [3]. Well, let me be slightly more precise. What J. Lawson found was really an example of a compact Hausdorff topological semilattice that does not have small semilattices (and I will come back to that further below), but this can easily be modified to an example of a T₀ deflationary topological semilattice that does not have small semilattices. That is no accident either: this is due to the so-called fundamental theorem of compact semilattices, which I will briefly mention near the end of this post.

Let me describe that counterexample. In passing, I will also simplify it somehow. J. Lawson uses a funny divergent series, but I think the functions I will introduce below make the argument slightly less obscure. Also, his spaces S_i are defined as certain finite products of {0, 1}, but I think it will be clearer to describe them as powersets.

We will build that counterexample in several steps: we build a very slowly growing function (Fact (i) below), which we use to show that a certain measure of finite sets has the property that the intersection of any two big sets, with respect to that measure, remains big enough (Fact (ii) below), and we will have what it takes to finally define our counterexample

A very slowly growing function

Let log₂ denote base 2 logarithm, so that log₂(2^x)=x for every non-negative real number x. This is a slowly growing function, which tends to infinity as its argument tends to infinity. The map x ↦ log₂ log₂ x is an even more slowly growing function. It maps [2, ∞[ to [0, ∞[, and also grows to infinity as its argument tends to infinity, but painfully slowly. Here is a plot of that function:

The point of choosing the log₂ log₂ function is that it grows so slowly to infinity that the following holds. That will be crucial to establishing Fact (ii) below, which in turn will be the key to the joint continuity of the semilattice operation.

• Fact (i). For every ε>0, there is a natural number k_ε≥2 such that for every k≥k_ε, log₂ log₂ (1+k) + ε > log₂ log₂ (1+2k).

Let us check this: the inequality log₂ log₂ (1+k) + ε > log₂ log₂ (1+2k) is equivalent to log₂ (1+k).2^ε > log₂ (1+2k), hence to 2^ε > log₂ (1+2k)/log₂ (1+k). As k tends to infinity, log₂ (1+2k)/log₂ (1+k) ~ 1+ 1/log₂ k, which will be smaller than any fixed number >1 (such as 2^ε) for k large enough. If you prefer an explicit definition, I will let you check that defining k_ε as 2^1/(2ε-1) works, as in that case we even have 2^ε > log₂ (2+2k)/log₂ (1+k) (=1/log₂ (1+k) + 1) for every k≥k_ε. ☐

We now define the finite set N_i as the set of all positive natural numbers strictly smaller than 2²ⁱ, for every natural number i, namely:
N_i ≝ {1, …, 2²ⁱ–1}.

Let S_i be the powerset of N_i, seen as a dcpo with the inclusion ordering. Notice that S_i is an incredibly well-behaved dcpo. In fact it is a finite dcpo, in particular an algebraic dcpo, in which all elements are themselves finite. It is also a (complete) lattice.

Measuring elements of S_i in such a way that the intersection of any two big sets remains big enough

We define a kind of measure of size of the elements E of S_i, by defining the following extended non-negative real number σ_i(E): if E is the whole of N_i, then σ_i(E)=∞; otherwise, we let:
σ_i(E) ≝ i–log₂ log₂ (2²ⁱ–|E|),

where |E| denotes the cardinality of E. Alternatively, σ_i(E) = i–log₂ log₂ (1+|N_i–E|).

Notice that, when E is equal to N_i minus one element, then |N_i–E|=1, so σ_i(E) = i–log₂ log₂ 2 = i; while if E is empty, σ_i(E) = i–log₂ log₂ (2²ⁱ) = 0.

It is clear that σ_i is a monotonic map from S_i to [0, ∞], hence a Scott-continuous map (since S_i is finite!). In fact, we have the stronger property that if E and F are any two elements of S_i such that |E| ≤ |F|, then σ_i(E) ≤ σ_i(F).

Here is a plot of σ_i(E) as a function of |E| when i=3, growing from σ₃(∅)=0 to σ₃(E)=3 when |E|=254. (The value σ₃(E)=∞ when |E|=255, namely when E=N₃, is not shown.)

They key property of the maps σ_i is the following.

• Fact (ii). For all real numbers τ > ε > 0, there is a natural number i_τ,ε such that for every i≥i_τ,ε, for all elements E and F of S_i such that σ_i(E) > τ and σ_i(F) > τ, we have σ_i(E ∩ F) > τ–ε.

In other words, the intersection of any two “big” subsets of S_i (as measured through the function σ_i) will remain “big”, up to some small ε, and this for every index i that is large enough.

Let us verify Fact (ii). Not that I would want to spoil the surprise, but we will define i_τ,ε as τ + log₂ log₂ (1+2k_ε), where k_ε was introduced in Fact (i). If E is equal to the whole of N_i, then E ∩ F = F, and Fact (ii) is trivial; similarly if F=N_i. Hence we will assume that |N_i–E|≥1 and |N_i–F|≥1. We also reason by contradiction, and we assume that σ_i(E ∩ F) ≤ τ–ε for some i≥i_τ,ε. This means that i – log₂ log₂ (1+|N_i–(E ∩ F)|) ≤ τ–ε.

We have |N_i–(E ∩ F)| = |(N_i–E) ∪ (N_i–F)| ≤ |N_i–E| + |N_i–F| ≤ 2k, where k ≝ max (|N_i–E|, |N_i–F|). Therefore τ–ε ≥ i – log₂ log₂ (1+|N_i–(E ∩ F)|) ≥ i – log₂ log₂ (1+2k). Since i ≥ i_τ,ε = τ + log₂ log₂ (1+2k_ε), it follows that τ–ε ≥ τ + log₂ log₂ (1+2k_ε) – log₂ log₂ (1+2k), namely that log₂ log₂ (1+2k) ≥ log₂ log₂ (1+2k_ε) + ε. In particular, k≥k_ε.

This enables us to use Fact (i): so log₂ log₂ (1+k) + ε > log₂ log₂ (1+2k). Recalling that |N_i–(E ∩ F)| ≤ 2k, it follows that σ_i(E ∩ F) = i – log₂ log₂ (1+|N_i–(E ∩ F)|) ≥ i – log₂ log₂ (1+2k), so σ_i(E ∩ F) > i – log₂ log₂ (1+k) – ε. Now k = max (|N_i–E|, |N_i–F|), so k is equal to |N_i–E| or to |N_i–F|. Let us assume that k=|N_i–E|. We have just shown that σ_i(E ∩ F) > i – log₂ log₂ (1+|N_i–E|) – ε = σ_i(E) – ε > τ – ε, which is impossible. The case where k=|N_i–F| is symmetric. ☐

Building the counterexample L

We will build our counterexample L as a subspace of the following nice space K: K is the product of [0, ∞] (with its Scott topology) with S₀, S₁, …, and so on. Recall that S_i is the powerset of N_i, also in its Scott topology, and that N_i={1, …, 2²ⁱ–1}. K is a product of continuous dcpos, all of which have a least element, so K is itself a continuous dcpo, and the Scott topology on K is also the product topology of the Scott topologies on [0, ∞], S₀, S₁, … (see Proposition 5.1.56 in the book.) K is also a complete lattice, in which suprema and infima are computed componentwise. This is a space with very nice properties.

We define L as the subspace of K consisting of those tuples (s, A₀, A₁, …) such that s≤σ_i(A_i) for every i≥0.

Please mind the fact that the topology of L is the subspace topology, not the Scott topology of the componentwise ordering.
In order to understand what the points of L are, let me suggest the following explanation. (Please refer to the picture below.) In order to choose a point in L, one way is to pick an arbitrary element A₀ of S₀, an arbitrary element A₁ of S₁, and so on; then we pick s ∈ [0, ∞] in such a way that s≤σ₀(A₀), s≤σ₁(A₁), etc. In the example below, I have chosen A₀ as the top element {1} of S₀, and the associated constraint s≤σ₀(A₀) is vacuous (equivalent to s≤∞). I have then chosen A₁ as {1, 3} in S₁; then σ₁(A₁)=1, so that forces s≤1. Next, I have chosen A₂ as {1,3,4,6,7} in S₂; then σ₂(A₂) is roughly 0.21, and s has to be at most that. And so on, and so forth.

The specialization ordering of K is componentwise: (s, A₀, A₁, …) ≤ (t, B₀, B₁, …) if and only if s≤t, A₀⊆B₀, A₁⊆B₁, etc. Therefore the specialization ordering of L is also componentwise, and I will write ≤ for it.

Let me observe that, with that ordering, L is an inf-semilattice. The infimum (s, A₀, A₁, …) ⋀ (t, B₀, B₁, …) is (u, C₀, C₁, …) where:

• C_i ≝ A_i ∩ B_i for every i≥0;
• u ≝ min(s, t, inf_i≥0 σ_i(C_i)).

The term inf_i≥0 σ_i(C_i) in the definition of u is, of course, due to the fact that we require u to be smaller than or equal to every σ_i(C_i), if (u, C₀, C₁, …) is to be an element of L.

With that operation, it is clear that (L, ⋀) is a deflationary semilattice, and we must show that it is a topological semilattice, namely that ⋀ is jointly continuous from L × L to L.

The ⋀ operation is jointly continuous on L

In order to show this, we realize that a base of the topology of K is given by sets of the form U × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i where U is Scott-open in [0, ∞] (hence equal to the whole of [0, ∞] or of the form ]r, ∞] for some r>0), n≥0, and each E_i is a (necessarily finite) subset of S_i for each i, 0≤i≤n. Hence a base of the topology of L is given by the corresponding sets L ∩ (U × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i).

Let therefore (s, A₀, A₁, …) and (t, B₀, B₁, …) be two points of L, whose infimum (u, C₀, C₁, …), given as above, lies in some basic open set W ≝ L ∩ (U × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i). We wish to find an open rectangle around the pair of points (s, A₀, A₁, …) and (t, B₀, B₁, …), and which would be mapped into W by ⋀.

• If U = [0, ∞], this is easy: we simply take W × W for that open rectangle. We do not need to care about the first components, which will always fall inside U; for every i≥0 with i≤n, C_i contains E_i, so both A_i and B_i contain E_i (since C_i = A_i ∩ B_i), and conversely, the intersection of any two sets that both contain E_i still contains E_i.
• We now examine the case where U is equal to ]r, ∞] for some r>0. Our assumption that (u, C₀, C₁, …) is in W can be rephrased as, first:
  (a) for every i with 0≤i≤n, A_i and B_i contain E_i.
  Second, u = min(s, t, inf_i≥0 σ_i(C_i)) > r, which means that there is a positive number τ such that r < τ < min(s, t, inf_i≥0 σ_i(C_i)), or explicitly:
  (b) r<τ (c) τ<s,t (d) for every i≥0, τ<σ_i(A_i ∩ B_i).
  Let ε>0 be such that r < τ–ε (thanks to (b)). Considering the number i_τ,ε obtained in Fact (ii), we let m ≝ max(n, i_τ,ε). The desired open rectangle is the product of V₁ ≝ L ∩ (]τ, ∞] × ↑A₀ × ↑A₁ × … × ↑A_m × Π_i>mS_i) and of V₂ ≝ L ∩ (]τ, ∞] × ↑B₀ × ↑B₁ × … × ↑B_m × Π_i>mS_i). It is clear that (s, A₀, A₁, …) is in V₁ and that (t, B₀, B₁, …) is in V₂, using (b). It remains to verify that for every pair of points (s‘, A‘₀, A‘₁, …) in V₁ and (t‘, B‘₀, B‘₁, …) in V₂, their infimum (u‘, C‘₀, C‘₁, …) is in W.
  By definition, C‘_i = A‘_i ∩ B‘_i for every i≥0, and u‘ ≝ min(s‘, t‘, inf_i≥0 σ_i(C‘_i)). We have:
  • for every i≥0, A‘_i contains A_i, B‘_i contains B_i, so C‘_i = A‘_i ∩ B‘_i contains A_i ∩ B_i, which contains E_i by (a);
  • s’>τ and t’>τ; for every i≥0, either i≤m, and then σ_i(C‘_i) ≥ σ_i(A_i ∩ B_i) (since σ_i is monotonic) > τ by (d); or i>m, in particular i≥i_τ,ε, so by Fact (ii) (this is the whole point of the construction!), and using the fact that τ<σ_i(A_i ∩ B_i)≤σ_i(A_i),σ_i(B_i), we obtain that σ_i(A_i ∩ B_i)>τ–ε. Putting all those inequalities together, u‘ = min(s‘, t‘, inf_i≥0 σ_i(C‘_i)) > min(τ, τ–ε) = τ–ε > r. And we are done!

This finishes the proof of:

Lemma. (L, ⋀) is a deflationary topological semilattice.

L is also T₀. In fact, with some extra effort, one can show that L is stably compact. (Hint: K is a product of stably compact spaces, hence is itself stably compact. Then L is a patch-closed subset of K, owing to the fact that each of the maps σ_i is patch-continuous… for the trivial reason that the patch topology on each S_i is the discrete topology.) Also, one can show that L is not just an inf-semilattice, but in fact a complete lattice. (Beware, though, that its topology is most likely not the Scott topology. I have thought about it, and while I have no proof of it, I doubt that the topology would be the Scott topology.)

No small semilattices for L

Despite this niceness, we have:

Lemma. (L, ⋀) does not have small semilattices.

Proof. Let us consider any sub-semilattice A of (L, ⋀) whose interior contains the top element ⊤ ≝ (∞, N₀, N₁, …). That interior contains a basic open subset W ≝ L ∩ (U × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i), for some Scott-open subset of [0, ∞], etc. For every k ∈ N_n+1, let z_k be the point (∞, N₀, N₁, …, N_n, N_n+1–{k}, N_n+2, …); namely it differs from the top element only at coordinate n+1, where we remove the single number k. The point z_k is in W. Moreover, since N_n+1 is finite (that is crucial!), the sum (=infimum) of all the points z_k, k ∈ N_n+1, is a finite sum. Since all the points z_k are in W, hence in A, that sum is also in A. Let us write (s, A₀, A₁, …) for that sum. We see that A_i=N_i for every i≠n+1, but that does not matter. What does matter is that A_n+1 is the intersection of the sets N_n+1–{k} when k ranges over N_n+1, hence is empty. It follows that s≤σ_n+1(∅), namely that s=0.

Hence we have just proved that any sub-semilattice A whose interior contains ⊤ must contain an element (s, A₀, A₁, …) such that s=0. It follows that no basic open subset of the form L ∩ (]r, ∞] × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i) with r>0 can contain any sub-semilattice A that would be a neighborhood of ⊤. Hence (L, ⋀) does not have small semilattices. ☐

As a consequence, (L, ⋀) cannot be equipped with the structure of a Q_fin-algebra. Let us (re)explain why. If there were a structure map α : Q_fin(L) → L of a Q_fin-algebra, α would have to map every non-empty finitary compact subset ↑{z₁, …, z_n} of L to the sum, namely to the infimum z₁ ⋀ … ⋀ z_n. But that map is not continuous (despite the fact that binary infimum ⋀ is continuous). Explicitly, if α were continuous, then the inverse image of the open set L ∩ (]r, ∞] × Π_i≥0 S_i) would be an open neighborhood of ↑{⊤}. Hence it would contain an open subset of the form ☐W, for some open neighborhood W of ⊤ in L. Taking z_k as in the proof of the Lemma above, which are all in W, ↑{z₁, …, z_n} would be in ☐W hence in α^–1(L ∩ (]r, ∞] × Π_i≥0 S_i)); but α(↑{z₁, …, z_n}) = z₁ ⋀ … ⋀ z_n is of the form (s, A₀, A₁, …) where s=0, which is impossible.

Lawson’s counterexample

The stably compact, deflationary topological semilattice (L, ⋀) is, in fact, not Lawson’s counterexample [3], but it is not far away!

Jimmie Lawson had proposed his counterexample in order to answer a conjecture of Dyer and Shields in the negative. I will refer you to [3] for that aspect of things. It is obtained as a Rees quotient of another topological semilattice (L’, ≤), which is a variant of (L, ≤), and which I will describe below.

In order to motivate (L’, ≤), I should say that perhaps the most interesting aspect of the theory of semilattices with small semilattices is the so-called fundamental theorem of compact semilattices [2, Theorem VI-3.4], which states an equivalence of categories between the category of compact Hausdorff semilattices with small semilattices on the one hand, and the category of bc-domains on the other hand. I will perhaps explain that one another time. That was found way earlier than everything else I have mentioned here, even before the term “bc-domain” was even coined. It is a pretty amazing result, too, in the sense that it says that a special form of continuous dcpo (bc-domains) arises from a situation with a priori no domain theory in it: a class of (compact) Hausdorff spaces, with a certain algebraic operation + on it.

Nowadays, Lawson’s counterexample serves mostly to show that there are compact Hausdorff semilattices that do not have small semilattices. Here it is.

Let L’ consist of the same set of points as L, but with the subspace topology induced by the inclusion in K^patch. I have already hinted that K is stably compact. Hence its patch space (K^patch, ≤) is a compact pospace (see Section 9.1 in the book). I have also hinted that L’ is closed in K^patch. Therefore (L’, ≤) is also a compact pospace. In particular, L’ is a compact Hausdorff space.

One can show that (L’, ⋀) is also a topological semilattice, but beware that ⋀ is no longer the infimum operation corresponding to the specialization ordering of L’! In fact, since L’ is Hausdorff, the specialization ordering of L’ is simply equality, and therefore no two distinct elements have an infimum. For the same reason, (L’, ⋀) is certainly not a deflationary topological semilattice.

In order to see that (L’, ⋀) is a topological semilattice, one needs to show that ⋀ is jointly continuous on L’. The easiest way to show this is to realize that the topology of L’ is generated by the open subsets of L, plus sets of the form L ∩ (U × ↓E₀ × ↓E₁ × … × ↓E_n × Π_i>n S_i) where U is of the form [0, r[ or equal to the whole of [0, ∞]. Those sets L ∩ (U × ↓E₀ × ↓E₁ × … × ↓E_n × Π_i>n S_i) generate a dual topology on L, and I will call L^op the space L with that dual topology: that is simply the Scott topology of the opposite ordering. (As a side note, the patch topology on K is the product of the patch topologies on each component, and it has basic open subsets of the form I × {E₀} × {E₁} × … × {E_n} × Π_i>n S_i, where I is an interval of the form [0, b[, ]a, b[, ]b, ∞], or [0, ∞], but using that kind of base will force you to replay most of the arguments we have already done. Instead, it is easy to see that the topology generated by the sets I × {E₀} × {E₁} × … × {E_n} × Π_i>n S_i coincides with that generated by the sets U × ↑E₀ × ↑E₁ × … × ↑E_n × Π_i>n S_i where U is open and upwards-closed in [0, ∞], and by the sets U × ↓E₀ × ↓E₁ × … × ↓E_n × Π_i>n S_i where U is open and downwards-closed in [0, ∞].) Then, since we already know that ⋀ is jointly continuous on L, it suffices to show that it is also jointly continuous on L^op. I will let you verify this. This is much easier than the continuity of ⋀ on L, and in particular does not require fancy arguments such as Fact (ii).

Hence (L’, ⋀) is a compact Hausdorff, topological semilattice. By similar arguments as with (L, ⋀), it does not have small semilattices; to be precise, the argument is the same, simply replace ↑E_i by {E_i}.

Conclusion

The notion of semilattices with small semilattices occurs naturally in several places. The last place I have mentioned here is the fundamental theorem of compact semilattices, which I may talk about at more length another time. My first motivating example was Schalk’s characterization of the algebras of the finitary Smyth hyperspace monad Q_fin on Top₀.

Here is another case where small semilattices come into play.

For a sober space X, the sobrification of Q_fin(X) is the space Q_s(X) of so-called strong non-empty compact saturated subsets of X, not Q(X) as one might expect [1, Proposition 7.33]. Strong compact sets were initially introduced by Reinhold Heckmann [4], and I should definitely say a word about them some time! Then Q_s also defines a monad on the category of sober spaces, and its algebras are exactly the deflationary sober semilattices with small semilattices [1, Theorem 7.37]. The proof is a pretty simple extension of what we have done with Q_fin.

1. Andrea Schalk. Algebras for Generalized Power Constructions. PhD Thesis, TU Darmstadt, 1993.
2. 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.
3. Jimmie D. Lawson. Lattices with no Interval Homomorphisms. Pacific Journal of Mathematics 32(2):459–465, 1970
4. Reinhold Heckmann. An upper power domain construction in terms of strongly compact sets. In: Brookes, S., Main, M., Melton, A., Mislove, M., Schmidt, D. (eds) Mathematical Foundations of Programming Semantics. MFPS 1991. Lecture Notes in Computer Science, vol 598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55511-0_14

— Jean Goubault-Larrecq (April 19th, 2022)