Stone duality for preordered topological spaces II. Ad-frames

A preordered topological space is a topological space X with a preordering ≤. We do not assume any relationship between the topology and the preordering. I wish to explain a form of Stone duality for such preordered topological spaces. The idea is pretty simple: last time, we have seen a form of Stone duality for preordered sets, based on the premises that a preordered set is a topological space with an Alexandroff-discrete topology; hence a preordered topological space is a bitopological space, where one of the topologies is Alexandroff-discrete… and we have a Stone-like duality for bitopological spaces, Jung-Moshier duality, based on d-frames, a concept that we have seen in the June 2025 post.

This is basically all that has to be known about it. I will explain in a bit more detail how this works below, notably there are a few slight differences. However, and contrarily to my usual way of proceeding, I will not include proofs. For that, I will direct you to the arXiv report [1].

Ad-frames, a first try

In Jung-Moshier duality, the d-frames are tuples (L⁺, L_^–, tot, con) where L⁺ and L_^– are frames, each one being an abstraction of a topology, and the totality and consistency predicates tot and con are there to enforce enough constraints between L⁺ and L_^– that we could believe that the two topologies are on the same set of points, in a certain sense. Please have a look back at the June 2025 post: it makes no sense for me to reproduce what I have explained there.

What makes sense is to say how one can associate a d-frame with a bitopological space (X, τ⁺, τ_^–) (τ⁺ and τ_^– are the two topologies on the set X.): L⁺ is the frame O⁺X of subsets of X that are open in τ⁺, L^– is the frame O^–X of subsets of X that are open in τ^–; a pair (U⁺, U_^–) in O⁺X × O^–X is total, namely in tot, if and only if X = U⁺ ∪ U_^–, and it is consistent, or in con, if and only if U⁺ and U_^– are disjoint. This construction defines a d-frame dOX ≝ (O⁺X, O^–X, tot, con). We have seen that this is the object part of a functor dO from the category of bitopological spaces to the category of d-frames, and that dO is left adjoint to a functor dpt in the other direction.

We do a very similar thing here. We start with a preordered topological space (X, ≤), and we build:

• a frame Ω ≝ OX, the frame of open subsets of X;
• a completely distributive complete lattice L consisting of the upwards-closed subsets of X: those are the open subsets of X with the Alexandroff topology of ≤;
• a totality relation tot ⊆ Ω × L, defined by (U, A) ∈ tot if and only if X = U ∪ A;
• a consistency relation con ⊆ Ω × L, defined by (U, A) ∈ con if and only if U and A are disjoint.

Before we continue: ≤ is the preordering of the preordered topological space (X, ≤), and has in general no connection to the specialization preordering of X as a topological space. I will in fact never speak of the specialization preordering in this post.

We note that L is not just a frame. Just like with Bonsangue-Jacobs-Kok duality, L is a completely distributive complete lattice. This subtle difference will have some consequences in the definition of morphisms of ad-frames (we will see the definition later on), and it will also justify why we will actually make a second and then a third try at defining ad-frames (we make our first try right away).

Abstracting away from preordered topological spaces, we could therefore define an ad-frame (first try) as a tuple (Ω, L, tot, con) where:

• Ω is a frame;
• L is a completely distributive complete lattice;
• tot, con ⊆ Ω × L satisfy similar properties as with Jung-Moshier duality:
  • tot is an ⊑-upwards-closed bounded sublattice of Ω × L ordered by the logical ordering ⊴ defined as ≤ × ≥; ⊑ is the information ordering ≤ × ≤, and we write ≤ for both orderings on Ω and on L;
  • con is a ⊑-Scott-closed bounded sublattice of Ω × L ordered by ⊴;
  • tot and con satisfaction the interaction law: for all (u, a) ∈ con and (v, b) ∈ tot such that either u=v or a=b, (u, a) ⊑ (v, b).

Oh, right, this “⊑-upwards-closed bounded sublattice” and “⊑-Scott-closed bounded sublattice” mumbo-jumbo is hard to parse. I have given fuller explanations in the June 2025 post, but let us have a look at the condition that tot should be a ⊑-upwards-closed bounded sublattice of Ω × L ordered by ⊴. This is two properties, one with respect to the information ordering ⊑ = (≤ × ≤), one with respect to the logical ordering ⊴ = (≤ × ≥). We explain them while illustrating them in the case of the ad-frame (first try) associated with a preordered topological space (X, ≤).

• First, tot is ⊑-upwards-closed. For an ad-frame (first try) associated with a preordered topological space (X, ≤), and writing U, V, etc. for open subsets and A, B, etc. for upwards-closed subsets (with respect to the preordering ≤), this means that if U and A are total (their union covers the whole of X) and U ⊆ V and A ⊆ B, then V and B are still total.
• Second, tot is a bounded sublattice of Ω × L ordered by ⊴. This means that tot contains tt ≝ (⊤, ⊥), the largest element with respect to ⊴, it contains ff ≝ (⊥, ⊤), the least element with respect to ⊴, it is closed under binary supremum ⊻ with respect to ⊴ (where (u, a) ⊻ (v, b) = (u ⋁ v, a ⋀ b)) and under binary infimum ⊼ with respect to ⊴ (where (u, a) ⊼ (v, b) = (u ⋀ v, a ⋁ b)). But wait, here we can impose more.
  • In the case of a preordered topological space, tot is closed under arbitrary suprema with respect to ⊴, not just finite suprema. Given any family (u_i, a_i)_i∈I in Ω × L, its supremum is (⋁_i∈I u_i, ⋀_i∈I a_i). In the case of the ad-frame (first try) of a preordered topological space, this means that the supremum of a family (U_i, A_i)_i∈I is (∪_i∈I U_i, ∩_i∈I A_i). Now, if every pair (U_i, A_i) is total, namely if every U_i ∪ A_i = X for every i ∈ I, then the union of ∪_i∈I U_i with ∩_i∈I A_i contains the whole of X: otherwise there would be a point x in neither; x would then be in the complement of some A_i, and therefore it would have to be in the corresponding U_i since U_i ∪ A_i = X; and that is impossible since it is not in ∪_i∈I U_i.
  • Why didn’t we (and Tomáš Jakl) impose closure on arbitrary suprema, just finite suprema, in d-frames, then? The reason is that, with the d-frame associated with a bitopological space, the supremum of a family (U⁺_i, U^–_i)_i∈I is (⋁_i∈I U⁺_i, ⋀_i∈I U^–_i), which would be (∪_i∈I U⁺_i, int^– (∩_i∈I U^–_i)), where int^– is interior in the second topology. In that case, it may well be that U⁺_i ∪ U^–_i = X for every i ∈ I, but that the union of ∪_i∈I U⁺_i and of int^– (∩_i∈I U^–_i) is not equal to X. For example, let X be the set of real numbers R, I be the set N of natural numbers, q_n be an enumeration of the rational numbers, one for each n ∈ N, U^–_n be R–{q_n}, U⁺_n be ]q_n–1/2ⁿ, q_n+1/2ⁿ[. We take the usual metric topology on R for both topologies. Then U⁺_n ∪ U^–_n = R, but ∪_n∈N U⁺_n is a subset of Lebesgue measure at most 2 of R, hence does not cover the whole of R, and int^– (∩_n∈N U^–_n) is the complement of the closure of Q, hence is empty.

Hence we will reinforce the conditions on tot and con. This will be our second try.

Ad-frames, second try

We could therefore define an ad-frame (second try) as a tuple (Ω, L, tot, con) where (I am emphasizing the changes with respect to the first try):

• Ω is a frame;
• L is a completely distributive complete lattice;
• tot, con ⊆ Ω × L satisfy the following properties:
  • tot is an ⊑-upwards-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the logical ordering ⊴ = (≤ × ≥)—it is an ⊑-upwards-closed ⊴-frame;
  • con is a ⊑-Scott-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the logical ordering ⊴—it is a ⊑-Scott-closed ⊴-frame;
  • tot and con satisfaction the interaction law: for all (u, a) ∈ con and (v, b) ∈ tot such that either u=v or a=b, (u, a) ⊑ (v, b).

We have argued that, if we start from a preordered topological space (X, ≤), tot is closed under arbitrary suprema. This is also the case for con. Indeed, the supremum of a family (U_i, A_i)_i∈I is (∪_i∈I U_i, ∩_i∈I A_i). If every pair (U_i, A_i) is consistent, namely if every U_i is disjoint from A_i, then ∪_i∈I U_i is disjoint from ∩_i∈I A_i: otherwise there would be a point x in both, which would then be in some U_i and also in the corresponding A_i, which is impossible. (We could also have required that con be closed under arbitrary suprema for d-frames, contrarily to tot.)

A morphism f of ad-frames (second try) from (Ω, L, tot, con) to (Ω’, L’, tot’, con’) would be a pair of:

• a frame homomorphism φ : Ω → Ω’
• a morphism of complete lattices p : L→ L’
• that preserve consistency and totality in the sense that:
  • for every pair (v, b) ∈ tot, (φ (v), p (b)) ∈ tot’;
  • for every pair (v, b) ∈ con, (φ (v), p (b)) ∈ con’.

Note that p has to be a morphism of complete lattices, just as in Bonsangue-Jacobs-Kok duality. In other words, p must preserve all suprema and all infima. In Jung-Moshier duality, p would only have to be a frame homomorphism, namely to preserve all suprema and finite infima.

Order-reversal

Our second try at defining ad-frames is perfectly fine. Every preordered topological space (X, ≤) gives rise to an ad-frame (second try) O^adX ≝ (Ω, L, tot, con) where Ω ≝ OX, L is the collection of upwards-closed subsets of X with respect to ≤, (U, A) ∈ tot if and only if X = U ∪ A, and (U, A) ∈ con if and only if U and A are disjoint. This defines the object part of a functor O^ad (second try) from preordered topological spaces to ad-frames (second try), which is left adjoint to a certain functor pt^ad (second try).

However, this second try exhibits a curious asymmetry: L is an abstraction of the notion of upwards-closed sets. What about downwards-closed sets?

It is in principle very easy to talk about downwards-closed subsets of X: they are the complements of upwards-closed subsets. Hence we can represent any downwards-closed subset D as its complement, which is upwards-closed. Fine.

But let us think about the following problem. You are given an open subset U of X and a downwards-closed subset D with respect to ≤, and you wish to express that U and D are disjoint. Oh, wait: you are not given D, rather its representation X–D, and you are only allowed to use tot and con in addition to the usual union and intersection operations on Ω = OX and on the lattice L of upwards-closed subset of X (but no mixed union or intersection between open sets and upwards-closed sets, for example).

You could try and say that U and D are disjoint if and only if U is included in X–D. But neither tot nor con allows you to express such an inclusion. To cut it short, I don’t think you can express disjointness of an open set with a downwards-closed subset in the language of tot and con.

The easy solution I took is to consider that a preordered topological space is really a tritopological space, with: (1) its original topology OX, (2) the Alexandroff topology of its preordering ≤ and (3) the Alexandroff topology of its opposite preordering ≥. And we have to impose constraints on all three.

Accordingly, we will define an ad-frame (final try—in [1], the first try is not mentioned, then second try is shown in red, and this final try is what you get if you put on goggles that let you ignore the various colors) as a tuple (Ω, L, tot, con, sup, sub) where:

• Ω is a frame;
• L is a completely distributive complete lattice;
• tot, con ⊆ Ω × L satisfy the same properties as in our second try;
• sup, sub ⊆ Ω × L satisfy dual properties (see below);
• six (instead of just one! yes, that is really hairy) interaction laws are satisfied (see even further below).

sup and sub

Just like (Ω, L, tot, con) is meant to be represent (an abstract, pointfree version of) the topology Ω of a preordered topological space (X, ≤) together with its collection L of upwards-closed subsets and their interactions, (Ω, L, sup, sub) represents the same topology Ω, but the collection of downwards-closed subsets and their interactions with the topology. As above, we reuse upwards-closed subsets as representations of their complements.

I should probably have called sup something like tot^op and sub something like con^op, but I will explain why I eventually chose sup and sub.

We have already seen what the consistency relation (“con^op“) should be between open subsets U and downwards-closed subsets D: we represent D as X–D, and then U and D are consistent (disjoint) if and only if U is included in X–D. Hence the desired consistency relation is subset inclusion, whence the name sub.

Similarly, U and D are total (their union equals the whole space X) if and only if U ∪ D = X, if and only if U is a superset of X–D; whence the name sup.

Now, we define ad-frames (not just those arising from preordered topological spaces) so that:

• sup is an ⊴-upwards-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the information ordering ⊑ = (≤ × ≤);
• sub is a ⊴-Scott-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the information ordering ⊑;
• sup and sub satisfy the following interaction law: for all (u, a) ∈ sup and (v, b) ∈ sub such that either u=v or a=b, (u, a) ⊴ (v, b).

You can check that those hold on ad-frames obtained from preordered topological spaces, but it is simpler to observe that the conditions are obtained from the analogous conditions on tot and con by switching the roles of the logical ordering ⊴ = (≤ × ≥) and of the information ordering ⊑ = (≤ × ≤). This is the only reasonable thing to do if we wish to reverse orderings in the second component. Compare the conditions above on sup and sub with the conditions we gave on tot and con, and which I am reproducing below:

• tot is an ⊑-upwards-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the logical ordering ⊴ = (≤ × ≥);
• con is a ⊑-Scott-closed subset of Ω × L that is closed under finite infima and arbitrary suprema with respect to the logical ordering ⊴;
• tot and con satisfy the interaction law: for all (u, a) ∈ con and (v, b) ∈ tot such that either u=v or a=b, (u, a) ⊑ (v, b).

The new interaction laws

We have an interaction law that says how tot and con interact, another one that says how sup and sub interact, but we now have four binary operators, and it is only natural that there would be as many interaction laws as there are ordered pairs of operators—namely 4 choose 2 = 6. Accordingly, there are 4 remaining interaction laws. They all have a similar shape:

• between tot and sup: the only pairs (u, a) in tot ∩ sup are such that u=⊤;
• between tot and sub: the only pairs (u, a) in tot ∩ sub are such that a=⊤;
• between con and sup: the only pairs (u, a) in con ∩ sup are such that a=⊥;
• between con and sub: the only pairs (u, a) in con ∩ sub are such that u=⊥.

Let us check that these new laws hold for ad-frames stemming from preordered topological spaces (X, ≤).

• between tot and sup: if U and A are total (U ∪ A = X) and U ⊇ A, then U = U ∪ A = X, so U is the top element of Ω = OX;
• between tot and sub: if U and A are total (U ∪ A = X) and U ⊆ A, then A = U ∪ A = X, so A is the top element of L, the lattice of upwards-closed subsets of X;
• between con and sup: if U and A are consistent (disjoint) and U ⊇ A, then A = U ∩ A = ∅, the least element of L;
• between con and sub: if U and A are consistent (disjoint) and U ⊆ A, then U = U ∩ A = ∅, the least element of Ω.

The adjunction O^ad ⊣ pt^ad

A morphism f of ad-frames (final try) from (Ω, L, tot, con, sup, sub) to (Ω’, L’, tot’, con’, sup’, sub’) is a pair of:

• a frame homomorphism φ : Ω → Ω’
• a morphism of complete lattices p : L→ L’
• that preserve consistency, totality, sup and sub in the sense that:
  • for every pair (v, b) ∈ tot, (φ (v), p (b)) ∈ tot’;
  • for every pair (v, b) ∈ con, (φ (v), p (b)) ∈ con’;
  • for every pair (v, b) ∈ sup, (φ (v), p (b)) ∈ sup’;
  • for every pair (v, b) ∈ sub, (φ (v), p (b)) ∈ sub’.

This is the point where I will just state results and point you to [1] for details. Just arriving at the correct definition of ad-frames and their morphisms was probably exhausting already!

There is a functor O^ad from the category of preordered topological spaces and monotone continuous maps to the opposite of the category of ad-frames and their morphisms. On objects, O^ad (X, ≤) is (Ω, L, tot, con, sup, sub) where Ω ≝ OX, L is the lattice of upwards-closed subsets of X with respect to ≤, tot is totality (union equals X), con is disjointness, sup is superset and sub is subset. On morphisms, O^adf is (f^–1, f^–1), where the first f^–1 is inverse image of open subsets under f and the second one is inverse image of upwards-closed subsets under f.

O^ad has a right adjoint pt^ad. It is obtained by a well-known recipe: the points, namely the elements of pt^ad ΩL, for any given ad-frame ΩL ≝ (Ω, L, tot, con, sup, sub) are the morphisms of ad-frames from ΩL to O^ad1, where 1 is the preordered topological space with one element (its topology and its preordering need not be specified, since there is only one possible choice for each). After some simplifications, we arrive at the following definition.

The points of an ad-frame ΩL ≝ (Ω, L, tot, con, sup, sub), which will form the elements of pt^ad ΩL, are the pairs (x, s) where:

• x is a point of the frame Ω, namely a completely prime filter of elements of Ω (see Section 8.1 of the book);
• s is a completely prime complete filter of elements of L—a filter is complete if and only if it is closed under arbitrary infima, not just finite infima;
• the following conditions hold:
  • for every (u,a) ∈ tot, we have u ∈ x or a ∈ s;
  • for every (u,a) ∈ con, we have u ∉ x or a ∉ s;
  • for every (u,a) ∈ sup, we have u ∈ x or a ∉ s;
  • for every (u,a) ∈ sub, we have u ∉ x or a ∈ s.

The topology on pt^ad ΩL is described in extenso as follows. For every u ∈ Ω, we let O_u ≝ {(x, s) ∈ pt^ad ΩL | u ∈ x}. The family of sets O_u is closed under finite intersections and arbitrary unions, and form the desired topology. You may note a certain similarity with usual Stone duality between topological spaces and frames: the topology on pt Ω, for a frame Ω, is given by open subsets defined as {x ∈ pt ΩL | u ∈ x}.

The preordering on pt^ad ΩL is even simpler: (x, s) ≤ (y, t) if and only if s ⊆ t.

Ad-sobrification

In usual Stone duality, the monad obtained from the adjunction O ⊣ pt is pt o O, the sobrification monad. It is usually presented, up to natural isomorphism, as mapping every topological space X to its space of irreducible closed subsets with a certain topology.

Playing the same game here, the adjunction O^ad ⊣ pt^ad gives rise to an ad-sobrification monad _^ads. Its effect on preordered topological spaces (X, ≤) is a bit curious. The points of the ad-sobrification X^ads of X are pairs (C, [x]) where:

• C is an irreducible closed subset of X, namely a point of the sobrification SX (in the notation of the book; a more common notation is X^s);
• x is an ordinary point of X, and [x] is its equivalence class under the equivalence relation ≡ associated with the preordering ≤ on X (≡ is ≤ ⋂ ≥; hence if ≤ is a partial ordering, then [x] = {x}, and can be safely equated with x itself);
• the following conditions are met:
  • C intersects the downward closure ↓x of x with respect to ≤ (remember: ≤ is not the specialization preordering of X—if you are too much of a domain theorist, you will have confused the two);
  • C is included in the topological closure cl(↑x) of ↑x, where ↑x is the upward closure of x with respect to ≤ (again, not the specialization preordering);
  • C intersects ↑x;
  • C is included in the topological closure cl (↓x) of ↓x.

The topology on X^ads has as open subsets the sets ♢^adU ≝ {(C, [x]) ∈ X^ads | C intersects U}, where U ranges over the subsets of X.

The preordering on X^ads is given by (C, [x]) ≤ (C’, [x’]) if and only if x≤x’ (in X, with its preordering ≤—again, not the specialization preordering).

Ad-sobriety

The sobrification monad is idempotent, and so is the ad-sobrification monad. In practice, this means that an ad-sober preordered space, namely one that is isomorphic to the ad-sobrification of some preordered space, is isomorphic to its own ad-sobrification.

In usual Stone duality, a sober space is one that is isomorphic to the sobrification of some space, or equivalently it is a topological space in which every irreducible closed subset is the closure of a unique point. Similarly, here, a preordered topological space (X, ≤) is ad-sober if and only if every irreducible pair is of the form (cl ({x}), [x]) for some unique point x; an irreducible pair is just a point of X^ads, namely (let me recall it) a pair (C, [x]) where C is irreducible closed, x is a point of X, C intersects ↓x and ↑x and is included in the closure of both.

Let me stop here for this month. There are a few other things in [1], and notably a proof that the O^ad ⊣ pt^ad adjunction lifts the usual O ⊣ pt adjunction, at least when restricted to non-empty topological spaces and non-trivial ad-frames [1, Theorem 8.7]. (A non-trivial ad-frame is an ad-frame (Ω, L, tot, con, sup, sub) such that Ω is non-trivial, namely such that Ω has at least two elements, or equivalently one in which ⊥≠⊤.) This restriction is a bit annoying, but I don’t think it should be too much of a nuisance. At any rate, trying to get rid of this restriction seems to require a lot of effort for not much of a reward.

The nice thing about the fact that O^ad ⊣ pt^ad lifts the O ⊣ pt adjunction is that this is one point that Nesta van der Schaaf was pursuing (and had obtained) with his adjunction based on ordered locales, and which seems important in the pursuit of pointfree spacetimes in general relativity [2]. But I must admit my incompetence on this topic.

1. Jean Goubault-Larrecq. Stone Duality for Preordered Topological Spaces. http://arxiv.org/abs/2601.12932, January 21st, 2026.
2. Nesta van der Schaaf. Towards Point-Free Spacetimes. PhD thesis, University of Edinburgh, 2024. Available as arXiv report 2406.15406v2.

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