A Duality Theoretic View on Limits of Finite Structures

A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises — via Stone-Priestley duality and the notion of types from model theory — by enriching the expressive power of first-order logic with certain “probabilistic operators”. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.


Introduction
While topology plays an important role, via Stone duality, in many parts of semantics, topological methods in more algorithmic and complexity oriented areas of theoretical computer science are not so common.One of the few examples, the one we want to consider here, is the study of limits of finite relational structures.We will focus on the structural limits introduced by Nešetřil and Ossona de Mendez [32,35].These provide a common generalisation of various notions of limits of finite structures studied in probability theory, random graphs, structural graph theory, and finite model theory.The basic construction in this work is the so-called Stone pairing.Given a relational signature σ and a first-order formula ϕ in the signature σ with free variables v 1 , . . ., v n , define (the probability that a random assignment in A satisfies ϕ). (1.1) Nešetřil and Ossona de Mendez view the map A → -, A as an embedding of the finite σ-structures into the space of probability measures over the Stone space dual to the Lindenbaum-Tarski algebra of all first-order formulas in the signature σ.This space is complete and thus provides the desired limit objects for all sequences of finite structures which embed as Cauchy sequences.
Another example of topological methods in an algorithmically oriented area of computer science is the use of profinite monoids in automata theory.In this setting, profinite monoids are the subject of the extensive theory, based on theorems by Eilenberg and Reiterman, and used, among others, to settle decidability questions [36].In [7], it was shown that this theory may be understood as an application of Stone duality, thus making a bridge between semantics and more algorithmically oriented work.Bridging this semantics-versusalgorithmics gap in theoretical computer science has since gained quite some momentum, notably with the recent strand of research by Abramsky, Dawar and co-workers [2,3].In this spirit, a natural question is whether the structural limits of Nešetřil and Ossona de Mendez also can be understood semantically, and in particular whether the topological component may be seen as an application of Stone duality.
More precisely, recent work on understanding quantifiers in the setting of logic on finite words [10] has shown that adding a layer of certain quantifiers (such as classical and modular quantifiers) corresponds dually to a space-of-measures construction.The measures involved are not classical but only finitely additive and they take values in finite semirings rather than in the unit interval.Nevertheless, this appearance of measures as duals of quantifiers begs the further question whether the spaces of measures in the theory of structural limits may be obtained via Stone duality from a semantic addition of certain quantifiers to classical first-order logic.
The purpose of this paper is to address this question.Our main result is that the Stone pairing of Nešetřil and Ossona de Mendez is related by a retraction to a Stone space of finitely additive measures, which is dual to the Lindenbaum-Tarski algebra of a logic fragment obtained from first-order logic by adding one layer of probabilistic quantifiers, and which arises in exactly the same way as the spaces of semiring-valued measures in logic on words.That is, the Stone pairing, although originating from other considerations, may be seen as arising by duality from a semantic construction.
A foreseeable hurdle is that spaces of measures are not zero-dimensional and hence outside the scope of Stone duality.In fact, this issue stems from non-zero dimensionality of the unit interval [0, 1], in which the classical measures are valued.This is well-known to cause problems e.g. in attempts to combine non-determinism and probability in domain theory [22].However, in the structural limits of Nešetřil and Ossona de Mendez, at the base, one only needs to talk about finite models equipped with normal distributions and thus only the finite intervals Γ n := {0, 1  n , 2 n , . . ., 1} are involved.A careful duality theoretic analysis identifies a codirected diagram (i.e., an inverse limit system) based on these intervals compatible with the Stone pairing.The resulting inverse limit, which we denote Γ, is a Priestley space.It comes equipped with an algebra-like structure, which allows us to reformulate many aspects of the theory of structural limits in terms of Γ-valued measures as opposed to [0, 1]-valued measures.
Some interesting features of Γ, dictated by the nature of the Stone pairing and the ensuing codirected diagram, are that: • Γ is based on a version of [0, 1] in which the rationals are doubled; • Γ comes with section-retraction maps [0, 1] Γ [0, 1] ι γ ; • the map ι is lower semicontinuous while the map γ is continuous.These features are a consequence of general theory and precisely allow us to witness continuous phenomena relative to [0, 1] in the setting of Γ.
Let us stress that in the present paper we deal with finitely additive measures, as opposed to σ-additive ones.In fact, the Stone pairing construction naturally yields finitely additive measures on a Boolean algebra B. In turn, by Stone duality (cf.Section 2.1 below), B can be identified with the algebra of certain open subsets of a space X.As Nešetřil and Ossona de Mendez point out in [33,Fact 1 p. 373], if B is countable, there is a bijection between finitely additive probability measures B → [0, 1] and σ-additive (regular) probability measures on the space X equipped with the σ-algebra of measurable subsets generated by B. Thus, in this framework one can equivalently work with bona fide measures on X and this is the route that Nešetřil and Ossona de Mendez take.However, in general, the basic notion arising from the study of structural limits is that of finitely additive measure.
Our contribution.We show that the ambient space of measures for the structural limits of Nešetřil and Ossona de Mendez can be obtained via "adding a layer of quantifiers" in a suitable enrichment of first-order logic.The conceptual framework for seeing this is that of types from classical model theory.More precisely, we will see that a variant of the Stone pairing is a map into a space of measures with values in a Priestley space Γ.Further, we show that this map is in fact the embedding of the finite structures into the space of (0-)types of an extension of first-order logic, which we axiomatise.On the other hand, Γ-valued measures and [0, 1]-valued measures are tightly related by a retraction-section pair which allows the transfer of properties.These results identify the logical gist of the theory of structural limits and provide a new interesting connection between logic on words and the theory of structural limits in finite model theory.
Outline of the paper.In Section 2 we briefly recall Stone-Priestley duality, its application in logic via spaces of types, and the particular instance of logic on words (needed only to show the similarity of the constructions featuring in logic on words and in the theory of structural limits).
In Section 3 we introduce the Priestley space Γ with its additional operations, and show that it admits [0, 1] as a retract (the duality theoretic analysis justifying the structure of Γ 16:4 is deferred to the appendix).The spaces of Γ-valued measures are introduced in Section 4, and the retraction of Γ onto [0, 1] is lifted to the appropriate spaces of measures.
In Section 5, we introduce the Γ-valued Stone pairing and make the link with logic on words.Further, we compare convergence in the space of Γ-valued measures with the one considered by Nešetřil and Ossona de Mendez.In Section 6, we show that constructing the space of Γ-valued measures dually corresponds to enriching the logic with probabilistic operators.
To avoid overburdening the reader with technical details, we have decided to postpone the derivation of Γ and its structure until the appendix.This derivation relies on extended Priestley duality and the theory of canonical extensions, which we recall in Appendix A. This general theory is then applied in Appendix B to explain how the space Γ and its algebra-like structure are derived.

Preliminaries
Notation.Throughout this paper, if For a subset S ⊆ X, f S : S → Y is the obvious restriction.Given any set T , ℘ (T ) denotes its power-set.Further, for a poset P , P op is the poset obtained by turning the order of P upside down.
2.1.Stone-Priestley duality.In this paper, we will need Stone duality for bounded distributive lattices in the order-topological form due to Priestley [37].It is a powerful and well established tool in the study of propositional logic and semantics of programming languages, see e.g.[15,1] for major landmarks and [8] for an overview of the role of duality methods in logic and computer science.We briefly recall how this duality works.
A compact ordered space is a pair (X, ≤) consisting of a compact space X and a partial order ≤ on X closed in the product topology of X × X. (Note that such a space is always Hausdorff.)A compact ordered space is a Priestley space provided it is totally orderdisconnected.That is, for all x, y ∈ X such that x ≤ y, there is a clopen (i.e.simultaneously closed and open) set C ⊆ X that is an up-set for ≤ and satisfies x ∈ C but y / ∈ C. Next, we recall the construction of the Priestley space of a distributive lattice1 D. A non-empty proper subset F ⊂ D is a prime filter if it is (i) upward closed (in the natural order of D), (ii) closed under finite meets, and Denote by X D the set of all prime filters of D. By Stone's Prime Filter Theorem, the map is an embedding.Priestley's insight was that D can be recovered from X D , if the latter is equipped with the inclusion order and the topology generated by the sets of the form a and their complements.This makes X D into a Priestley space-the dual space of D-and the map -is an isomorphism between D and the lattice of clopen up-sets of X D .Conversely, any Priestley space X is the dual space of the lattice of its clopen up-sets.We call the latter the dual lattice of X.
This correspondence extends to morphisms: if h : a morphism of Priestley spaces, i.e. a continuous monotone map.
In the other direction, if f : X 1 → X 2 is a morphism of Priestley, then f −1 is a lattice homomorphism from the lattice of clopen up-sets of X 2 to the lattice of clopen up-sets of X 1 .
Priestley duality states that the ensuing functors are quasi-inverse, hence the category DLat of distributive lattices with homomorphisms is dually equivalent to the category Pries of Priestley spaces and continuous monotone maps.When restricting to Boolean algebras, we recover the celebrated Stone duality between the full subcategory BA of DLat defined by Boolean algebras, and the full subcategory BStone of Pries on those objects carrying the identity order.These can be identified with the Boolean (Stone) spaces, i.e. compact Hausdorff spaces in which the clopen subsets form a basis for the topology.

DLat
Pries op 2.2.Stone duality and logic: type spaces.The theory of types is an important tool for first-order logic.We briefly recall the concept as it is closely related to, and provides the link between, two otherwise unrelated occurrences of topological methods in theoretical computer science.Consider a first-order signature σ and a first-order theory T in this signature.Assume that we have a countably infinite sequence of distinct variables (v 1 , v 2 , v 3 , . . .).Then, for each n ∈ N, let Fm n denote the set of first-order formulas whose free variables are among v = (v 1 , . . ., v n ), and let Mod n (T ) denote the class of all pairs (A, α) where A is a model of T and α : {v 1 , . . ., v n } → A is an interpretation of v in A.

BA BStone op
Then the satisfaction relation, (A, α) |= ϕ(v), is a binary relation from Mod n (T ) to Fm n .It induces the equivalence relations of elementary equivalence and logical equivalence on these sets, respectively: The quotient FO n (T ) := Fm n /≈ carries a natural Boolean algebra structure and is known as the n-th Lindenbaum-Tarski algebra of T .Further, denoting by [(A, α)] the ≡-equivalence class of (A, α), Typ n (T ) := Mod n (T )/≡ is naturally endowed with a topology, generated by the sets {[(A, α)] | (A, α) |= ϕ(v)} for ϕ(v) ∈ Fm n , and is known as the space of n-types of T .Furthermore, the Lindenbaum-Tarski algebra is often defined as the quotient of Fm n with respect to provable equivalence, where ϕ(v) and ψ(v) are provably equivalent (modulo T ) if T ϕ(v) ↔ ψ(v).By Gödel's completeness theorem, FO n (T ) is isomorphic to the Lindenbaum-Tarski algebra or, equivalently, Typ n (T ) is the Stone dual of Fm n modulo provable equivalence.See e.g.[18, §6.3].
The Boolean algebra FO(T ) of all first-order formulas modulo logical equivalence over T is the directed colimit of the FO n (T ) for n ∈ N while its dual space, Typ(T ), is the codirected limit of the Typ n (T ) for n ∈ N and consists of (equivalence classes of) models equipped with interpretations of the full set of variables.Now, consider the set Fin(T ) consisting of isomorphism classes of finite T -models.Because two finite models are elementarily equivalent precisely when they are isomorphic, Fin(T ) can be identified with a subset of the space Typ 0 (T ) of 0-types of T .If we want to study finite models, there are two equivalent approaches: e.g. at the level of sentences, we can either consider the theory T fin of finite T -models, or the topological closure of Fin(T ) in the space Typ 0 (T ).This closure yields a space, which should tell us about finite T -structures.Indeed, it is equal to Typ 0 (T fin ), the space of pseudofinite T -structures.For an application of this, see [16].Below, we will see an application in finite model theory of the case T = ∅ (in this case we write FO(σ) and Typ(σ) instead of FO(∅) and Typ(∅)).
In light of the theory of types as exposed above, the Stone pairing of Nešetřil and Ossona de Mendez (equation (1.1)) can be seen as an embedding of finite structures into the space of finitely additive probability measures on Typ(σ), which set-theoretically are finitely additive functions FO(σ) → [0, 1].In fact, we will see that a finer grained variant of the Stone pairing corresponds precisely to embedding finite structures into the space of 0-types of an extension of first-order logic, cf.Theorem 6.6 and the discussion that follows the theorem.

Duality and logic on words.
As mentioned in the introduction, spaces of measures arise via duality in the study of semiring quantifiers in logic on words [10].In Section 5.1, we will show that basically the same construction yields the Stone pairing of Nešetřil and Ossona de Mendez.To make this analogy precise, here we provide a brief overview of logic on words and the role of measure spaces.As the technical development of the paper is independent from this material, the reader can safely defer the reading of this section.
Logic on words, as introduced by Büchi, see e.g.[29] for a recent survey, is a variation and specialisation of finite model theory where only models based on words are considered.I.e., a word w ∈ A * is seen as a relational structure on {1, . . ., |w|}, where |w| is the length of w, equipped with a unary relation P a for each a ∈ A, singling out the positions in w where the letter a appears, and the binary predicate < interpreted as the linear order 1 < • • • < |w|.Each sentence ϕ in a signature interpretable over these structures yields a language L ϕ ⊆ A * consisting of the words satisfying ϕ.Thus, logic fragments are considered modulo the theory of finite words and the Lindenbaum-Tarski algebras are subalgebras of ℘ (A * ) consisting of the appropriate L ϕ 's, cf.[16] for a treatment of first-order logic on words.
For lack of logical completeness, the duals of the Lindenbaum-Tarski algebras have more points than those given by models.Nevertheless, the dual spaces of types, which act as compactifications and completions of the collections of models, provide a powerful tool for studying logic fragments by topological means.The central notion is that of recognition, in which a Boolean subalgebra B ⊆ ℘ (A * ) is studied by means of the dual map Here β(A * ) is the Stone dual of ℘ (A * ), also known in topology as the Čech-Stone compactification of the discrete space A * , and X B is the Stone dual of B. The set A * embeds in β(A * ), and η is uniquely determined by its restriction η 0 : Anytime the latter is true for a map η and a language L as above, one says that 2 η recognises L.
When studying logic fragments via recognition, the following inductive step is central: given a notion of quantifier and a recogniser for a Boolean algebra of formulas with a free 2 Here, being beyond the scope of this paper, we are ignoring the important role of the monoid structure available on the spaces (in the form of profinite monoids or BiMs, cf.[16,10]).16:7 variable, construct a recogniser for the Boolean algebra generated by the formulas obtained by applying the quantifier.This problem was solved in [10], using duality theory, in a general setting of semiring quantifiers.The latter are defined as follows: let (S, +, •, 0 S , 1 S ) be a semiring, and k ∈ S. Given a formula ψ(v), the formula ∃ S,k v.ψ(v) is true of a word w ∈ A * if, and only if, where m is the number of assignments of the variable v in w satisfying ψ(v).If S = Z/qZ, we obtain the so-called modular quantifiers, and for S the two-element lattice we recover the existential quantifier ∃.
To deal with formulas with a free variable, one considers maps of the form where the extra bit in A × 2 is used to mark the interpretation of the free variable.In [10] (see also [11]), it was shown that L ψ(v) is recognised by f if, and only if, for every k ∈ S the language L ∃ S,k v.ψ(v) is recognised by the composite where S(X) is the space of finitely additive S-valued measures on X, and R maps w ∈ A * to the measure times.Here, n w,K is the number of interpretations α of the free variable v in w such that the pair (w, α), seen as an element of (A × 2) * , belongs to K. Finally, S(f ) sends a measure to its pushforward along f .

The space Γ
Nešetřil and Ossona de Mendez observed in [32] that the Stone pairing -, -(cf.equation (1.1)), with its second argument fixed, yields a finitely additive function FO(σ) → [0, 1].In other words, given two first-order formulas ϕ and ψ and a finite σ-structure A, and also f , A = 0 and t, A = 1, where f and t denote false and true, respectively.By Stone duality, we can identify FO(σ) with the Boolean algebra of clopen subsets of the space of types Typ(σ).By assuming that clopen (equivalently, first-order definable) subsets of Typ(σ) form the Boolean algebra of measurable subsets, we view Typ(σ) as a measurable space.Then, assigning to the second argument of the Stone pairing yields an embedding of Fin(σ), the set of isomorphism classes of finite σ-structures, into the space M I (Typ(σ)) of finitely additive measures: The main obstacle in being able to use duality theory to investigate the logical nature of the space M I (Typ(σ)) is the fact that this space is not zero-dimensional, and thus not amenable to the usual duality theoretic methods.This is a direct consequence of the fact that M I (Typ(σ)) is a closed subspace of the product [0, 1] FO(σ) , and thus inherits most of its topological properties from the unit interval [0, 1].Consequently, in order to exploit duality theory, e.g. in the form of Priestley duality, we need to find a suitable Priestley space Γ that will play the role of the unit interval.
The construction of Γ comes from the insight that the range of the Stone pairing -, A , for a finite σ-structure A and formulas restricted to a fixed number of free variables, can be confined to a chain of the form where n := |A| k if k is the number of free variables.Each Γ n , with the obvious total order and the discrete topology, is a Priestley space.To allow for formulas with any number of free variables, we require that Γ is obtained as a codirected limit of all finite chains Γ n .If the maps in the codirected diagram, say which send a nm to b n where b := a/m and b := a/m , respectively.Lastly, we require that Γ comes equipped with some algebraic structure which is rich enough so that we can define Γ-valued measures.Since Γ will be computed as a limit of the finite chains Γ n , the natural way to identify this algebraic structure is to analyse the arithmetic operations available on those finite chains.Given that we need to represent finite additivity, it is natural to consider partial addition 3 x + y on Γ n , defined whenever x + y ≤ 1.The partial addition on Γ n behaves well in many respects, and even has a right adjoint: where z − y is the partial subtraction defined whenever y ≤ z.However, partial addition is not compatible with floor and ceiling functions.Indeed, the operator dual to the partial addition on chains Γ n is not preserved by the lattice homomorphisms dual to the floor functions -: Γ nm → Γ n , nor by the duals of the ceiling functions -: Γ nm → Γ n .For more details, see Appendix B.
On the other hand, as we shall now see, the partial subtraction is compatible with floor functions.Note that, by the adjointness property stated above, addition and subtraction are interdefinable, and so we preserve our ability to express finite additivity.Let L n be the lattice dual to the Priestley space Γ n , i.e. the lattice of up-sets of Γ n .In other words, L n is the finite chain The following lemma provides a description of the operator on L n dual to partial subtraction on Γ n , and relies on extended Priestley duality between lattices with operators and Priestley spaces with partial operations (or relations), see e.g.[15] and also [12,13].The necessary background on extended Priestley duality, and a derivation of the following fact, are offered in Appendix B.
Lemma 3.1.The operator ⊕ : It is immediate to see that the lattice homomorphisms dual to floor functions are the lattice embeddings i n,nm : L n → L nm .Moreover, these embeddings preserve ⊕, i.e.
for all u, v ∈ L n .Note that this is not true for the duals of ceiling functions (see Appendix B for details).

3.1.
A concrete representation of Γ.In view of the discussion above, we define Γ as the limit of the codirected diagram where The lattice L is given by with ⊥ < L q and q ≤ L p for every p ≤ q in [0, 1] ∩ Q.We can then identify the points of Γ with the prime filters of L. Each q ∈ [0, 1] ∩ Q yields a prime filter q • := {p ∈ L | q ≤ p}, and each r ∈ (0, 1] yields a prime filter r − := {p ∈ L | r < p}.It is not difficult to see that all prime filters of L are of one of these types, and so Γ can be represented as based on the set The order of Γ is the unique total order that has 0 • as bottom element, satisfies r * < s * if and only if r < s for * ∈ {−, •}, and such that q • is a cover of q − for every rational q ∈ (0, 1] (i.e.q − < q • , and there is no element strictly in between).In a sense, the elements of the form q • are regarded as exact values, and those of the form r − as (lower) approximations.
Cf. Figure 2. The topology of Γ is generated by the sets of the form The second partial operation is definable in terms of − and is given by Explicitly, we have x ∼ x = 0 • and, whenever y < x, The proof of the correspondence between ⊕ and the pair (−, ∼) is included in Appendix B.
The two partial operations on Γ have specific topological and order-theoretic properties which are crucial in order to be able to reconstruct ⊕.Moreover, we need both operations in order to recover the classical [0, 1]-valued measures as retracts of Γ-valued measures (cf. the proof of Lemma 4.7 essential for Theorem 4.10).
Here, we collect only the necessary properties of − and ∼ needed in Section 4. Recall that a map between ordered topological spaces is lower (resp.Proof.Items 1-3 as stated above follow from the general theory of [12,13] and a direct proof is included in Appendix A. Item 4 requires extra care because we extended the natural domain of definition of ∼ for our application. The general theory entails that the partial map ∼ defined as the restriction of ∼ to {(x, y) | y < x} is upper semicontinuous and also that dom(∼ ) is an open up-set in Γ × Γ op .To deduce item 4, let q ∈ (0, 1] ∩ Q. Observe that x = y implies that x ∼ y = 0 • / ∈ ↑q • .Consequently, the preimage of ↑q • under ∼ is equal to (∼ ) −1 (↑q • ), which is open in the subspace topology of dom(∼ ).Since dom(∼ ) is open in Γ × Γ, the preimage (∼) −1 (↑q • ) is open in Γ × Γ and thus also in the subspace topology of dom(∼).

The retraction Γ
[0, 1].In this section we show that, with respect to appropriate topologies, the unit interval [0, 1] can be obtained as a topological retract of Γ, in a way that is compatible with the operation −.This will be important in Sections 4 and 5, where we need to move between [0,1]-valued and Γ-valued measures.Let us define the monotone surjection given by collapsing the doubled elements: The map γ has a right adjoint, given by Indeed, it is readily seen that, for all y ∈ Γ and x ∈ [0, 1], γ(y) ≤ x if and only if y ≤ ι(x).
The composition γ • ι coincides with the identity on [0, 1], i.e. ι is a section of γ.Moreover, as a consequence of the next lemma, this retraction lifts to a topological retract provided we equip Γ and [0, 1] with the topologies consisting of the open down-sets.
Proof.To check continuity of γ observe that, for q ∈ Q∩(0, 1), the sets γ −1 (q, 1] and γ −1 [0, q) coincide, respectively, with the open sets Also, ι is lower semicontinuous, for ι −1 (↓q − ) = [0, q) whenever q ∈ Q ∩ (0, 1]. It is easy to see that both γ and ι preserve the minus structure available on Γ and [0, 1] (the unit interval is equipped with the usual minus operation x − y defined whenever y ≤ x), that is, Remark 3.5.The map γ : Γ → [0, 1] has, in addition to the right adjoint ι, also a left adjoint ι − , given by The latter is also a section of γ, but in the following we shall mainly work with the map ι.The aim of this section is to replace [0, 1]-valued measures by Γ-valued measures.The reason for doing this is two-fold.First, the collection of all Γ-valued measures is a Priestley space (Proposition 4.3), and thus amenable to a duality theoretic treatment and a dual logic interpretation (cf.Section 6).Second, it retains more topological information than the space of [0, 1]-valued measures.Indeed, the former retracts onto the latter (Theorem 4.10).

16:12
Let D be a distributive lattice.Recall that, classically, a monotone function We denote by M Γ (D) the subset of Γ D consisting of the measures µ : D → Γ.
Note that, if we only required one of the two inequalities in item 3 above, we would essentially be considering finitely subadditive measures.
Since Γ is a Priestley space, so is Γ D equipped with the product order and topology.Hence, we regard M Γ (D) as an ordered topological space, whose topology and order are induced by those of Γ D .Explicitly, given measures µ, ν ∈ M Γ (D) we have µ ≤ ν if, and only if, µ(a) ≤ ν(a) for all a ∈ D.Moreover, the topology of M Γ (D) is generated by the sets of the form a < q , for all a ∈ D and q ∈ Q∩[0, 1].Note that the sets a < q are clopen down-sets, while the sets a ≥ q are clopen up-sets.It turns out that M Γ (D) is a Priestley space (Proposition 4.3).The proof of this fact relies on the following lemma: Lemma 4.2.Let X, Y be compact ordered spaces, f : X → Y a lower semicontinuous function, and g : X → Y an upper semicontinuous function.If X is a closed subset of X, then so is {x ∈ X | g(x) ≤ f (x)}.Then, the open set f −1 (U ) ∩ g −1 (V ) contains x and is disjoint from {x ∈ X | g(x) ≤ f (x)}.Since x was arbitrary, we conclude that the set E Proof.It suffices to show that M Γ (D) is a closed subspace of Γ D .Let Note that the evaluation maps ev a : Γ D → Γ, f → f (a) are continuous for every a ∈ D. Thus, the first set in the intersection defining C 1,2 is closed because it is the equaliser of the evaluation map ev 0 and the constant map of value 0 • .Similarly for the set {f ∈ Γ D | f (1) = 1 • }.The last set is the intersection of sets of the form ev a , ev b −1 (≤), which are Proof.First, we show that if h : D → E is a lattice homomorphism and µ : E → Γ is a measure, the composite map µ • h : D → Γ is also a measure.Since µ and h are monotone, so is µ • h.Further, µ(h(0)) = µ(0) = 0 • , and µ(h(1)) = µ(1) = 1 • .For the third condition in Definition 4.1 observe that, for all a, b ∈ D, where the middle inequality holds because µ is a measure.The inequality With respect to continuity, recall that the topology of M Γ (D) is generated by the sets of the form Hence, in view of Proposition 4.3, M Γ (h) is a morphism of Priestley spaces.It is not difficult to see that identities and compositions are preserved, therefore M Γ : DLat → Pries is a contravariant functor.Remark 4.5.We work with the contravariant functor M Γ : DLat → Pries because M Γ is concretely defined on the lattice side.However, by Priestley duality, DLat is dually equivalent to Pries, so we can think of M Γ as a covariant functor Pries → Pries (this is the perspective traditionally adopted in analysis, and also in the works of Nešetřil and Ossona de Mendez).From this viewpoint, Section 6 provides a description of the endofunctor on DLat dual to M Γ : Pries → Pries.
Next, we establish a property of the functor M Γ : DLat → Pries which is very useful when approximating a fragment of a logic by smaller fragments (see, e.g., Section 5.1).
Proof.Let D be the colimit in DLat of a directed diagram with colimit maps ι i : D i → D (where i and j vary in a directed poset I).We must show that the cone {M Γ (ι i ) : Let Y be a Priestley space and {g i : Y → M Γ (D i )} i∈I a set of Priestley morphisms such that g i = M Γ (h i,j ) • g j , for every i ≤ j.We need to prove that there is a unique Priestley morphism ξ : Y → M Γ (D) such that M Γ (ι i ) • ξ = g i for every i.Note that, since directed colimits in DLat are computed in the category of sets, for every element a of D there are an index i ∈ I and an element a i ∈ D i such that ι i (a i ) = a.Define where a i satisfies ι(a i ) = a.This definition is independent of the choice of a i : for any other a j ∈ D j such that ι j (a j ) = a, we have that h i,k (a i ) = h j,k (a j ) for some k ≥ i, j, and so for all y ∈ Y .Observe that, for every y ∈ Y , ξ(y) is indeed a measure D → Γ.For example, let a, b ∈ D.Then, because I is directed, there is an index i such that a = ι i (a i ) and b = ι i (b i ) for some a i , b i ∈ D i .Since g i (y) is a measure, we have that The other properties are proved in the same spirit.Next, we show that ξ is a Priestley morphism.Monotonicity is immediate: if y ≤ y in Y and a i ∈ D i , then For continuity, for all a i ∈ D i and q ∈ Q ∩ [0, 1], we have and, similarly, ξ −1 ( ι i (a i ) ≥ q ) = g −1 i ( a i ≥ q ).Lastly, we show that ξ is unique with these properties.Let λ : Y → M Γ (D) be a Priestley morphism such that g i = M Γ (ι i ) • λ for every i ∈ I.For every a ∈ D, and element a i ∈ D i such that ι i (a i ) = a, we get for all y ∈ Y .Hence, λ = ξ.
Recall the maps γ : Γ → [0, 1] and ι : [0, 1] → Γ from equations (3.1)-(3.2).In Section 3.3 we have shown that this is a retraction-section pair.In Theorem 4.10 we will lift this retraction to the spaces of measures.We start with an easy observation: Lemma 4.7.Let D be a distributive lattice.The following statements hold: Proof.(1) The only non-trivial condition to verify is finite additivity.As pointed out after Lemma 3.3, the map γ preserves both minus operations on Γ.Hence, for all a, b ∈ D, the inequalities µ(a (2) The first two conditions in Definition 4.1 are immediate.The third condition follows from the fact that ι(r − s) = ι(r) − ι(s) whenever s ≤ r in [0, 1], and x ∼ y ≤ x − y for every (x, y) ∈ dom(−).
In view of the previous lemma, there are well-defined functions Finally, we see that the set-theoretic retraction of M Γ (D) onto M I (D) lifts to the topological setting, provided we restrict to the down-set topologies.If (X, ≤) is a partially ordered topological space, write X ↓ for the space with the same underlying set as X and whose topology consists of the open down-sets of X. Proof.It suffices to show that γ # and ι # are continuous.It is not difficult to see, using Lemma 4.8, that γ # : M Γ (D) ↓ → M I (D) ↓ is continuous.For the continuity of ι # , note that the topology of M Γ (D) ↓ is generated by the sets of the form a < q , for a ∈ D and q ∈ Q ∩ (0, 1].We have which is an open set in M I (D) ↓ .This concludes the proof.

The Γ-valued Stone pairing and limits of finite structures
In the work of Nešetřil and Ossona de Mendez, the Stone pairing -, A is [0, 1]-valued, i.e. an element of M I (FO(σ)).In this section, we show that basically the same construction for the recognisers arising from the application of a layer of semiring quantifiers in logic on words (cf.Section 2.3) provides an embedding of finite σ-structures into the space of Γ-valued measures.It turns out that this embedding is a Γ-valued version of the Stone pairing.Hereafter we make a notational difference, writing -, -I for the (classical) [0, 1]-valued Stone pairing.
The main ingredient of the construction are the Γ-valued finitely supported functions.To start with, we point out that the partial operation − on Γ uniquely determines-by adjointness-a partial "plus" operation on Γ. Define + : dom(+) → Γ, where dom(+) = {(x, y) | x ≤ 1 • − y} , by the following rules (whenever the expressions make sense): Then, for every x ∈ Γ, the function (-) + x sending y to y + x is left adjoint to the function (-) − x sending y to y − x.This is the content of the next lemma (where, for all x, y ∈ Γ, we let [x, y] := {z ∈ Γ | x ≤ z ≤ y}); for a proof, see the end of Appendix B.4. Lemma 5.1.For every x ∈ Γ, the monotone map (-) This plus operation allows us to define the notion of finitely supported Γ-valued function: To improve readability, if the sum y 1 +• • •+y m exists in Γ, we denote it m i=1 y i .Finitely supported functions in the above sense always determine measures over the power-set algebra: Lemma 5.3.Let X be any set.There is a well-defined mapping : F(X) → M Γ ( ℘ (X)), assigning to every f ∈ F(X) the measure Proof.We wish to prove that f is a measure for every f ∈ F(X).First, observe that ∅ f = 0 • and X f = 1 • by definition.Moreover, M → M f is monotone because x ≤ x + y for all x, y ∈ Γ for which x + y is defined.16:17 It remains to show that, for any M, N ⊆ X, To simplify these expressions, set m := M \N f , n := N \M f , and i := M ∩N f .Then, we see that Consequently, the inequalities in (5.1) can be rewritten, respectively, as By Lemma 5.1, these are equivalent, respectively, to Next, observe that, for all x, y ∈ Γ for which x + y is defined, (x + y) ∼ y is always of − -type or equal to 0 • , and so (x + y) ∼ y = ι − (γ(x)) (where the map ι − : [0, 1] → Γ is defined as in Remark 3.5).Therefore, ((m + i) ∼ i) 5.1.The Γ-valued Stone pairing and logic on words.Fix a countably infinite sequence of distinct variables (v 1 , v 2 , v 3 , . . .).Recall that FO n (σ) is the Lindenbaum-Tarski algebra of first-order formulas with free variables among {v 1 , . . ., v n }.The dual space of FO n (σ) is the space of n-types Typ n (σ).Its points are the elementary equivalence classes of pairs (A, α), where A is a σ-structure and α : {v 1 , . . ., v n } → A is an interpretation of the variables.Also, recall that Fin(σ) denotes the set of isomorphism classes of finite σ-structures.
We would like to define a function Fin(σ) → F(Typ n (σ)) which, intuitively, associates with a finite σ-structure A the (normalised) characteristic function of the finite set and 0 otherwise.However, we need to slightly modify this definition to take into account the fact that elements of Typ n (σ) are elementary equivalence classes of pairs (B, β), and such an equivalence class may contain several pairs whose first component is A.
Define the map Fin(σ) → F(Typ n (σ)) which sends the isomorphism class of a finite σ-structure A to the finitely supported function f A n whose value at an equivalence class for every interpretation α of the free variables s.t.(A, α) is in the equivalence class).
It is not difficult to see that this definition does not depend on the choice of a representative in the isomorphism class of A.
Restricting f A n to FO n (σ), we get a measure Summing up, we have the composite map Essentially the same construction is featured in logic on words, cf.equation (2.1): • The set of finite σ-structures Fin(σ) corresponds to the set of finite words A * .
• The collection Typ n (σ) of (equivalence classes of) σ-structures with interpretations corresponds to (A × 2) * or, interchangeably, β(A × 2) * (in the case of one free variable).• The fragment FO n (σ) of first-order logic corresponds to the Boolean algebra of languages, defined by formulas with a free variable, dual to the Boolean space X appearing in (2.1).• The first map in the composite (5.2) sends a finite σ-structure A to the measure f A n which, evaluated on K ⊆ Typ n (σ), counts the (proportion of) interpretations α : {v 1 , . . ., v n } → A such that (A, α) ∈ K, similarly to R from (2.1).
On the other hand, the assignment A → µ A n defined in (5.2) is also closely related to the classical Stone pairing.Indeed, for every formula ϕ in FO n (σ), where -, -I is the classical [0, 1]-valued Stone pairing (cf. the beginning of Section 5).In this sense, µ A n can be regarded as a Γ-valued Stone pairing, relative to the fragment FO n (σ).Next, we show how to extend this to the full first-order logic FO(σ).First, we observe that-as in the classical case-the construction is invariant under extensions of the set of free variables: Proof.Every variable assignment α : {v 1 , . . ., v n } → A has |A| m−n possible extensions α : {v 1 , . . ., v m } → A, to account for the m − n unused variables.Moreover, for every ϕ ∈ FO n (σ), (A, α) |= ϕ if, and only if, (A, α ) |= ϕ.Because µ A m and µ A n take only values of • -type, and p • + q • = (p + q) • , we see that The Lindenbaum-Tarski algebra of all first-order formulas FO(σ) is the directed colimit of the chain of its Boolean subalgebras Since the functor M Γ turns directed colimits into codirected limits (Proposition 4.6), the Priestley space M Γ (FO(σ)) is the limit of the diagram where, for every measure µ : FO m (σ) → Γ, the measure q n,m (µ) is the restriction of µ to FO n (σ).In view of Lemma 5.4, for every A ∈ Fin(σ), the tuple (µ A n ) n∈N is compatible with the restriction maps.Thus, recalling that limits in the category of Priestley spaces are computed as in sets, by universality of the limit construction this tuple yields a measure -, A Γ : FO(σ) → Γ in the space M Γ (FO(σ)) whose restriction to FO n (σ) coincides with µ A n , for every n ∈ N.This we call the Γ-valued Stone pairing associated with A. As in the classical case, using the fact that any two non-isomorphic finite structures are separated by a first-order sentence, it is not difficult to see that the mapping A → -, A Γ gives an embedding -, -Γ : Fin(σ) → M Γ (FO(σ)).
The following theorem illustrates the relation between the classical Stone pairing, namely -, -I : Fin(σ) → M I (FO(σ)), and the Γ-valued one.Proof.Fix an arbitrary finite σ-structure A ∈ Fin(σ).Let ϕ be a formula in FO(σ) with free variables among {v 1 , . . ., v n }, for some n ∈ N. By construction, ϕ, A Γ = µ A n (ϕ).Therefore, by equation (5.3), ϕ, A Γ = ( ϕ, A I ) • and thus the statement follows at once.Remark 5.6.The construction in this section works, mutatis mutandis, also for smaller logic fragments, i.e. for sublattices D ⊆ FO(σ).This corresponds to composing the embedding Fin(σ) → M Γ (FO(σ)) with the restriction map M Γ (FO(σ)) → M Γ (D) sending a measure µ : FO(σ) → Γ to µ D : D → Γ.The only difference is that, in general, the ensuing map Fin(σ) → M Γ (D) need not be injective.In Theorem 6.4 below we will see that, in fact, PL D is complete with respect to this measure-theoretic semantics.In other words, PL D is the logic of the space of measures M Γ (D).To start with, we let P(D) be the Lindenbaum-Tarski algebra of PL D , that is the quotient of the free distributive lattice on the set with respect to the congruence generated by the conditions (L1)-(L5).In particular, recall from Section 2.2 that elements of P(D) are equivalence classes of formulas ϕ, ψ with respect to equiprovability in the logic PL D .Moreover, considering representatives, we have ϕ ≤ ψ in P(D) if, and only if, ϕ ψ holds in the logic PL D .
In order to show that every prime filter of P(D) yields a measure in M Γ (D), we need the following lemma: Lemma 6.2.The following statements hold: (1) for every x, y ∈ Γ such that y ≤ (2) for every x ∈ Γ, the map x − (-) : [0 • , x] → Γ sends suprema to infima.
Proof.(1) Recall that, for every x, y ∈ Γ such that y ≤ x, So, it suffices to show that, whenever q • ≤ x, and it is precisely r • − q • .Thus, suppose x = r − for some r ∈ (0, 1].To show the non-trivial inequality, pick z ∈ Γ such that p • −q • ≤ z whenever q • ≤ p • ≤ x.Note that γ(x)−q ≤ γ(z), for otherwise we could find p ∈ Q ∩ [0, 1] satisfying q ≤ p < γ(x) and γ(z) < p − q, and so z < p • − q • .It follows that (2) Fix x ∈ Γ.We must prove that, for every subset x − i∈I By Lemma 3.2, the map x − (-) : [0 • , x] → Γ is lower continuous and order reversing.Since the interval [0 • , x] is closed in Γ, it is a compact ordered space with respect to the induced order and topology (in fact, it is a Priestley space).The supremum i∈I y i then coincides with the limit of {y i | i ∈ I}, regarded as a net in [0 • , x] (cf.[14,Proposition VI.1.3]).By lower continuity of x − (-), we have that x − i∈I y i , regarded as an element of Γ op , is less than or equal to the supremum (computed in Γ op ) of the x − y i 's.That is, in Γ.The other inequality follows at once from the fact that x − (-) is order reversing.the axiomatisation given by Heckmann [17] in terms of a certain geometric propositional logic.The work of Heckmann was later streamlined by Vickers [40].For another infinitary duality theoretic approach to logics with probabilistic quantifiers see e.g.[28,5], where an axiomatisation based on the signature of σ-complete Boolean algebras is provided.Logics with probabilistic quantifiers have been studied also in the setting of finite model theory.In fact, Friedman's probabilistic quantifiers (cf.[38,39]), which express that the probability is > 0, are the main ingredient in the proof of one of the main results in the theory of structural limits, see [34, Theorem 3.2 and Corollary 4.3].Furthermore, Kontinen [27], Keisler and Lotfallah [24], and Knyazev [25] showed that probabilistic quantifiers behave reasonably well with respect to zero-one laws.
Probabilistic quantifiers akin to ours were extensively studied already in the 1970s, see Keisler's survey [23].The main difference, compared to our approach, is that infinitary conjunctions or rules need to be added in order to obtain completeness, and therefore the compactness theorem is lost.Our logic, on the other hand, does not allow for nesting of probabilistic quantifiers, is finitary and satisfies the compactness theorem.The latter follows at once from duality theory and the fact that the space of Γ-valued measures dual to PL * D is topologically compact.More recently, Zhou [41] showed that a propositional version of Keisler logic can be made finitary; however, this logic still fails to satisfy the compactness theorem.

Conclusion
Types are points of the dual space of a logic (identified with its Lindenbaum-Tarski algebra).In classical first-order logic, 0-types are just the models modulo elementary equivalence.But when there are not "enough" models, as in finite model theory, the spaces of types provide completions of the sets of models.
In [10], it was shown that for logic on words and various quantifiers we have that, given a Boolean algebra of formulas with a free variable, the space of types of the Boolean algebra generated by the formulas obtained by quantification is given by a space-of-measures construction.Here we have shown that a suitable enrichment of first-order logic gives rise to a space of measures M Γ (FO(σ)) closely related to the space M I (FO(σ)) used in the theory of structural limits.Indeed, Theorem 5.5 tells us that the ensuing Stone pairings interdetermine each other.Further, the Stone pairing for M Γ (FO(σ)) is just the embedding of the finite models in the completion/compactification provided by the space of 0-types of the enriched logic (Theorem 6.6).
These results identify the logical gist of the theory of structural limits, and provide a new and interesting connection between logic on words and the theory of structural limits in finite model theory.But we also expect that it may prove a useful tool in its own right.
For instance, for structural limits, it is an open problem to characterise the closure Fin(σ) of the image of the [0, 1]-valued Stone pairing [33].The same question translates equivalently to the Γ-valued setting (Theorem 5.7), native to logic and where we can use duality.The closure Fin(σ), seen as a subspace of M Γ (FO(σ)), is also a Priestley space and this embedding dually corresponds to a quotient of the Lindenbaum-Tarski algebra of the enriched logic.Understanding this quotient is crucial in order to characterise Fin(σ).One would expect that this is the subspace M Γ (FO(T fin )) of M Γ (FO(σ)) given by the quotient FO(σ) FO(T fin ) onto the theory of pseudofinite structures.The purpose of such a characterisation would be to understand the points of the closure as "generalised models".16:27 Another subject that we would like to investigate is that of zero-one laws.The zero-one law for first-order logic states that the sequence of measures for which the nth measure, on a sentence ψ, yields the proportion of n-element structures satisfying ψ, converges to a {0, 1}-valued measure.Over Γ this will no longer be true as 1 is split into its "limiting" and "achieved" personae.Yet, we expect the above sequence to converge also in this setting and, by Theorem 5.5, it will converge to a {0 • , 1 − , 1 • }-valued measure.Understanding this finer grained measure may yield useful information about the zero-one law.
Finally, it would be interesting to investigate whether the limits for schema mappings introduced by Kolaitis et al. [26] may be seen also as a type-theoretic construction.Also, we would want to explore the connections with other semantically inspired approaches to finite model theory, such as those recently put forward by Abramsky, Dawar et al. [2,3].example (in the case of Boolean algebras) is the celebrated Jónsson-Tarski duality [20,21], from which most of the theory originates.
We present an algebraic approach to extended Priestley duality which is based on the theory of canonical extensions.Its main feature is that both the space X and its dual lattice A embed (order-theoretically) into a complete lattice A δ , the canonical extension of A. This allows us to capture Priestley duality in purely algebraic terms, and is an incredibly powerful approach for dealing with additional operations on lattices.For example, if A comes equipped with a binary operation h : A × A → A then, by studying the relationship between the graph of h and the points of X in the product A δ × A δ × A δ , we can identify two partial binary operations on X that fully capture h.An instance of this situation is studied in this section.Our main inspiration comes from [12,13] and also [6].
A.1.Basic notation and terminology.The canonical extension of a (bounded) distributive lattice A is an embedding e : A → A δ of A into a complete lattice A δ that is dense and compact: (Dense) Every element of A δ is both a join of meets and a meet of joins of elements in the image of e.
(Compact) Given subsets S and T of A with e(S) ≤ e(T ), there are finite sets S ⊆ S and T ⊆ T such that e(S ) ≤ e(T ).
Canonical extensions of distributive lattices always exist and are unique, up to isomorphism.In fact, A δ is isomorphic to the lattice of all up-sets of the Priestley space dual6 to A. In the following, we will always identify A with a sublattice of its canonical extension A δ .Open elements of A δ are those of the form S, for some S ⊆ A. The set of all open elements of A δ is denoted I(A δ ).In fact, there is a bijection between open elements of A δ and ideals of A, given by i ∈ I(A δ ) → ↓i ∩ A and I ⊆ A → I.The latter restricts to a bijection between the set M ∞ (A δ ) of completely meet-irreducible elements of A δ and prime ideals of A. Recall that p ∈ A δ is completely meet-irreducible if, for all S ⊆ C, S ≤ p implies a ≤ p for some a ∈ S. The set F(A δ ) of closed elements of A δ , and the set J ∞ (A δ ) of completely join-irreducible elements of A δ are defined order-dually.
Note that M ∞ (A δ ) does not include the top element of A δ .However, it is sometimes useful to add it, and so we will denote by M ∞ 1 (A δ ) the set M ∞ (A δ ) ∪ {1} and, similarly, J ∞ 0 (A δ ) denotes the set J ∞ (A δ ) ∪ {0}.
For a distributive lattice A, every element of A δ is a meet of completely meet-irreducible elements and a join of completely join-irreducible elements.Moreover, if we equip J ∞ (A δ ) and M ∞ (A δ ) with the partial orders induced by A δ , we have an order-isomorphism κ : J ∞ (A δ ) → M ∞ (A δ ), j → {a ∈ A | j ≤ a} satisfying, for every m ∈ M ∞ (A δ ), j ∈ J ∞ (A δ ) and u ∈ A δ , u ≤ κ(j) ⇐⇒ j u and κ −1 (m) ≤ u ⇐⇒ u m.
This corresponds to the fact that the complement of a prime filter is a prime ideal, and vice versa.For more details, we refer the reader to [9].
A.2. Priestley duality.Thanks to the correspondence between prime filters (resp.prime ideals) of a distributive lattice A and completely join-irreducible (resp.completely meetirreducible) elements of A δ , we can retrieve the Priestley space X = (X, ≤, τ ) dual to A by endowing either the set M ∞ (A δ ), or the set J ∞ (A δ ), with an order and topology.Both choices result in the same space, but we opt for the former as it provides a minor technical advantage in our applications.The order on M ∞ (A δ ) is inherited from A δ and the subbase of the topology is formed by the sets {m | a m} and their complements, for every a ∈ A.
Remark A.1.We warn the reader that, with the definition of dual Priestley space given in Section 2.1, the Priestley space X just defined is, in fact, dual to the lattice A op and not A.
Just observe that, for m, n ∈ M ∞ (A δ ), where I m := {a ∈ A | a ≤ m} is the prime ideal corresponding to m ∈ M ∞ (A δ ) and F j := {a ∈ A | j ≤ a} is the prime filter corresponding to j ∈ J ∞ (A δ ).When working with canonical extensions, it is convenient to order prime filters by reverse inclusion, so that both X and its dual lattice A order-theoretically embed into A δ .However, we opted for the inclusion order in the main text because it gives the usual pointwise order between measures.
To apply the general theory to Γ, in Appendix B we will take order-duals on the lattice side.
To also capture the duality between lattice homomorphisms and Priestley morphisms, we recall how maps between lattices extend to maps between their canonical extensions.For a monotone map f : A → B between distributive lattice, its sigma extension f σ : A δ → B δ is defined by The two extensions are closely related as they agree on open and closed elements.Moreover, if f preserves binary joins then f σ preserves all non-empty joins and, similarly, if f preserves binary meets then f π preserves all non-empty meets.
Whenever f σ = f π , we say that f is smooth and denote the unique extension by f δ .This happens, for example, when f preserves finite (possibly empty) joins or finite meets.Recall that a monotone map f between complete lattices preserves all joins if and only if it has a right adjoint f # and, dually, it preserves all meets precisely when it has a left adjoint f .Moreover, if h : A → B is a lattice homomorphism, the right adjoint h δ # of h δ : A δ → B δ preserves completely meet-irreducible elements and, dually, the left adjoint h δ preserves completely join-irreducible elements.We can compute the continuous monotone map dual to a homomorphism h : A → B as the restriction of h δ # : B δ → A δ to the completely meetirreducible elements.Note that, if we defined dual Priestley spaces as based on J ∞ (A δ ), we would have to take the restriction of h δ : B δ → A δ to completely join-irreducible elements.

Figure 2 :
Figure 2: The Priestley space Γ and its dual lattice L.

3. 2 .
The algebraic structure on Γ.Because the embeddings i n,nm : L n → L nm preserve the operators ⊕ : L n × L n → L n , the colimit lattice L is naturally equipped with the operator ⊕ : L × L → L obtained by gluing together all the operators on the finite sublattices L n .By extended Priestley duality, the operator ⊕ on L is dual to a pair of partial operations 4 − and ∼ on Γ.The first one, − : dom(−) → Γ, has domain dom(−) = {(x, y) ∈ Γ × Γ | y ≤ x} and is defined by upper ) semicontinuous provided the preimage of any open down-set (resp.open up-set) is open.The next lemma shows, in particular, that − and ∼ are semicontinuous, a key observation which will allows us to show that the space of Γ-valued measures is a Priestley space (Proposition 4.3).Lemma 3.2.If dom(−) is regarded as an ordered subspace of Γ × Γ op , the following hold: (1) dom(−) is a closed up-set in Γ × Γ op ; (2) − : dom(−) → Γ and ∼ : dom(−) → Γ are monotone, i.e. if (x, y) ∈ dom(−), x ≤ x and y ≤ y, then x − y ≤ x − y and x ∼ y ≤ x ∼ y ; (3) − : dom(−) → Γ is lower semicontinuous; (4) ∼ : dom(−) → Γ is upper semicontinuous.

Proof.
Whenever g(x) ≤ f (x) for some x ∈ X, there are an open down-set U and open up-set V of Y that are disjoint and satisfy f (x) ∈ U and g(x) ∈ V .See e.g.[31, Theorem 4 p. 46].

FromLemma 4 . 4 .
semicontinuity of − and ∼ (Lemma 3.2), and Lemma 4.2, we conclude that M Γ (D) is a closed subspace of Γ D .It turns out that the construction D → M Γ (D) is functorial: The assignment D → M Γ (D) can be extended to a contravariant functor M Γ : DLat → Pries by setting, for all lattice homomorphisms h : D → E,

Lemma 4 . 8 .Corollary 4 . 9 .
γ # : M Γ (D) → M I (D) is a continuous and monotone map.Proof.The topology of M I (D) is generated by the sets of the form {m ∈ M I (D) | m(a) ∈ O}, for a ∈ D and O an open subset of [0, 1].In turn,(γ # ) −1 {m ∈ M I (D) | m(a) ∈ O} = {µ ∈ M Γ (D) | µ(a) ∈ γ −1 (O)} is open in M Γ (D) because γ : Γ → [0, 1] is continuous by Lemma 3.3.This shows that γ # : M Γ (D) → M I (D) is continuous.Monotonicity is immediate.Note that γ # : M Γ (D) → M I (D)is surjective, since it admits ι # as a (set-theoretic) section.It follows that M I (D) is a compact ordered space: For each distributive lattice D, M I (D) is a compact ordered space.Proof.The surjection γ # : M Γ (D) → M I (D) is continuous (Lemma 4.8).Since M Γ (D) is compact by Proposition 4.3, so is M I (D).The order of M I (D) is clearly closed in the product topology, thus M I (D) is a compact ordered space.

Proposition 6 . 3 .
Let F ⊆ P(D) be a prime filter.The assignment a → {q • | P ≥q a ∈ F } defines a measure µ F : D → Γ.
are monotone, we are guaranteed that Γ is a Priestley space, see e.g.[19, Corollary VI.3.3].There are two natural candidates for such maps, namely the floor and ceiling functions -: Γ nm Γ n and -: Γ nm Γ n