Positive Modal Logic Beyond Distributivity

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $\Pi_1$-persistence and show that every weak positive modal logic is $\Pi_1$-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist correspondence result.


Introduction
Duality between modal algebras and modal spaces on the one hand and Heyting algebras and Esakia spaces on the other have been central to the study of modal and intermediate logics, [6,8]. Many important results such as Sahlqvist canonicity and correspondence are based on duality techniques [36]. In [7], duality between modal algebras and modal spaces has been extended to a duality between modal distributive lattices (i.e. with distributive lattices taking the role of Boolean algebras) and modal Priestley spaces. Among other things, this led to a Sahlqvist theory for positive distributive modal logic.
When the algebraic side of a duality is based on Boolean algebras or distributive lattices, in the spatial side of the duality one works with the space of all prime filters of a given lattice. This is no longer the case when the base lattice is non-distributive. There have been many attempts to extend a duality for Boolean algebras and distributive lattices to the setting of all lattices, e.g. by Urquhart [37], Hartonas [23,24], Gehrke and van Gool [16], and Goldblatt [21]. Each of these uses either a ternary relation, or two-sorted frames. While this has proven a fruitful and interesting approach towards duality, it is quite different from known dualities for propositional logics like Stone and Priestley dualities. As a consequence, it can be difficult to modify existing tools and techniques from other propositional bases for these dualities.
An approach towards duality for (non-distributive) meet-semilattices was developed by Hofmann, Mislove and Stralka (HMS) [27], along the same lines of the proof of the Van Kampen-Pontryagin duality for locally compact abelian groups given in [35]. This was later generalised to a duality for lattices by Jipsen and Moshier [34]. In this approach the dual space is based not on prime filters, but all (proper) filters of a lattice. This perspective is closely related to the possibility semantics of modal logic [29] and to choice-free duality for Boolean algebras [4], where again one works with the space of all proper filters. Such an approach was also developed for ortholattices by Goldblatt [19] and extended later by Bimbo [5]. We also refer to the very recent work [28] for the possibility semantics for modal ortholattices.
Our aim in this paper is to investigate not necessarily distributive positive logics. We introduce these logics and call them simply positive (modal) logics. The duality we use is a restriction of HMS duality. Our choice of duality is based on the fact that, in our view, this duality is closest to dualities used in distributive cases. This is also demonstrated by the results in this paper which show that classic results such as Sahlqvist theory can be adapted from the distributive case to a non-distributive one.
Our approach is analogous to Esakia duality for Heyting algebras. We recall that a Priestley space is a partially ordered compact space satisfying the Priestley separation axiom x y implies that there is a clopen upset U such that x ∈ U and y / ∈ U .
These spaces provide duality for distributive latices via the lattice of clopen upsets. In the HMS duality we work with similar structures, but instead of a partially ordered compact space we have a meet-semilattice with a compact topology and instead of clopen upsets we work with clopen filters. Then the HMS analogue of the Priestley separation axiom is x y implies that there is a clopen filter U such that x ∈ U and y / ∈ U .
These spaces provide duality for meet-semilatices via clopen filters. Recall that an Esakia space is a Priestley space where for every clopen upsets U and V the Heyting implication U → V is also a clopen upset (a more standard condition equivalent to the former states that ↓U is clopen for every clopen U ). In analogy with this an HMS space is a Lattice space if for every clopen filters U and V their join in the lattice of filters (the least filter containing U and V ), i.e. {x | x ≥ a ∧ b for a ∈ U and b ∈ V } is a clopen filter.
This also allows us to define a new Kripke like semantics for positive logics. The analogue of a Kripke frame is a meet-semilattice and formulas are interpreted as filters. This new semantics is a generalisation of the team semantics of [26] and of the modal information semantics of [3].
In the case of distributive lattices and Heyting algebras the lattice of all upsets is isomorphic to the canonical extension. The representation of lattices as clopen filters leads to two kinds of completions of a lattice: (1) by taking point generated upsets of the dual space of L we obtain the filter completion of L and (2) by taking all filters we obtain a new completion of L that we call the F 2 -completion. The canonical extension of L is a completion which is situated between these two, although as we notice it is not a sublattice of F 2 -completion. Our main results are preservation and correspondence results. Using a duality technique similar to that of Sambin and Vaccaro [36] we show that every sequent is preserved by filter completions and that every Sahlqvist formula is preserved by the double F 2 -completion. The former provides a purely topological proof of the result by Baker and Hales [2] that every variety of lattices is closed under ideal completions. An alternative approach to Sahlqvist correspondence and canonicity for non-distributive logics has been undertaken in [10]. But this approach is purely algebra based and is not concerned with the relational semantics and duality developed in this paper.
We extend our results to a modal extension of (not necessarily distributive) positive logic. We extend the base logic with two unary modalities, and , that are interpreted via a relation in the usual way. Interestingly, as a consequence of the non-standard interpretation of joins, while distributes over finite meets, does not distributive over finite joins but is merely monotone. A similar phenomenon in the context of modal intuitionistic logic has been exposed in [31]. The two modalities are related via similar axioms as Dunn's duality axioms for distributive positive modal logic [14].
We extend the results for positive logic to the modal setting, and obtain a duality for its algebraic semantics, and Sahlqvist correspondence and canonicity results. Using this, we obtain a sound and complete semantics for the extension of the logic that legislates to distributive over finite joins.
With this paper we hope we are laying a groundwork for the theory of non-distributive modal logics. As discussed in the conclusion, there are many interesting directions for future research, ranging from intermediate positive logics (that lie between non-distributive and distributive positive logic) to deriving more results for the modal logic presented in this paper to extending non-distributive positive logic with different types of modalities. As such, we view this paper merely as the beginning of an adventure, rather than a completed one.

Duality for Lattices
In [27] Hofmann, Mislove and Stralka proved a Stone-type duality for meet-semilattices with a top element. The dual spaces are also given by meet-semilattices with top, but equipped with a Stone topology. Variations of this duality for the categories of bounded meet-semilattices (with both top and bottom) and unbounded meet-semilattices were given in [11,9].
In this section, we recall the definition of a meet-semilattice and of several types of filters. We give the duality for them, reformulated in a way resembling Priestley duality. We then investigate the restriction to a duality for the category of lattices, similar to the restriction of Priestley duality to Esakia duality. Finally, we use this duality to describe three types of completions of a lattice.

Meet-Semilattices and Bounded Meet-Semilattices
We recall the definitions of meet-semilattices (which we shall generally refer to as simply semilattices) and bounded meet-semilattices, homomorphism, and the categories they form. We explore their connection by exposing a dual adjunction between these categories.

Definition.
A meet-semilattice is a poset (X, ≤) in which every pair of elements x and y have a greatest lower bound, denoted by x∧y. We sometimes write (X, ∧) for a meet-semilattice. The underlying poset order can then be recovered via x ≤ y if and only if x ∧ y = x. A (meetsemilattice) homomorphism from (X, ∧) to (X ′ , ∧ ′ ) is a function f : X → X ′ that satisfies f (x ∧ y) = f (x) ∧ ′ f (y) for all x, y ∈ X. We write SL for the category of meet-semilattices and homomorphisms.
Since all semilattices we work with in this paper are meet-semilattices, we will omit the prefix "meet-" and simply refer to them as semilattices. Next we equip a semilattice with an upper and lower bound.

Definition.
A semilattice (X, ∧) is called bounded if there exist elements ⊤ and ⊥ in X, called top and bottom, such that x ∧ ⊤ = x and x ∧ ⊥ = ⊥ for all x ∈ X. We denote a bounded semilattice by (X, ⊤, ⊥, ∧). A bounded semilattice homomorphism is a top-and bottom-preserving semilattice homomorphism. We write BSL for the category of bounded semilattices and their homomorphisms.
We sometimes identify a (bounded) semilattice with its state-space, and refer to X instead of the tuple (X, ⊤, ⊥, ∧). It will then be clear from context whether we are dealing with a semilattice or a bounded one.
Let (X, ∧) be a semilattice and a ⊆ X. Then we define the upward closure of a by ↑a := {y ∈ X | x ≤ y for some x ∈ a} and we say that a is upward closed or and upset if ↑a = a. If a = {x} is a singleton we write ↑x instead of ↑{x}. We define similarly the downward closure and downsets.

Definition.
A filter in a semilattice (X, ∧) is an upset a ⊆ X that is closed under ∧. It is called proper if a = X. A filter is called principal if it is empty, or of the form ↑x for some x ∈ X.
2.4 Remark. Observe that we view the empty filter as a principal filter as well. This will streamline notation when discussing frame semantics in Section 3. ⊳ The collection of filters of (X, ∧) forms a bounded semilattice with intersection as meet and X and ∅ as top and bottom elements. We denote this bounded semilattices by F(X, ∧). Furthermore, if f : (X, ∧) → (X ′ , ∧ ′ ) is a semilattice homomorphism, then f −1 : F(X ′ , ∧ ′ ) → F(X, ∧) is a bounded semilattice homomorphism, and setting Ff = f −1 yields a contravariant functor F : SL → BSL.
Conversely, if (A, ⊤, ⊥, ∧) is a bounded semilattice then the collection of proper non-empty filters with intersection forms a semilattice, denoted by F b A. Again, defining F b f = f −1 for bounded homomorphisms yields a contravariant functor F b : BSL → SL.
2.5 Proposition. The functors F and F b establish a dual adjunction between SL and BSL.
Proof. Define the units η : It is easy to see that both η X (x) and θ X (x) are filters. Moreover, θ X (x) is proper and non-empty because ∅ / ∈ θ X (x) and X ∈ θ X (x) for all x. A routine verification shows that both η and θ define natural transformations. In order to show that F and F b form a dual adjunction it suffices to verify that they satisfy the following triangle equalities: We verify the left one, the right being similar. Let X be a semilattice and a ∈ FX. Then we have: so θ FX (Fη X (a)) = a. This completes the proof.
2.6 Remark. Observe that filters of a semilattice correspond bijectively to homomorphisms into 2 = {⊤, ⊥}, the two-element chain. If a is a filter in (X, ∧) then the characteristic map χ a : X → 2 given by χ a (x) = ⊤ iff x ∈ a is a homomorphism, and for every such homomorphism the preimage of ⊤ is a filter. Similarly, proper non-empty filters of a bounded semilattice correspond to bounded homomorphisms into 2. So the dual adjunction between SL and BSL is given by the dualising object 2. ⊳ If (X, ∧) is a semilattice that happens to have all finite joins (including a bottom element), then we have a second way of obtaining a bounded semilattice. Namely, the principal filters of (X, ∧) form a sublattice of F(X, ∧). The empty filter is principal by convention and X is principal because it is of the form ↑⊥, where ⊥ is the bottom element of (X, ∧). Furthermore, if p = ↑x and q = ↑y are principal filters, then their intersection is principal as well, because p ∩ q = ↑(x ∨ y).
This observation will help us elucidate the connection between the duality for lattices (whose dual spaces have finite joins) and the filter extension of a lattice (in Section 2.4). Moreover, we will show that using principal filters are interpretants for positive formulae yields stronger canonicity results (albeit with a more restrictive semantics) in Section 4.5.

Duality for Bounded Meet-Semilattices
We now work our way towards a duality for bounded semilattices. We do so by taking the dual adjunction between semilattices and bounded semilattices, and equipping semilattices with a Stone topology. This duality has appeared before in [11,Sections 2.4] in a more abstract desguise and is a variation of the duality given by Hofmann, Mislove and Stralka in Chapter I of [27].
(M 3 ) (X, ∧, ⊤) satisfies the HMS separation axiom: for all x, y ∈ X, if x ≤ y then there exists a clopen filter a such that x ∈ a and y / ∈ a.
An M-morphism is a continuous semilattice homomorphism. We write MSpace for the category of M-spaces and M-morphisms.
Condition (M 3 ) is a variation of the Priestley separation axiom. It immediately implies that any M-space is Hausdorff. Furthermore, it can be shown that every M-space is zerodimensional in the same way as for Priestley spaces. For future reference, we derive some other useful properties of M-spaces. Proof. If c is the empty filter then it is principal by convention, and the empty principal filter is automatically closed because any empty set in a topological space is closed.
So let c be a non-empty closed filter and suppose towards a contradiction that it is not principal. Then for each x ∈ c there exists some y ∈ c such that x ≤ y. Therefore, using (M 3 ), for each x ∈ c we can find a clopen filter containing x such that c ⊆ b x . Then c ⊆ x∈c b x is an open cover of c, and by compactness we can find a finite subcover, say c ⊆ b x1 ∪ · · · ∪ b xn . By construction, for each b xi we can find a y i ∈ a such that y i / ∈ b xi . But this implies y 1 ∧· · ·∧y n ∈ c because c is a filter, while y 1 ∧ · · · ∧ y n / ∈ b xi for all 1 ≤ i ≤ n, a contradiction. Next, suppose c is non-empty and principal, i.e. c = ↑x for some x ∈ X. If y / ∈ c then x ≤ y, so there exists a clopen filter a containing x and not containing y. But then c = ↑x ⊆ a, so X \ a is an open neighbourhood of y disjoint from c. Therefore c is closed.
2.9 Lemma. Every closed filter of an M-space X = (X, ∧, τ ) is the intersection of clopen filters.
Proof. If c is the empty filter then the statement is trivial. If c is non-empty, then by Lemma 2.8 it is of the form ↑x for some x ∈ X. Suppose y / ∈ c, then x ≤ y so we can find a clopen filter a containing x but not y. Since x ∈ a we have c ⊆ a. Since y was chosen arbitrarily, the claim follows.
2.10 Lemma. Let c be a closed subset of an M-space X = (X, ∧, τ ). Then ↑c is closed as well.
Proof. If y / ∈ ↑c then for each x ∈ c we have x ≤ y, hence a clopen filter a x containing x but not y. Then c ⊆ x∈c a x , so by compactness we find a finite subcover, say, c ⊆ a 1 ∪ · · · ∪ a n . Since all the a i are upward closed, we have ↑c ⊆ a 1 ∪ · · · ∪ a n . By construction, none of the a i contain y, so X \ (a 1 ∪ · · · ∪ a n ) is an open neighbourhood of y disjoint from ↑c.
We claim that every bounded semilattice gives rise to an M-space. We now denote a bounded semilattice by A, since we think of them as (a variety of) algebras.
2.11 Proposition. Let A be a bounded semilattice and write F b A for the set of non-empty proper filters of X. Generate a topology Proof. We already know that (F b A, ∩) is a semilattice. Condition (M 3 ) follows easily: If x, y ∈ F b A are such that x ⊆ y then there is an element a ∈ A such that a ∈ x and a / ∈ y, and θ A (a) is a clopen filter separating x and y.
To show that (F b A, τ A ) is compact, it suffices to show that every open cover of F b A of subsets in the subbase (1) has a finite subcover. So suppose where I and J are index sets and a i , b j ∈ A. Consider the filter F = ↑{b j1 ∧ · · · ∧ b jn | n ∈ ω and j 1 , . . . , j n ∈ J}. Note that F is nonempty because the case n = 0 entails that the empty meet (which is ⊤) is in F . If F is not proper then we must have ⊥ ∈ F , so there must exist j 1 , . . . , j n ∈ J such that b j1 ∧ · · · ∧ b jn = ⊥. Since filters are closed under meets, no proper filter can contain all of b j1 , . . . , b jn , and hence is a finite subcover of the one in (2). Indeed, any filter that is in none of θ A (b j1 ) c , . . . , θ A (b jn ) c must contain b j1 , . . . , b jn , and hence also a.
If A is a bounded semilattice, then we denote the M-space constructed in Proposition 2.11 by F top A. This assignment extends to a contravariant functor Conversely, if X = (X, ∧, τ ) is an M-space, then we denote by F clp X the collection of clopen filters. This forms a bounded semilattice with meet given by intersection, and X and ∅ as top and bottom. It gives rise to a contravariant functor 2.12 Theorem. The functors F clp and F top establish a dual equivalence MSpace ≡ op BSL.
Proof. We prove that η : id MSpace → F top F clp and θ : id BSL → F clp F top , defined on components by η X (x) = {a ∈ F clp X | x ∈ a} and θ A (a) = {x ∈ F top A | a ∈ x}, are natural isomorphisms.
Let us start with the former. Naturality is routine again, so we focus on showing that for each X ∈ MSpace, η X is an isomorphism. So let X be an arbitrary M-space. If x, y ∈ X and x = y then either x ≤ y or y ≤ x, so by (M 3 ) we find η X (x) = η X (y), and hence η X is injective. To see that η X is surjective, let F ∈ F top F clp X. Then F is a closed filter of X, because it is the intersection of clopen filters, so by Lemma 2.8 it is of the form ↑x for some x ∈ X. We claim that F = η X (x). By construction of x we have x ∈ a for all a ∈ F , so F ⊆ η X (x). Now if a is any clopen filter with x ∈ a, then we have F = ↑x ⊆ a and by a straightforward compactness argument we find b 1 , . . . , b n ∈ F such that b 1 ∩ · · · ∩ b n ⊆ a. Since F is a filter we have b 1 ∩ · · · ∩ b n ∈ F , and hence a ∈ F . Finally, η X preserves meets because and it is continuous because for each a ∈ F clp X we have Since any bijective homomorphism preserves and reflects meets, and a bijective continuous function between Stone spaces is a homeomorphism, we find that η X is an isomorphism in MSpace.
Next we establish that θ is an isomorphism on components, again leaving the verification of naturality to the reader. Let A be a bounded semilattice. It is easy to see that θ A is well defined. If a = b, then without loss of generality we may assume a ≤ b. We find that ↑a ∈ F top A is a filter such that ↑a ∈ θ A (a) while ↑a / ∈ θ A (b). Therefore θ A is injective. Preservation of meets is similar to preservation of meets for η X . Moreover, θ A (⊤) = F top A (the top element of F clp F top A) and θ A (⊥) = ∅ (the bottom element of F clp F top A) because the filters in F top A are non-empty and proper, so they all contain ⊤ and none contains ⊥. For surjectivity, let F ∈ F clp F top A be any clopen filter. Then using an argument similar to the proof of Lemma 2.9, Compactness yields a finite number a 1 , . . . , a n ∈ A such that F = θ A (a 1 ) ∩ · · · ∩ θ A (a n ). But then F = θ A (a 1 ∧ · · · ∧ a n ), so θ A is indeed surjective. Thus we have found that θ A is a bijective BSL-morphism, and since BSL is a variety of algebras this implies that it is an isomorphism.
2.13 Corollary. Let X be an M-space. Then X has arbitrary non-empty meets. In particular, X has a bottom element.
Proof. By Theorem 2.12 X is isomorphic to (F top A, ∩, τ A ) for some bounded semilattice A.
Since the arbitrary intersection of filters in F top A is a filter again, it follows that X has all non-empty meets. In particular, the meet of all elements in X is a bottom element.

Restriction to Lattices
Our next goal is to restrict the duality for bounded semilattices to a duality for lattices. By a lattice we will always mean a bounded lattice, i.e. a bounded semilattice with binary joins. The restriction of HMS duality to lattices was first studied by Jipsen and Moshier [34]. Here we follow a different version first explored in [13], since we find the interpretation of join that we discuss below simpler than in [34]. We also point out that [34] formulate their duality in terms of spectral spaces whereas we prefer to work with M-spaces.
A different restriction of HMS duality was used in [22], where it was restricted to a duality for implicative semilattices and used to study modal extensions of the meet-implication fragment of intuitionistic logic.
We start by making the observation that, if (X, ∧) is a semilattice, then F(X, ∧) is not only a bounded semilattice, but it is also complete. Indeed, the arbitrary intersection of filters forms a filter again. Therefore, in particular, it is a complete lattice, with joins defined as the meet of all upper bounds. That is, if F ⊆ F(X, ∧) is a collection of filters, then F = {a ∈ F(X, ∧) | b ⊆ a for all b ∈ B}. We can characterise joins of non-empty filters as follows.
2.14 Lemma. Let (X, ∧) be a semilattice. The join of a non-empty set F ⊆ F(X, ∧) of non-empty filters of (X, ∧) can be defined as Proof. By definition the right hand side is the smallest filter containing all filters in F .
In particular, if F only contains the empty filter then we find that F = ↑∅ = ∅. We abbreviate {a, b} by a b. Note that a b = ↑{x ∧ y | x ∈ a, y ∈ b} (provided a and b are not the empty filter), and this is simply the smallest filter containing a and b. Moreover, for any set F of filters we have F = {a 1 · · · a n | a 1 , . . . , a n ∈ F }.
If the semilattice F(X, ∧) happens to be distributive then we may omit the upward closure ↑, but we shall not be in this situation in general. While this yields a restriction of the functor F : SL → BSL to SL → Lat on objects, it does not yet work for morphisms. Indeed, we need extra conditions on the morphisms between semilattices to ensure that their inverses preserve joins.
The category LFrm will be used in Section 3 to interpret non-distributive positive logic. We verify that the inverse of an L-morphisms is indeed a lattice homomorphism.
Proof. We already know that f −1 is a bounded semilattice homomorphism, so we only have to show that it preserves binary joins. That is, we show that for arbitrary filters If a ′ or b ′ is the empty filter, then the equality is trivial, so suppose both are non-empty. The inclusion ⊇ follows from the fact that f −1 (a ′ b ′ ) is a filter that contains both f −1 (a ′ ) and . By definition of L-frame morphism, we can find y, z ∈ X such that y ′ ≤ f (y) and z ′ ≤ f (z) and y ∧ z ≤ x. This means that y ∈ f −1 (a ′ ) and z ∈ f −1 (b ′ ), and So F restricts to a contravariant functor F : LFrm → Lat. Interestingly, the converse holds as well; we can restrict F b to a contravariant functor Lat → LFrm.
2.17 Proposition. Let h : L → L ′ be a lattice homomorphism. Then h −1 : Proof. We already know that h −1 is a semilattice homomorphism, so we only have to show that it satisfies the additional condition from Definition 2.15. Let p ′ ∈ FL ′ and q, r ∈ FL and suppose q ∩ r ⊆ h −1 (p ′ ). Let q ′ := ↑h[q] and r ′ := ↑h[r]. Then it is easy to verify that q ′ and r ′ are filters (because q and r are), and by construction q ⊆ h −1 (q ′ ) and r ⊆ h −1 (r ′ ). It remains to show that q ′ ∩ r ′ ⊆ p ′ . Let a ′ ∈ L ′ be such that a ′ ∈ q ′ ∩ r ′ . Since a ′ ∈ q ′ there exists a ∈ q such that h(a) ≤ a ′ . Since a ′ ∈ r ′ there exists b ∈ r such that h(b) ≤ a ′ . But then a ∨ b ∈ q ∩ r, so by assumption h(a ∨ b) ∈ p ′ . This implies a ′ ∈ p ′ , because h(a ∨ b) = h(a) ∨ h(b) ≤ a ′ and p ′ is a filter (hence up-closed).
The relation between the categories SL and LFrm is similar to that between the categories Pos, of posets and order-preserving functions, and IntKrip, of posets and bounded morphisms. (The latter category is called Krip because the posets in it are often referred to as intuitionistic Kripke frames.) See also figure 1.

SL BSL
LFrm Lat The functor up takes a poset to its lattice of upsets, and pf takes a lattice to its ordered set of prime filters.
We now define the spaces and morphisms that will give a duality for lattices. They are a variation on the "PUP spaces" from [13].
2.18 Definition. A lattice space, or L-space for short, is an M-space X = (X, ∧, τ ) such that a b is a clopen filter whenever a and b are clopen filters. An L-space morphism is an M-space morphism that is simultaneously an L-morphism. We write LSpace for the category of L-spaces and L-space morphisms.
We prove the following lemma for future reference. x ∈ b we can find a clopen filter a such that x ∈ a ⊆ b. As a consequence b = x∈b a x , and since b is the smallest filter containing all of the a x we have b = x∈b a x . Conversely, suppose b = i∈I a i , where each a i is a clopen filter. Then b is a filter by definition. As a consequence of (3) we have b = {a i1 · · · a in | n ∈ ω and i 1 , . . . , i n ∈ I}.
Since X is an L-space this is the union of clopen sets, hence it is open.
2.20 Theorem. The duality for bounded semilattices from Theorem 2.12 restricts to a duality LSpace ≡ op Lat.
Proof. We only have to verify that the restriction of F clp to LSpace lands in Lat, and the restriction of F top to Lat lands in LSpace. The former follows from the fact that the clopen filters of an L-space are closed under , together with Proposition 2.16.
For the latter, suppose that L is a lattice and let θ L (a) and θ L (b) be two arbitrary clopen filters of F top L. Writing x, y, z for elements in F top L, we have Let us elaborate on the last equality: If x ∈ θ L (a) and y ∈ θ L (b) then a ∈ x and b ∈ y, so a∨b ∈ x∩y. So z ⊇ x∩y implies a∨b ∈ z, and therefore we have "⊆". Conversely, if z ∈ θ L (a∨b) then we need to find x ∈ θ L (a) and y ∈ θ L (b) such that x ∩ y ⊆ z. Let x = ↑a ∈ θ L (a) and this implies d ∈ z, and therefore x ∩ y ⊆ z. This proves "⊇". The restriction on morphisms follows from Proposition 2.17.
An overview of the dualities for semilattices and lattices and the analogy with intuitionistic logic is given in Figure 2, in Section 3.3.

Lattice Completions
In the final subsection of this section, we investigate how L-spaces give rise to several types of lattice completions. First we show how the well-known filter completion and canonical extension can be obtained as lattices of suitable filters of an L-space. Afterward, we discuss a third extension which arises from L-spaces in a natural way. To the best of our knowledge, this extension has not appeared in the literature before. The following diagram gives an overview of the situation.   Let X = (X, ∧, τ ) be an M-space. Let F k X denote the collection of closed filter of X. (Recall from Lemma 2.8 that these are precisely the principal filters.) Since the arbitrary intersection of a collection of closed filters is again a closed filter, F k X forms a complete semilattice with meet . The top and bottom element are given by X and ∅, respectively. The join can then be defined as the smallest upper bound in the inclusion order. Specifically, the join of two closed filters c 1 and c 2 is given by c 1 c 2 . Writing c i = ↑x i , this is equal to c 1 c 2 = ↑(x 1 ∧ x 2 ). Interestingly, the join of an arbitrary collection F ⊆ F k X is not necessarily given by F , because this need not be closed. Rather, the join in F k X, denoted by , is given by if the join does not include the empty set. The join of a set {∅} ∪ {↑x i | i ∈ I} that does contain the empty set is given by where we presume the meet on the right hand side to give the empty set if I = ∅. Note that we make use of the fact that the semilattice underlying an M-space has all non-empty meets.
2. 23 Theorem. The filter completion of a lattice is isomorphic to the complete lattice of closed filters of its dual L-space. That is, for any lattice L, we have Clearly this is well defined, since ξ(p) is a principal filter for each p ∈ F ∂ L, hence a closed filter. Suppose p = q are two filters in F ∂ L. Without loss of generality assume p ⊆ q. Then q / ∈ ξ(p) and q ∈ ξ(q), so ξ(p) = ξ(q) and therefore ξ is injective.
To see that ξ is surjective, suppose given a closed filter c ∈ F k F top L. If c is non-empty, then by Lemma 2.8 it is of the form ↑x for some x ∈ F top L. But F top L contains all non-empty proper filters, so x is also in F ∂ L. It now follows from the definition that c = ξ(x). If c is empty then c = ∅ = ξ(L). Indeed, ξ(L) is empty because no filter in F top L contains the bottom element of L, hence none of them are supersets of L. This proves surjectivity.
Next we claim that ξ is a complete lattice homomorphism. It preserves the top element because ξ({⊤ L }) = F top L, and the bottom element since ξ(L) = ∅. To see that ξ preserves all meets, recall that the meet of elements in F ∂ L is given by . For all F ⊆ F ∂ L such that L / ∈ F we have Finally, note it follows immediately from (5) and (6) that so ξ also preserves all joins. It follows that ξ is a complete lattice homomorphism.
The next completion we investigate is the well-known canonical extension. A canonical extension of L is a completion that is dense and compact.
It is known that every lattice has a canonical extension [15,Proposition 2.6], and that any two canonical extension of a lattice L are isomorphic by a unique isomorphism that commutes with the embeddings of L [15, Proposition 2.7]. So we can talk of the canonical extension of a lattice. We investigate how to obtain a description of the canonical extension of a lattice L using its dual L-space. This is similar to the topological description of canonical extensions found in [34].
2.25 Definition. Let X = (X, ∧, τ ) be an M-space. By a saturated filter we mean a filter b on X that is the intersection of all open filters that contain it. We write F sat X for the collection of saturated filters. Clearly, the arbitrary intersection of saturated filters is again saturated, so F sat X forms a complete semilattice with top X and bottom ∅.
Note that every open filter is saturated. Every closed filters is the intersection of all clopen filters containing it, so closed filters are saturated as well. Since F sat X is a complete semilattice it is also a complete lattice, and the join of a set F of saturated filters is given by

entirely of open filters, then
F is open again, hence saturated, so joins of open filters in F sat X are computed using . We now prove that the saturated filters can be used to describe the canonical extension of a lattice. This is the L-space counterpart of the characterisation of the canonical extension using spectral spaces, as given in [34,Theorem 4 We first show that the completion is dense. By definition, every element in F sat F top L is the intersection of open elements. Since ↑x is closed for each x ∈ F top L, it follows that each filter d in F sat F top L is the union of all closed elements it contains. This automatically implies that d is the smallest saturated filter containing the set of closed filters contained in d, hence d is the join of closed elements. Therefore (θ L , F sat F top L) is dense.
Next we show that F sat F top L is compact. By [15,Lemma 2.4] it suffices to show that for any (Note that we can use here because we are taking the join of opens.) The implication from right to left is obvious. So suppose Since the join of opens is computed as usual, we can rewrite the right hand side to obtain an open cover Using a similar argument as in the proof of Lemma 2.9, we find a single set on the right hand side containing the left hand side. This means that there exists a finite Our final completion is defined in terms of L-spaces, rather than lattices.
2.27 Definition. The double filter completion or F 2 -completion of a lattice L is the lattice FF b L with embedding θ. In other words, it is the complete lattice of all filters of an L-space (not just the clopen or closed or saturated ones).
An immediate question is whether we can characterise the double filter completion algebraically. We leave this as an interesting direction for further research.

Non-Distributive Positive Logic
We use the duality and dual adjunction from Section 2 to give a Kripke-style semantics for nondistributive positive logic. Inspired by the fact that the filters of a semilattice form a lattice, we use semilattices as frames and (principal) filters as valuations. We use these to study (non necessarily distributive) positive logic.
We start this section by giving an axiomatisation of our logic. By design the algebraic semantics is simply given by lattices. Then, in Section 3.2 we formally define frames and models, and give several (classes of) examples. We prove that the frame semantics is sound. In Section 3.3 we use the duality from Section 2 to derive completeness of the basic logic with respect to several classes of frames. We give the standard translation into a suitable firstorder logic and prove Sahlqvist correspondence in Section 3.4, where we also work out specific examples of correspondence results. Finally, we prove Sahlqvist canonicity in Section 3.5. This gives rise to completeness results and a new proof of Baker and Hales' theorem which states that every variety of lattices is closed under filter extensions. Moreover, it follows that every variety of lattices is also closed under F 2 -extensions.
To distinguish the various notions of entailment each has their own notation, which are summarised in Table 1. We denote the interpretation of a formula ϕ in a lattice and in a frame M by ϕ and ϕ M , respectively.

Notation
Purpose Location

Logic and Algebraic Semantics
Let L(Prop) denote the language of positive propositional logic. That is, L(Prop) is generated by the grammar ϕ : where p ranges over some arbitrary but fixed set Prop of proposition letters. If no confusion arises we will omit reference to Prop and simply write L.
For lack of a strong enough implication, we define the minimal logic L based on L as a collection of consequence pairs. This is based on Dunn's axiomatisation of positive modal logic [14], leaving out the modal axioms and distributivity. Formally, a consequence pair is simply an expression of the form ϕ ψ, where ϕ and ψ are formulae in L. The intuitive reading of ϕ ψ is: "If ϕ holds, then so does ψ." 3.1 Definition. Let L be the smallest set of consequence pairs closed under the following axioms and rules: top and bottom ϕ ⊤, ⊥ ϕ, reflexivity and transitivity ϕ ϕ, ϕ ψ ψ χ ϕ χ , the conjunction rules and the disjunction rules If Γ is a set of consequence pairs then we let L(Γ) denote the smallest set of consequence pairs closed under the axioms and rules mentioned above and those in Γ. We write ϕ ⊢ Γ ψ if ϕ ψ ∈ L(Γ) and ϕ ⊣⊢ Γ ψ if both ϕ ⊢ Γ ψ and ψ ⊢ Γ ϕ. If Γ is the empty set then we simply write ϕ ⊢ ψ and ϕ ⊣⊢ ψ.
The algebraic semantics of the logic L are simply lattices. We establish this formally.

Definition. Let
A be a lattice with operations ⊤ A , ⊥ A , ∧ A , ∨ A , and induced order ≤ A . A lattice model is a pair = (A, σ) consisting of a lattice A and an assignment σ : Prop → A of the proposition letters. We define the interpretation ϕ of an L-formula ϕ in recursively via We say that a lattice A validates a consequence pair ϕ ψ if ϕ ≤ A ψ for all lattice models based on A, notation: A ϕ ψ. If Γ is a set of consequence pairs then we write Lat(Γ) for the full subcategory of Lat whose objects validate all consequence pairs in Γ.
Observe that ⊣⊢ Γ is an equivalence relation on L. Write L(Γ) for the set of ⊣⊢ Γ -equivalence classes of L, and denote by [ϕ] the equivalence class of ϕ in L(Γ). Then it follows from the rules in Proof. The direction from left to right holds by definition. Conversely, if A ∈ Lat(Γ) and = (A, σ A ) is a lattice model, then the assignment [p] → σ A (p) extends to a lattice homomorphism i : Proof. As a consequence of Lemma 3.4 it suffices to show that ϕ ⊢ Γ ψ if and only if ϕ ℒΓ ≤ L ψ ℒΓ . It follows from the disjunction rules, identity and transitivity that ϕ

Frame Semantics
As we have seen, the collection of filters of a semilattice forms a lattice. Therefore we can use semilattices as a generalisation of (intuitionistic) Kripke frames to interpret non-distributive positive logic. As interpretants of the formulae we choose either all filters of the semilattice, or the principal filters.
3.6 Definition. A lattice Kripke frame, or L-frame for short, is a pair (X, ≤) where X is a set and ≤ is a partial order on X such that each pair of elements x, y ∈ X has a greatest lower bound, denoted by x ∧ y. A valuation for an L-frame (X, ≤) is a function V : Prop → F(X, ≤) which assigns to each proposition letter a filter of (X, ≤). An L-frame together with a valuation is called an L-model. The interpretation of an L-formula ϕ at a state x in an L-model M = (X, ≤, V ) is defined recursively via If the frame is fixed and we want to emphasise the role of the valuation in the interpretation, we will write If Γ is a set of consequence pairs, then we let LFrm(Γ) denote the full subcategory of LFrm whose objects validate all consequence pairs in Γ. We write ϕ Γ ψ if X ϕ ψ for all X ∈ LFrm(Γ). If Γ = ∅ then we write ϕ ψ instead of ϕ ∅ ψ.
Indeed, L-frames are simply semilattices. When used as frame semantics, we usually write them as a set with a partial order, rather than a set with a conjunction, to stress that they can be viewed as relational structures.
For any L-frame F = (X, ≤), the collection F * := F(X, ≤) forms a lattice, called the complex algebra of F. Since valuations of F correspond bijectively to assignments of F * , we can define the complex algebra of an L-model M = (X, ≤, V ) by M * = (F(X, ≤), V ). A routine induction on the structure of ϕ then proves the following lemma.
As an immediate corollary we obtain the following persistence result. This is similar to persistence of intuitionistic formulae in intuitionistic Kripke frames, except for the fact that we require formulae to be interpreted as filters rather than upsets.

Proposition
Proof. Immediate consequence of Lemma 3.7.

Theorem
Since M was chosen arbitrarily, this proves ϕ Γ ψ.
3.10 Example. We list some examples of (classes of) L-frames.
1. Any linearly ordered set is an L-frame. Filters in such frames are simply upsets. For example, N, Z and R with the natural ordering are all L-frames.
2. Any Scott domain is a semilattice, hence an L-frame.
3. If X is a set, then the collections P + X, P ω X and P + ω X of non-empty, finite, and finite non-empty subsets of X, respectively, form semilattices. In each case, the meet given by set-theoretic union. The latter is the free semilattice over X, and filters of P + ω X correspond bijectively with subsets of X. ⊳ Before discussing more examples, we identify an important subclass of L-frames and Lmodels. In some cases, we can interpret every formula as a principal filter. (Recall that we legislated the empty filter to be principal as well.) These are significant because, as we have seen in Section 2.4, the principal filters of an L-space X give rise to the filter extension of the lattice dual to X.

3.11
Definition. An L-frame (X, ≤) is called principal if it has all finite joins, including a bottom element which is denoted by 0. A principal valuation V for an L-frame (X, ≤) is a valuation such that V (p) is a principal filter of (X, ≤) for each proposition letter p. A principal L-model is a principal L-frame together with a principal valuation.
We have the following persistence results.
3.12 Proposition. Let M = (X, ≤, V ) be an L-model. Then ϕ M a principal filter for every ϕ ∈ L.
Proof. It follows from Lemma 3.7 that ϕ M is a filter for each formula ϕ, so we only have to show that it is principal. For the base cases this is obvious: V (p) is principal by definition and ⊥ M = ∅ and ⊤ M = ↑0. For the step cases it suffices to observe that if ϕ 1 M = ↑x 1 and . In other words, the collection of principal filters of a closed L-model is closed under ∩ and .
We give some examples of classes of principal L-models. Since very closed L-model is in particular an L-model, this simultaneously extends our collection of examples of L-models.
3.13 Example. The L-frame (N, ≤) from Example 3.10(1) is principal but (Z, ≤) and (R, ≤) are not, because do not have minimal elements. Every filter of (N, ≤) is principal, so every L-model based on (N, ≤) is principal. An example of a non-principal filter of (R, 3.14 Example. Another interesting class of examples of principal L-frames is given by rooted trees of finite depth. Filters in such frames are always principal, and a valuation indicates that a property p is true at a node x and henceforth. As usual, a node satisfies ϕ ∧ ψ if it satisfies both ϕ and ψ. In such structures, a node x satisfies ϕ ∨ ψ if it has two cousins y and z whose "youngest" common ancestor is also an ancestor of x, such that y satisfies ϕ and z satisfies ψ. ⊳ 3.15 Example. We briefly recall a simplified version of team semantics for propositional logics. This underlies many versions of modal dependence and independence logics, such as the ones studied in [25,32,38,39]. Consider the language T(Prop) given by ϕ : These can be interpreted in models consisting of a set X and a valuation Π : Prop → PX of the proposition letters. However, rather than assigning truth of formulae to elements of X, truth is defined for subsets of X (the teams). Let M = (X, Π) be such a model and T ⊆ X a team, then we let We can add ⊤ and ⊥ by defining them to be always true and always false, respectively.
Interestingly, the interpretation looks a lot like that in Definition 3.6. Let us make this precise. For a set Prop of proposition letters, let ¬Prop = {¬p | p ∈ Prop}. Then, given a team model M = (X, Π), we can define a principal L-model Then the meet of M ′ is given by set-theoretic union, and it is easy to see that V is a principal valuation. Moreover, L(Prop ∪ ¬Prop) is simply the extension of T(Prop) with a top and bottom element, and for each team model M, team T , and formula ϕ ∈ L(Prop ∪ ¬Prop) we have We investigate the relation between non-distributive positive logic interpreted in L-frames and modal information logic [3]. Modal information logic is the extension of propositional classical logic with two binary modal operators inf and sup . These are interpreted in Kripke models M = (X, R, V ) where R is a pre-order on X as follows: ) and M, y ϕ and M, z ψ Note that we need not require that every pair of states has an infimum and a supremum, nor that it is unique. The definition simply uses the fact that they might exist. Observe that we can recover the usual modal diamond via ϕ = inf (ϕ, ⊤). Moreover, we can define a temporal diamond as ϕ = sup (ϕ, ⊤). Clearly, every L-model is a model for modal information logic. Interestingly, the interpretation of inf is closely aligned to our interpretation of joins; the only difference is that the infimum is allowed to be below the state under consideration. Taking this into account, our interpretation of joins in an L-model M = (X, ≤, V ) coincides with where ∨ is the non-classical join. ⊳ 3.17 Remark. The partial translation of L into modal information logic may give rise to an analogue of the Gödel-McKinsey-Tarski translation [17,33]. The role of can possibly be replaced by inf (−, −), as this ensures that the interpretation of a formula is a filter. We flag this as an interesting direction for future research. ⊳ We write LMod for the category of L-models and L-model morphisms, and cLMod for its full subcategory of principal L-models.
As desired, L-model morphisms preserve and reflect truth of L-formulae. Proof. Routine induction on the structure of ϕ.

Descriptive Frames and Completeness
We have already seen a duality for lattices, by means of L-spaces. L-spaces can be viewed as topologised L-frames. In fact, since every L-space X is of the form F top A for some lattice A, it follows that X has a bottom element (given by {⊤ A }) and binary joins (given by ), so that we may also view them as topologised principal L-frames. In this subsection we define how to interpret L-formulae in L-spaces, and show how this gives rise to completeness for L. We denote L-spaces and L-spaces with a valuation by X and M. Their non-topologised counterparts are L-frames and L-models, and are denoted by X and M. If X is an L-space, then we write κX for its underlying (principal) L-frame.
3.20 Definition. A clopen valuation for an L-space X is an assignment V : Prop → F clp X, which assigns to each proposition letter a clopen filter of X. We call a pair M = (X, V ) of an L-space and a clopen valuation an L-space model.
The interpretation ϕ M of an L-formula ϕ in an L-space model M = (X, V ) is defined as in the underlying L-model (κX, V ). The L-space model M validates a consequence pair ϕ ψ if ϕ M ⊆ ψ M , notation: M ϕ ψ. We say that an L-space X validates ϕ ψ if every L-space model based on it validates ϕ ψ. Finally, we write ϕ LSpace ψ if every L-space validates ϕ ψ.
3.21 Lemma. Let X be an L-space, A its dual lattice, and ϕ, ψ ∈ L. Then Proof. The first item follows from the fact that clopen valuations of X correspond bijectively to assignments of the proposition letters for A, together with a routine induction on the structure of ϕ. The second item follows immediately from the first.
We can now prove completeness of L with respect to several classes of frames. 3. all L-frames.
Proof. Soundness follows from Theorem 3.9. For completeness, we show that ϕ ⊢ ψ implies ϕ ψ. So suppose ϕ ⊢ ψ, then by Theorem 3.5 we can find a lattice A that does not validate ϕ ψ. As a consequence of Lemma 3.21 the L-space X dual to A does not validate ϕ ψ, so there must exist a clopen valuation V such that (X, V ) ϕ ψ. But this implies that (κX, V ) ϕ ψ. Since κX is a (principal) L-frame and every clopen filter of an L-space is principal (due to Lemma 2.8) we have found a (principal) L-model not validating ϕ ψ.
It is now natural to wonder weather we can prove similar theorems for extensions of L with a set Γ of consequence pairs. We will answer this question positively in Section 3.5.
3.23 Remark. We can alternatively describe L-spaces as descriptive L-frames. That is, as Lframes with extra structure, similar to descriptive intuitionistic Kripke frames. Since we prefer to work with L-spaces, we only briefly sketch this perspective.
A general L-frame is a tuple (X, ≤, A) such that (X, ≤) is an L-frame and A is a collection of filters of (X, ≤) containing X and ∅, and closed under ∩ and . The sets in A are called admissible filters.
• refined if for all x, y ∈ X such that x ≤ y there exists an a ∈ A such that x ∈ a and y / ∈ a; • compact if C = ∅ for each C ⊆ A ∪ −A with the finite intersection property; • descriptive if it is refined and compact.
Write D-LFrm for the category of descriptive L-frames and general L-morphisms. Then we have Proof sketch. We sketch the isomorphism on objects; the correspondence on morphisms is obvious. Let (X, ≤, A) be a descriptive L-frame, write ∧ for the conjunction induced by ≤ and let τ A be the topology on X generated by A ∪ −A. Then it follows immediately from the definition of a descriptive L-frame that (X, ∧, τ A ) is an L-space. Conversely, an L-space X = (X, ∧, τ ) gives rise to the descriptive L-frame (X, ≤, F clp X).
In order to show that these assignments define a bijection we need to show that for any descriptive L-frame (X, ≤, A), the clopen filters of (X, ∧, τ A ) are exactly the filters in A. This is easy: if b is any clopen filter, then in particular it is closed, hence by the intersection of all filters in a that contain it (by refinedness). But then compactness entails that b is equal to one of the clopen filters in A containing it.
Continuing the analogy with intuitionistic logic, descriptive L-frames relate to L-spaces in the same way intuitionistic Kripke frames relate to Esakia spaces. The various dualities are depicted in Figure 2, together with the analogous diagram from intuitionistic logic.

The First-Order Translation and Sahlqvist Correspondence
In this section we define the standard translation of L into a suitable first-order logic. We use this to derive a Sahlqvist correspondence result. We prove that for every consequence pair ψ χ, the collection of L-frames validate ψ χ are first-order definable. Our proof of the correspondence result follows a standard proof from normal modal logic, such as found in [6, Section 3.6]. Thus, it showcases how our duality for lattices allows us to transfer classical techniques to the positive non-distributive setting. However, it is complicated (or rather, made more interesting) by the non-standard interpretation of disjunctions.
3.24 Definition. Let FOL be the single-sorted first-order language which has a unary predicate P p for every proposition letter p, and a binary relation symbol R.
Intuitively, the relation symbol of our first-order language accounts of the poset structure of L-frames. It is used in the translation of disjunctions. If x, y and z are first-order variables, then we can express that x is above every lower bound of y and z in the ordering induced by the relation symbol R using a first-order sentence. In order to streamline notation we abbreviate this as follows: abovemeet(x; y, z) := ∀w((wRy ∧ wRz) → wRx). If x, y 1 , . . . , y n is a finite set of variables and n ≥ 1 then we define abovemeet(x; y 1 , . . . , y n ) in the obvious way.
We are now ready to define the standard translation.
Furthermore, we define the standard translation of a consequence pair ϕ ψ as Every L-model M = (X, ≤, V ) gives rise to a first-order structure for FOL: ≤ accounts for the interpretation of the binary relation symbol, and the interpretation of the unary predicates is given via the valuations of the proposition letters. We write M • for the L-model M conceived of as a first-order structure for FOL.
3.26 Proposition. We have Proof. This follows from a routine induction on the structure of ϕ. The propositional case holds by definition of the P i . The cases ϕ = ⊤ and ϕ = ⊥ hold by definition. The case ϕ = ψ 1 ∧ ψ 2 is trivial. For ϕ = ψ 1 ∨ ψ 2 , we have This completes the proof.
3.27 Corollary. Let M be an L-model. Then we have: In order to obtain similar results as in Corollary 3.27 for frames, we need to quantify the unary predicates in FOL corresponding to the proposition letters. We can do so in a secondorder language, say, SOL. However, getting a second-order correspondent for a consequence pair ϕ ψ that is satisfied in a frame if and only if ϕ ψ is, is not as easy as simply quantifying over all possible interpretations of the unary predicates. That is, we cannot simply add ∀P 1 · · · ∀P n in front of st x (ϕ ψ). Indeed, we wish to only take those interpretations into account that arise from a valuation of the proposition letters as filters.
Thus we wish to quantify over interpretations of the unary predicates corresponding to filters in the underlying frame. We can force this by adding conditions that ensure that the P 's are interpreted as filters in the antecedent of the implication st x (ϕ) → st x (ψ). Then the implication is vacuously true for "illegal" interpretations of the unary predicates. This intuition motivates the following definition of the second-order translation of a consequence pair.
Since all unary predicates in so(ϕ) are bounded, it can be interpreted in a first-order structure with a single relation. Therefore, every L-model X gives rise to a structure X • for SOL in which we can interpret second order translations.

Lemma.
For all L-frames X = (X, ≤) and all consequence pairs ψ χ we have Proof. Suppose X ψ χ. Then for every valuation V of the proposition letters we have V (ψ) ⊆ V (χ). If any of the P i is interpreted as a subset of X that is not a filter, then the implication inside the quantifiers in (15) is automatically true, because the antecedent is false. If all P i are interpreted as filters, then the implication holds because of the assumption.
The converse is similar.
Next, we show how one can use the second-order translation to obtain local correspondence results. We first define what we mean by local correspondence.
3.30 Definition. Let ϕ ψ be a consequence pair and α(x) a first-order formula with free variable x. Then we say that ϕ ψ and α(x) are local frame correspondents if for any L-frame X and any state w we have Since our language is positive, every formula is upward monotone. That is, extending the valuation increases the truth set of formulae.
3.31 Lemma. Let X be an L-frame and let V and V ′ be valuations for X such that V (p) ⊆ V ′ (p) for all p ∈ Prop. Then for all ϕ ∈ L we have V (ϕ) ⊆ V ′ (ϕ).
Proof. Straightforward induction on the structure of ϕ.
As a consequence of this lemma, a frame validates ⊤ χ if and only if it validates χ ′ , where χ ′ is obtained from χ by replacing all proposition letters with ⊥. This, in turn, implies that so(⊤ χ ′ ) is a first-order correspondent of ⊤ χ, since the lack of proposition letters in χ ′ implies that there are no second-order quantifiers in so(⊤ χ ′ ).
Furthermore, consequence pairs of the form ⊥ χ are vacuously valid on all frames. Both these cases (with either ⊤ or ⊥ as the antecedent of the consequence pair) are not very interesting. In the next theorem we prove that all other consequence pairs have a local first-order correspondent as well. Since we have already discussed what happens if the antecedent is ⊤ or ⊥, we may preclude these cases from the proof of our theorem.
3.32 Theorem. Any consequence pair ψ χ of L-formulae locally corresponds to a first-order formula with one free variable.
Proof. We know that X ψ χ if and only if X • |= so(ψ χ). Our strategy for obtaining a first-order correspondent is to remove all second-order quantifiers from the second-order translation. We assume that this expression has been processed such that no two quantifiers bind the same variable.
If the antecedent is equivalent to either ⊤ or ⊥ then we already know that the statement is true, so we assume that it is not. Moreover, in this case we can replace the antecedent with an equivalent formula that does not contain ⊤ or ⊥. So we may assume that the antecedent does not involve ⊤ or ⊥.
Let p 1 , . . . , p n be the propositional variables occurring in ψ, and write P 1 , . . . , P n for their corresponding unary predicates. We assume that every proposition letter that occurs in χ also occurs is ψ, for otherwise we may replace it by ⊥ to obtain a formula which is equivalent in terms of validity on frames.
Step 1. We start by pre-processing the formula so(ψ χ) some more. We make use of the fact that, after applying the second-order translation, we have classical laws such as distributivity.
Step 1A. Use equivalences of the form to pull out all quantifiers that arise in st x (ψ). Let Y := {y 1 , . . . , y m } denote the set of (bound) variables that occur in the antecedent of the implication from the second-order translation. We end up with a formula of the form ∀P 1 · · · ∀P n ∀x∀y 1 · · · ∀y m (isfil(P 1 ) ∧ · · · ∧ isfil(P n ) ISFIL ∧ψ) → st x (χ) .
In this formula,ψ is made up of formulae of the form P i z and abovemeet(z; z ′ , z ′′ ) by using ∧ and ∨, where z, z ′ , z ′′ ∈ Y ∪ {x}.
Step 1B. Use distributivity (of first-order classical logic) to pull out the disjunctions from ISFIL ∧ψ. That is, we rewrite ISFIL ∧ψ as a (finite) disjunction All the ys with any sort of subscript come from the set Y ∪ {x}.
Step 2. Next we focus on each of the formulae of the form given in (8) individually. We read off minimal instances of the P i making the antecedent true. Intuitively, these correspond to the smallest valuations for the p i making the antecedent true. For each proposition letter P i , let P i y i1 , . . . , P i y i k be the occurrences of P i in AT in the antecedent of (8). Intuitively, we define the valuation of p i to be the filter generated by the (interpretations of) y i1 , . . . , y i k . Formally, σ(P i ) := λu.abovemeet(u; y i1 , . . . , y i k ).
(If k = 0, i.e. there is no variable y with P i y, then we let σ(P i ) = λu.(u = u).) Then for each L-model M and states x ′ , y ′ 1 , . . . , y ′ m in M we have If we replace each unary predicate P in (16) with σ(P ), then all of the "isfil" formulas become true, as do all of the formulae in AT. Writing [σ(P )/P ] st x (χ) for the formula obtained from st x (χ) by replacing each instance of a unary predicate P with σ(P ), we arrive at the first-order formula ∀x∀y 1 · · · ∀y m REL → [σ(P i )/P i ] st x (χ) Step 3. Finally, we claim that for every L-frame X, X • validates (8) if and only if it validates (9). The implication from left to right is simply an instantiation of the quantifiers as filters.
For the converse, assume that M is some model based on X, so that M • is an extension of X • giving the interpretations of the unary predicates as filters. We may disregard the case where any of them is not a filter as that would make the antecedent in (16) false, hence the whole implication true. Let x ′ , y ′ 1 , . . . , y ′ m be states in M and assume that We need to show that It follows from the assumption that (9) . Moreover, as a consequence of (10) we have M • |= ∀y(σ(P )(y) → P y) for all P ∈ {P 1 , . . . , P n }. Using Lemma 3.31 it follows that M • |= st x (χ)[x ′ , y ′ 1 , · · · , y ′ m ], as desired.
Let us work out some explicit examples so we can see the proof of the theorem in action.
3.33 Example. Consider the formula p ∧ (q ∨ q ′ ) (p ∧ q) ∨ (p ∧ q ′ ). This corresponds to distributivity; the reverse consequence pair is always valid. We temporarily abbreviate χ := (p ∧ q) ∨ (p ∧ q ′ ). The second-order translation of this formula is As per instructions, we rewrite this to Tor each of the conjoints we can find a first-order correspondent. Let us focus first on the part, ∀P ∀Q∀Q ′ ∀x∀y∀y ′ (ISFIL ∧P x ∧ Qx) → st x (χ) . Then we have σ(P ) = σ(Q) = λu.x ≤ u and σ(Q ′ ) = λu.(u = u). The standard translation of χ is which is vacuously true because the first disjoint is true. As a consequence, the first conjoint of (11) is vacuously true. Hence it does not contribute non-trivially to the local correspondent, and we may ignore it. Similarly, the second conjoint in (11) is vacuously true.
For the third conjoint we obtain σ(P ) = λu.x ≤ u, σ(Q) = λu.y ≤ u and σ(Q ′ ) = λu.y ′ ≤ u. Plugging these into the antecedent yields Thus we find the following first-order correspondent: This can be reformulated as follows: If the frame is principal this reduces further. If y ≤ x then taking z = x and z ′ = x ∨ y ′ yields states above y and y ′ respectively whose meet is x. Similar if y ′ ≤ x. We get: if y ∧y ′ ≤ x then there are z, z ′ such that y ≤ z and y ′ ≤ z ′ and z ∧ z ′ = x. In other words, the lattice is distributive. ⊳ 3.34 Example. Next consider the modularity axiom Writing χ for the right hand side of the consequence pair, after applying step 1 of the proof of Theorem 3.32 we have We now compute the σ(P i ) of the individual disjoints, and instantiate these in the standard translation of χ, which is We begin with the first conjoint. Here σ(P 1 ) = σ(P 3 ) = λu.x ≤ u and σ(P 2 ) = ∅. Substituting this in st x (χ) automatically makes it true, because the first conjoint of [σ(P i )/P i ] st x (χ) then reads (x ≤ x) ∧ (x ≤ x). Since this is automatically true we can ignore it in our first-order correspondent. Similar reasoning allows us to disregard the second conjoint of (12).
In the third conjoint we obtain σ(P 1 ) = λu.y ≤ u, σ(P 2 ) = λu.z ≤ u and σ(P 3 ) = λu.abovemeet(u; x, y). Substituting these in the third conjoint gives Leaving out everything that is trivially true, this reduces to In words, an L-frame (X, ≤) validates modularity if for all x, y, z ∈ X such that y ∧ z ≤ x either • there exist s, t ∈ X above y and z, respectively, such that s ∧ t ≤ x and x ∧ y ≤ t. ⊳

Sahlqvist Canonicity
In this section we use the Sahlqvist correspondence result from the previous section to prove automated completeness results. Our approach resembles the one taken by Sambin and Vaccaro [36], and again demonstrates how classical techniques can be used in our non-distributive setting. We use the well-known notion of d-persistent to obtain completeness proofs of extensions of L in a similar way as in Theorem 3.22. Recall that we denote the L-frame underlying an L-space X by κX.

Definition. A consequence pair ϕ ψ is called d-persistent if
X ϕ ψ implies κX ϕ ψ for all descriptive frames X.
Clearly, if all consequence pairs in a set Γ are d-persistent, then the same proof as for Theorem 3.22 yields completeness of L(Γ) with respect to all (principal) L-frames validating all consequence pairs in Γ.
The main result of this section states that all consequence pairs of L-formulae are dpersistent. Before proving this, we collect some preliminary lemmas. These are essentially reformulations of Lemmas 4.2.7 and 4.2.8 from [13]. We fix an arbitrary L-space X. If U is a valuation for X then we denote by U (ϕ) the truth set of ϕ in (X, U ). Furthermore, if U and V are valuations then U ∩ V is the valuation defined by (U ∩ V )(p) = U (p) ∩ V (p). Finally, we say that V is a closed valuation for κX if for each proposition letter p, the set V (p) is a closed filter of X.
3.36 Lemma. Let ϕ be any formula, and U, V any valuations for an L-frame X = (X, ≤).
Proof. We use induction on the structure of ϕ. If ϕ = ⊤ or ϕ = ⊥ then the statement is trivial. If ϕ = p then it holds by definition. If ϕ = ϕ 1 ∧ ϕ 2 then we have The first inclusion follows from the induction hypothesis. The second inclusion follows from the fact that the two filters on the third line both contain the sets generating the filter on the second line.
The following lemma is a version of the intersection lemma [36].
3.37 Lemma. Let X = (X, ≤, τ ) be an L-space. Let V be any closed valuation for X and write V <· U if U is a clopen valuation extending V . Then for all ϕ ∈ L we have Proof. We prove this by induction on the structure of ϕ. If ϕ = ⊤ or ϕ = ⊥ then the result is obvious. If ϕ = p then the result follows from Lemma 2.9. Suppose ϕ = ϕ 1 ∧ ϕ 2 . Then we have Lastly, suppose ϕ = ϕ 1 ∨ ϕ 2 . For this induction step we need to work a bit harder. View X as the dual of the lattice A and recall that each clopen filter of X is of the form θ A (a) = {p ∈ F top A | a ∈ p} for some a ∈ A. To avoid unnecessary notational clutter, we suppress the subscript A from θ A .
We now have all the tools to prove a Sahlqvist canonicity theorem.
3.38 Theorem. Any consequence pair ψ χ of L-formulae is d-persistent.
Proof. If the antecedent is equivalent to ⊥ then the statement is vacuously true. If the antecedent is equivalent to ⊤ then (in terms of validity on frames) the consequence pair ⊤ χ is equivalent to a consequence pair without proposition letters, so that the statement is true as well. In all other cases, the antecedent is equivalent to a formula that does not contain ⊤ or ⊥, so we assume that ψ does not contain ⊤ or ⊥. Let X be an L-space that validates ψ χ. We wish to show that κX ψ χ. As a consequence of Theorem 3.32, it suffices to show that (κX) • |= so(ϕ ψ). As in Step 1 in the proof of Theorem 3.32, we can rewrite so(ϕ ψ) to a conjunction of formulae of the form ∀P 1 · · · ∀P n ∀x∀y 1 · · · ∀y m (ISFIL ∧ AT ∧ REL → st x (χ)), where P 1 , . . . , P n are the unary predicates corresponding to proposition letters in ψ χ and y 1 , . . . , y m are the variables arising from the second-order translation. So it suffices to show that these are all validated in (κX) • . Observe that valuations of κX correspond bijectively with interpretations of the unary predicates that make ISFIL true. So it suffices to show that for all (not necessarily clopen) valuations V for κX and all x ′ , y ′ 1 , . . . , y ′ m we have Here, by (κX, V ) • we mean the first-order structure (κX) • that interprets the unary predicates via the valuation V and variables via the assignment given in square brackets. So suppose we have an instantiation x ′ , y ′ 1 , . . . , y ′ m of the variables. For each proposition letter p i , let V m (p i ) be the filter generated by the set {y | P i y occurs in AT }. Then V m defines a principal, hence closed valuation of the proposition letters in ψ and χ. We claim that The direction from right to left is trivial. For the converse, suppose the LHS holds and suppose (κX, follows that V is an extension of V m . Since validity of REL is independent of the valuation, and the valuation V m assigns a filter to each proposition letter, we have (κX, So the left to right condition of (13) also holds.
So it suffices to prove (κX, , so our goal further reduces to proving Observe that the variables y 1 , . . . , y m do not occur in st x (χ) (we have not substituted anything yet), so (14) holds if and only if (κX, V m ), x ′ χ. That is, if and only if x ′ ∈ V m (χ). As a consequence of Lemma 3.37 it suffices to show that x ′ ∈ U (χ) for all clopen valuations of X extending V m . But this follows from our assumptions: Since U is a clopen valuation . . , y ′ m ] (recall thatψ is the reformulation of st x (ψ) with all quantifiers pulled out), and the instantiations x ′ , y ′ 1 , . . . , y ′ m now witness the fact that (κX, U ) • |= st x (ψ). This implies (κX, U ) ψ. So by the assumption that X validates ψ χ we get (κX, U ) χ, as desired.
In particular, this shows that we get a sound and complete semantics for distributive and modular lattices. We also obtain the following generalisation of Baker and Hales' result [2], which states that every variety of lattices is closed under ideal completions. In order not to overload the paper we do not define ideal completions here and refer the reader to [13,Section 4.3] for all the details including on how the next theorem implies the Baker and Hales's result.
3.40 Theorem. Every variety of lattices is closed under taking filter completions and F 2completions.
Proof. Suppose a lattice A satisfies an equation ϕ = ψ. Then it validates ϕ ψ and ψ ϕ. Writing X for the L-space dual to A, Lemma 3.21 implies that X ϕ ψ and ψ ϕ. Since every consequence pair is d-persistent the lattice of (principal) filters of X validate ϕ ψ and ψ ϕ. But this precisely means that the filter extension of A and the double filter extension of A validate ϕ = ψ.
3.41 Remark. The d-persistence that we discuss here allows us to move from valuations that interpret proposition letters as clopen filters to valuation that assign to each proposition letter an arbitrary filter. It is analogous to d-persistence in intuitionistic and modal logics [6,8]. In the classical setting d-persistence allows one to move from clopen valuations to arbitrary valuations, and in the intuitionistic case from valuations into clopen upsets to valuations into all upsets. We point out that, while in the distributive setting this corresponds algebraically to canonical extensions, in our setting the corresponding algebraic structure is the F 2 -completion. ⊳

Normal Modal Extensions
We investigate the extension of non-distributive positive logic with two modal operators, and , interpreted via a relation in the usual way (see e.g. [6, Definition 1.20]). As our point of departure we take L-frames with an additional relation. We stipulate sufficient conditions on the relation to ensure persistence, but we do not enforce any axioms on the modalities. It will turn out that preserves finite conjunctions as usual. However, as a consequence of the non-standard interpretation of disjunctions, is non-normal. This is reminiscent of the modal extension of intuitionistic logic investigated by Kojima [31], where also is normal and is not. The interaction axioms relating and are closely aligned to Dunn's axioms for (distributive) positive modal logic [14], see Remark 4.7.
After investigating the modal logic from a semantic point of view, we use our newly developed intuition to give syntactic definition of the logic in Section 4.2. This is sound by design, and the algebraic semantics is given by lattices with operators. In Section 4.3 we provide a duality for the algebraic semantics by means of modal L-spaces. We show that each modal L-space has an underlying modal L-frame, and as a consequence we obtain completeness for the basic logic.
Subsequently, in Section 4.4 we prove Sahlqvist correspondence. The methods used are analogous to those in Sections 3.4, but it is no longer the case that any consequence pair is Sahlqvist. We identify as Sahlqvist consequence pairs precisely the negation-free Sahlqvist formulae from normal modal logic [6, Definition 3.51], where the implication is replaced by . We then use this to prove Sahlqvist canonicity in Section 4.5, in the same way as in 3.5. As a consequence of the more restricted notion of a Sahlqvist consequence pair, we now distinguish two types of persistence: p-persistence and d-persistence. The former is persistence to the underlying principal frame, where proposition letters are interpreted as principal filters. For d-persistence we do not impose any extra conditions on the valuation. It will turn out that all consequence pairs are p-persistent, and all Sahlqvist consequence pairs are d-persistent.
One may wonder whether it would be more natural to insist that be normal as well. We do not because the additional conditions required to ensure that is normal complicate the presentation of the semantics and duality. Moreover, in order to make normal we only need to extend our basic system with the consequence pair which is Sahlqvist! So we obtain a local first-order correspondent and automated completeness via our Sahlqvist theorems from Sections 4.4 and 4.5. We explicitly compute the local correspondent in Example 4.37.

Relational Meet-Frames
Let L be the language generated by the grammar where p ranges over some set Prop of proposition letters. A modal consequence pair is an expression of the form ϕ ψ, where ϕ, ψ ∈ L . We derive an appropriate notion of modal L-frame, such that the truth set of each formula is guaranteed to be a filter.

Definition.
A modal L-frame is a tuple (X, ≤, R) where (X, ≤) is an L-frame and R is a binary relation on X such that: 1. If x ≤ y and yRz then there exists a w ∈ X such that xRw and w ≤ y; 2. If x ≤ y and xRw then there exists a z ∈ X such that yRz and w ≤ z; 3. If (x ∧ y)Rz then there exist v, w ∈ X such that xRv and yRw and v ∧ w ≤ z; 4. If xRv and yRw then (x ∧ y)R(v ∧ w).
A modal L-model is a a modal L-frame together with a valuation V that assigns to each proposition letter a filter of (X, ≤).
The interpretation of L -formulae in a modal L-model M is defined via the clauses from Definition 4.1 and M, x ϕ iff ∀y ∈ X, xRy implies M, y ϕ M, x ϕ iff ∃y ∈ X such that xRy and M, y ϕ Satisfaction and validity of formulae and modal consequence pairs are defined as expected. In particular, if is a class of modal L-frames and ϕ ψ is a modal consequence pair, then we write ϕ ψ if the consequence pair ϕ ψ is valid on all frames in . The four conditions of a modal L-frame are depicted in Figure 3. If (X, ≤, R) is a modal L-frame, then we denote the set of R-successors of a state x ∈ X by R[x] := {y ∈ X | xRy}. Also, for a subset a ⊆ X we define Next we prove persistence.

Proposition.
Let M = (X, ≤, R, V ) be a modal L-model. Then for each ϕ ∈ L the set ϕ M is a filter in (X, ≤).
Proof. The proof proceeds by induction on the structure of ϕ. The only non-trivial cases are the modal cases. We prove the statement for ϕ = ψ, the case ϕ = ψ is similar.
So suppose ϕ = ψ, M, x ψ and x ≤ y. Then there exists an R-successor z of x satisfying ψ, and by (2) we can find an R-successor w of y such that z ≤ y. By the induction hypothesis we then find M, w ψ and therefore M, y ψ. Next, suppose that both x and y satisfy ψ. Then there exist v ∈ R [x] and w ∈ R[y] both satisfying ψ. By (4) we then have (x ∧ y)R(v ∧ w) and as a consequence of the induction hypothesis M, v ∧ w ψ. Therefore M, x ∧ y ψ. We conclude that ψ M is a filter in (X, ≤).

Remark.
We could have slightly weakened condition 4 by requiring the existence of some (x ∧ y)-successor above v ∧ w. We use the current formulation because it aligns more closely to the notion of a modal L-space. ⊳ Morphisms between modal L-frames and -models are a combination of L-morphisms and bounded morphisms.
is an L-morphism and for all x, y ∈ X and z ′ ∈ X ′ : 2. If f (x)R ′ z ′ then there exists a z ∈ X such that xRz and f (z) ≤ z ′ ; 3. If f (x)R ′ z ′ then there exists a w ∈ X such that xRz and z ′ ≤ ′ f (w).
A bounded L-morphism between models is bounded L-morphism between the underlying frames that preserves and reflects truth of proposition letters.
The bounded L-morphism conditions are depicted in Figure 4. Proof. This follows from a routine induction on the structure of ϕ. We showcase the modal cases of the proof. Suppose ϕ = ψ. It follows immediately from Definition 4.4(1) that then there exists some z ∈ X such that xRz and f (z) ≤ y ′ . This implies M, z ψ and by induction M ′ , f (z) ψ. Persistence then yields M ′ , y ′ ψ. Therefore M ′ , f (x) ψ. Next, let ϕ = ψ. Then the preservation from left to right follows from Definition 4.4 (1).
ψ, then there exists a y ′ ∈ X ′ such that f (x)Ry ′ and M ′ , y ′ ψ. By (3) we can find some w ∈ X such that xRw and y ′ ≤ f (w). Persistence implies M ′ , f (w) ψ and induction yields M, w ψ. Therefore M, x ψ.
We give a number of modal consequence pairs that are valid in every modal L-frame. These motivate the definition of a modal lattice in Section 4.2.
4.6 Lemma. Let (X, ≤, R) be a modal L-frame. Then the following consequence pairs are all valid: Proof. All of these follow immediately from the definition of the interpretation of and . In particular, they do not rely on any of the conditions from Definition 4.1.

Remark.
In Lemma 4.6 we have found two duality axioms relating and . These are similar to Dunn's duality axioms for positive modal logic [14], which are Indeed, the left hand one is exactly the same. It is easy to see (using e.g. a semantic argument in Kripke frames, the frame semantics used in [14]) that in the distributive case that the two right hand side duality axioms are equivalent. ⊳ Just like in the propositional case in Section 3.2, we can identify a class of frames where formulae can be interpreted exclusively as principal filters.

Definition.
A principal modal L-frame is a modal L-frame (X, ≤, R) such that 0. (X, ≤) has binary joins and all non-empty meets; 1. If x ≤ y and yRz then there exists a w ∈ X such that xRw and w ≤ y; 2. If x ≤ y and xRw then there exists a z ∈ X such that yRz and w ≤ z; 3. If (x ∧ y)Rz then there exist v, w ∈ X such that xRv and yRw and v ∧ w ≤ z; where the i range over some index set I, then there exist z i such that x i Rz i for all i ∈ I and z i ≤ z; 4. If x i Ry i , where i ranges over some non-empty index set I, then ( x i )R( y i ).
A principal modal L-model is a principal modal L-frame together with a principal valuation, and L -formulae are interpreted in the same way as in modal L-models. As for modal L-frames, if is a class of principal modal L-frames and ϕ ψ is a modal consequence pair, then we write ϕ ψ if the consequence pair ϕ ψ is valid on all frames in .
Observe that these conditions subsume the ones from Definition 4.1, so principal modal L-frames form a subclass of the modal L-frames (and similar for the corresponding models). As desired, the truth set of a L -formula in a principal modal L-model is always given by a principal filter.
4.9 Proposition. Let M = (X, ≤, R, V ) be a principal modal L-model. Then for all ϕ ∈ L the truth set ϕ M is a principal filter in (X, ≤).
Proof. The proof proceeds by induction on the structure of ϕ. The propositional cases are the same as in Proposition 3.12. We give the induction step for ϕ = ψ and leave the case ϕ = ψ to the reader.
So let ϕ = ψ and suppose that ψ M is a principal filter (this is the induction hypothesis). If ψ M is empty then we are done. If not, then we can prove that it is upward closed in the same way as in Proposition 4.2. Moreover, it follows immeidately from (4) that ψ M ∈ ψ M , so ψ M is indeed a principal filter.

Logic and Modal Lattices
Guided by the validities from Lemma 4.6, we define the logic L as follows.
If Γ is a set of modal consequence pairs then we let L (Γ) denote the smallest set of modal consequence pairs closed under the axioms and rules mentioned above and those in Γ. We write ϕ ⊢ Γ ψ if ϕ ψ ∈ L (Γ) and ϕ ⊣⊢ Γ ψ if both ϕ ⊢ Γ ψ and ψ ⊢ Γ ϕ. If Γ is the empty set then we simply write ϕ ⊢ ψ and ϕ ⊣⊢ ψ.
Observe that Becker's rule together with linearity for implies that is a normal modal operator. The logic gives rise to the following algebraic semantics, given by modal lattices.

Definition.
A modal lattice is a tuple (A, , ) consisting of a lattice A and two maps , : A → A satisfying for all a, b ∈ A: Formulae ϕ ∈ L can be interpreted in a modal lattice = (A, , ) with an assignment σ : Prop → A. Analogous to Section 3.1, the interpretation of proposition letters is given by the assignment, and the connectives and modalities as interpreted via their counterparts in . This gives rise to validity of formulae and modal consequence pairs in a modal lattice .
If M = (X, V ) is a modal L-model then V is an assignment for X ‡ and we write M ‡ = (X ‡ , V ). If M = (X, V ) is a principal modal L-model then V is an assignment for X † and we let M † = (X † , V ). We obtain the following counterpart of Lemma 3.7.
4.13 Lemma. Let M be a modal L-model and N a principal modal L-model. Then We write ϕ Γ ψ if any modal lattice that validates all consequence pairs in Γ also validates ϕ ψ. Then we can prove the next theorem in the same way as in Section 3.1.

Modal L-spaces and Duality
We prove a duality for modal lattices by means of L-spaces with an additional relation. First we show that such structures have an underlying (principal) modal L-frame, and then we proceed to the duality. 3. For all x, y ∈ X we have xRy if and only if for all a ∈ F clp X: • If x ∈ [R]a then y ∈ a; • If y ∈ a then x ∈ R a.
Truth and validity in modal L-spaces is defined as usual, using clopen valuations.
The second item is a condition often seen in the definition of general frames. Item (3) is our counterpart of the tightness condition, and has previously been used in [7,Section 2].
Next, we prove that each modal L-space has an underlying (principal) modal L-frame. We need the following lemma for this.
4.16 Lemma. Let X = (X, ≤, τ, R) be a modal L-space. Then R [x] is closed for all x ∈ X.
Proof. Suppose y / ∈ R [x]. Then there exists a clopen filter a such that either x ∈ [R]a and y / ∈ a, or y ∈ a and x / ∈ R a. In the first case X \ a is a clopen neighbourhood of y disjoint from R [x]. In the second case a is a clopen neighbourhood of y disjoint from R [x].
4.17 Proposition. Let X = (X, ≤, τ, R) be a modal L-space. Then (X, ≤, R) is a principal modal L-frame. Proof. We know that L-spaces have all non-empty meets, and hence also binary joins, so (0) is satisfied. We verify the other conditions. We start with (4), so that we can use it when proving the others. Condition 4. Suppose x i Ry i , where i ranges over some non-empty index set I. By the tightness condition of modal L-spaces, in order to prove ( x i )R( y i ) it suffices to show that for all clopen filters a, x i ∈ [R]a implies y i ∈ a and y i ∈ a implies x i ∈ R a.
First assume x i ∈ [R]a. Since [R]a is a clopen filter and x i ≤ x j for each j ∈ I we have By assumption x j Ry j so y j ∈ a for all j ∈ I. Since a is a clopen filter it is principal, and therefore y i ∈ a.
Next, suppose y i ∈ a. Since a is a filter and y i ≤ y j we have y j ∈ a for all j ∈ I. This implies x j ∈ R a for all j ∈ I, and since R is a clopen filter, hence principal, we find x i ∈ R a.
Condition 1. Let x ≤ y and yRz. Suppose towards a contradiction that there exists no w ∈ X such that xRw and w ≤ z. Now let x ′ = R [x] be the minimal element in R [x] (which is an R-successor of x by (4)). Then x ′ ≤ z, so we can find a clopen filter a containing x ′ such that z / ∈ a. This implies R[x] ⊆ a, so that x ∈ [R]a, but y / ∈ [R]a because yRz and z / ∈ a. As x ≤ y this violates the fact that [R]a is a filter.
Condition 2. Let x ≤ y and xRw. Suppose towards a contradiction that there exists no z ∈ X such that yRz and w ≤ z. Then R[y] ∩ ↑z = ∅. Both R[y] and ↑z are closed, as a consequence of Lemmas 4.16 and 2.10 and the fact that singletons in a Stone spaces are always closed. Therefore we can find a clopen filter a containing ↑z which is disjoint from R[y]. This implies that x ∈ R a while y / ∈ R a. Since x ≤ y this contradicts the fact that R a is a filter.
Condition 3. Finally, let {x i | i ∈ I} be some non-empty collection of elements of X and suppose ( x i )Rz. Since x i ≤ x j for all j ∈ I, condition (2) implies that R[x j ] = ∅ for all j ∈ I. As a consequence of (4) there is a smallest element z j := R[x j ] in each R[x j ]. We claim that z j ≤ z. Suppose not, then there is a clopen filter a containing z j such that z / ∈ a. This implies x j ∈ [R]a for all j ∈ I, but x i / ∈ [R]a because ( x i )Rz and z / ∈ a. But this contradicts the fact that [R]a is principal filter.
Since we know that each modal L-space has an underlying modal L-frame, we can now conveniently define the morphisms between them as follows.
is a modal L-morphism. We denote the resulting category by MLSpace.
We work our way towards a duality between modal lattices and modal L-spaces.

4.19
Proposition. If X = (X, ≤, τ, R) is a modal L-space then (F clp X, [R], R ) is a modal lattice. Moreover, if f : X → X ′ is a modal L-space morphism, then is a modal lattice homomorphism.
Proof. The maps [R], R are functions on F clp X by definition. It follows from Proposition 4.17 and Lemma 4.6 that they satisfy the conditions from Definition 4.11.
If f is a modal L-space morphism then in particular it is an L-space morphism, so F clp f is a lattice homomorphism from F clp X ′ to F clp X. So we only have to show that f −1 preserves the modalities. This can be proven in the same way as in Proposition 4.5.
In order to show pR A −1 (p) we need to prove −1 (p) ⊆ −1 (p) ⊆ −1 (p). The left inclusion is trivial, so we only have to prove the right one. Suppose a ∈ −1 (p). Then a ∈ p. Since ⊤ ∈ p by assumption and because ⊤ ∧ a ≤ (⊤ ∧ a) = a, this implies a ∈ p. So a ∈ −1 (p). . Let (A, , ) be a modal lattice. Then for each a ∈ A we have

Lemma
. If there exists a filter q such that pR A q, then we have a ∈ q and q ⊆ −1 (p) (by definition of R A ). This implies a ∈ p and hence ⊤ ∈ p. It then follows from Lemma 4.21 that −1 (p) is a filter and pR A −1 (p). By assumption a ∈ −1 (p) so that a ∈ p and therefore p ∈ θ A ( a).
If p has no R A -successors, then for each filter q we have ¬(pR A q), so there exists an a such that −1 (p) ⊆ q or q ⊆ −1 (p). If a ∈ p then the filter q generated by −1 (p) and a is such that −1 (p) ⊆ q ⊆ −1 (p). (Clearly −1 (p) ⊆ q. To see that q ⊆ −1 (p), suppose c ∈ q. Then there exists a d ∈ −1 (p) such that d ∧ a ≤ c. Since d, a ∈ p and d ∧ a ≤ (d ∧ a) ≤ c this implies c ∈ −1 (p).) So a / ∈ p for all a ∈ A. In particular ⊤ / ∈ p. Since ⊤ = ⊤ = (⊤ ∨ a) ≤ (⊤ ∨ a) ∨ a this implies a ∈ p, as required. For the reverse inclusion, suppose p ∈ θ A ( a). Then a ∈ −1 (p), so pR A q implies a ∈ q, and hence p ∈ [R A ]θ A (a).
Next suppose p ∈ R A θ A (a). Then there exists a filter q such that pR A q and a ∈ q. By definition of R A this implies a ∈ −1 (p) and hence a ∈ p, so p ∈ θ A ( a). Conversely, suppose p ∈ θ A ( a). Let q be the filter generated by −1 (p) and a. Then pR A q and a ∈ q, so p ∈ R A θ A (a). 4.26 Definition. Let X = (X, ≤, τ, R) be a modal L-space. Then we write πX = (X, ≤, R) for the underlying principal modal L-frame, and κX = (X, ≤, R) for the underlying principal modal L-frame regarded as a modal L-frame.
While they may appear the same, the difference between πX and κX lies in the valuations they allow for. While valuations of πX necessarily interpret proposition letters as principal filters, a valuation for κX can assign any filter to a proposition letter. As a consequence, both frames differ in terms of validity. We will see in Section 4.5 that the move from X to πX preserves validity of all modal consequence pairs, and the move from X to κX preserves validity of all Sahlqvist consequence pairs.
As we did for F 2 -completions of lattices in Section 2.4, we define the F 2 -completion of a modal lattice via the duality. We also define the filter completion in this way. Recall that if X is a modal L-frame, then by X ‡ we denote its modal lattice of filters, and if X is a principal modal L-frame then X † is the modal lattice of principal filters. 1. The filter completion of is defined as (π * ) † .
In both cases the inclusion from into its completion is given by θ A , which sends a ∈ A to {p ∈ F top ( * ) | a ∈ p}. It follows from Lemma 4.22 that this map preserves and .

Sahlqvist Correspondence
In this section we extend the results from Section 3.4 to obtain Sahlqvist correspondence for modal L-frames. Interestingly, our definition of a Sahlqvist consequence pair is closely aligned to Sahlqivst formulae from normal modal logic (see e.g. [6, Definition 3.51]).
To account for the additional relation in the definition of a modal L-model, we work with a first-order logic with an extra binary relation symbol (compared to Section 3.4). That is, we let FOL 2 be the first-order logic with a unary predicate for each proposition letter and two binary predicates S (corresponding to the partial order) and R (corresponding to the modal relation). Every modal L-model M gives rise to a first-order structure M • for FOL 2 in the obvious way. We extend the standard translation from Definition 3.25 to a translation st x : L → FOL 2 by adding the clauses st x ( ϕ) = ∀y(xRy → st y (ϕ)) st x ( ϕ) = ∃y(xRy ∧ st y (ϕ)) We then have the following counterparts of Proposition 4.28 and Corollary 4.29 We obtain similar results for modal L-frames by extending the second-order translation. Write SOL 2 for the second-order logic with the same predicates as FOL 2 where we allow quantification over unary predicates.
4.30 Definition. Let p 1 , . . . , p n be the proposition letters occurring in the L -formuale ψ and χ, and write P 1 , . . . , P n denote their corresponding unary predicates. The second order translation of a consequence pair ψ χ by so(ψ χ) = ∀P 1 · · · ∀P n ∀x (isfil( Since all unary predicates in so(ϕ) are bounded, it can be interpreted in a first-order structure with two relations. So every modal L-model X gives rise to a structure X • for SOL 2formulae with no free predicates.
Proof. Similar to the proof of Lemma 3.29.
Finally, we still have monotonicity of all connectives of L , so the following analogue of Lemma 3.31 goes through without problems.
4.32 Lemma. Let X be a modal L-frame and let V and V ′ be valuations for X such that We are now ready to define Sahlqvist consequence pairs and prove a correspondence result. We make use of the following notion of a boxed atom.

Definition.
A boxed atom is a formula of the form n p := · · · n times p, where p is a proposition letter.
If R is a relation, then we write R n for the n-fold composition of R. That is, xR n y if there exist x 0 , . . . , x n+1 such that x = x 0 , y = x n+1 and x i Rx i+1 for all i ∈ {0, . . . , n}. With this definition, truth of n p in a modal L-model M = (X, ≤, R, V ) can be given as M, x n p iff ∀y ∈ X, xR n y implies M, y p.
We legislate xR 0 y iff x = y. Then the interpretation of 0 p simply coincides with p.

Definition.
A Sahlqvist antecedent is a formula made from boxed atoms, ⊤ and ⊥ by freely using ∧, ∨ and .
4. 35 Theorem. Let ψ ∈ L be a Sahlqvist antecedent, and let χ ∈ L be any formula. Then ψ χ locally corresponds to a first-order formula on frames that is effectively computable from the sequent.
Proof. We employ the same strategy as in the proof of Theorem 3.32. The second half of the proof is identical to that of Theorem 3.32, so we focus on the first half.
We assume that this expression has been processed such that no two quantifiers bind the same variable. Let p 1 , . . . , p n be the propositional variables occurring in ψ, and write P 1 , . . . , P n for their corresponding unary predicates. We assume that every proposition letter that occurs in χ also occurs is ψ, for otherwise we may replace it by ⊥ to obtain a formula which is equivalent in terms of validity on frames.
Step 1. We start by pre-processing the formula so(ψ χ) some more. We make use of the fact that, after applying the second-order translation, we have classical laws such as distributivity.
Step 1A. Use equivalences of the form to pull out all existential quantifiers that arise in st x (ψ). Let Y := {y 1 , . . . , y m } denote the set of (bound) variables that arise in the antecedent of the implication from the second-order translation. We end up with a formula of the form ∀P 1 · · · ∀P n ∀x∀y 1 · · · ∀y m (isfil(P 1 ) ∧ · · · ∧ isfil(P n ) In this formula,ψ is made up of • boxed atoms: formulae of the form ∀z(z ′ R n z → P i z) (where P i z falls under this umbrella as ∀z(z ′ R 0 z → P i z)); • top and bottom: formulae of the form (x = x) and (x = x); • relations of the form zRz ′ ; and • formulae of the form abovemeet(z; z ′ , z ′′ ) by using ∧ and ∨, where z, z ′ , z ′′ ∈ Y ∪ {x}.
Step 1B. Use distributivity (of first-order classical logic) to pull out the disjunctions from ISFIL ∧ψ. That is, we rewrite ISFIL ∧ψ as a (finite) disjunction where BOX-AT contains atoms and boxed atoms and REL contains relations of the form zRz ′ and abovemeet(z; z ′ , z ′′ ).
Step 1C. Finally, use equivalences of the form and to rewrite so(ϕ ψ) into a conjunction of formulae of the form ∀P 1 · · · ∀P n ∀y 1 · · · ∀y m ISFIL ∧ BOX-AT ∧ REL → χ .
Step 2. Next we focus on each of the formulae of the form given in (16) individually. We read off minimal instances of the P i making the antecedent true. Intuitively, these correspond to the smallest valuations for the p i making the antecedent true. For each proposition letter P i , let ∀y i1 (z i1 R ni 1 y i1 → P i y i1 ), . . . , ∀y i1 (z i k R n k y i k → P i y i k ) be the occurrences of P i in BOX-AT in the antecedent of (16). Intuitively, we define the valuation of p i to be the filter generated by the (interpretations of) y i1 , . . . , y i k . Formally, (If k = 0, i.e. there are no boxed atoms involving P i in the formula, then we let σ(P i ) = λu.(u = u).) If there are no R ni j -successors of z ij then the corresponding boxed atom is vacuously true, so it does not affect the "interpretation" of P i . In order to reflect this in the expression of σ(P i ), we take the join over all subsets of {i 1 , . . . , i k }.
The remainder of the proof is analogous to the proof of Theorem 3.32.
In the next example we apply the algorithm of the proof of Theorem 4.35 to two simple consequence pairs, p p and p p. The shows the mechanism of the proof in action. Moreover, it demonstrates that the duality between and is weaker than in the classical case, because the formulae locally correspond to different frame conditions. 4.36 Example. The second-order translation of p p is ∀P ∀x(isfil(P ) ∧ P x → ∃y(xRy ∧ P y)) This is already of the desired shape, so we proceed to Step 2. We find σ(P ) = λu.x ≤ u. Substituting this gives the first-order formula ∀x(isfil(P ) ∧ (x ≤ x) → ∃y(xRy ∧ (x ≤ y))). Note that the antecedent of the formula is always true, so the (simplified) local correspondent of p p is ∀x∃y(xRy ∧ x ≤ y).
Then σ(P ) = λu.∃y(xRy ∧ y ≤ u). Instantiating this makes the antecedent of the outer implication vacuously true, so that we get the local correspondent ∀x∃y(xRy ∧ y ≤ x). ⊳ Next, we use Theorem 4.35 to enforce normality for the diamond operator. It follows from Lemma 4.6 that the consequence pair p ∨ q (p ∨ q) is valid in every modal L-frame, so we focus on its converse. We arrive at a frame condition closely related to the one identified in [ to our system. This is a Sahlqvist pair, so we can use the algorithm from Theorem 4.35 to the first-order frame condition ensuring its validity. The standard translation of the antecedent is st x ( (p ∨ q)) = ∃y(xRy ∧ (P y ∨ Qy ∨ ∃z∃z ′ (abovemeet(y; z, z ′ ) ∧ P z ∧ Qz ′ ))) Processing the formula, we obtain the following second-order translation: ∀P ∀Q∀x∀y∀z∀z ′ (ISFIL ∧xRy ∧ P y → st x (χ)) ∧ ∀P ∀Q∀x∀y∀z∀z ′ (ISFIL ∧xRy ∧ Qy → st x (χ)) ∧ ∀P ∀Q∀x∀y∀z∀z ′ (ISFIL ∧xRy ∧ abovemeet(y; , z, z ′ ) ∧ P z ∧ Qz ′ → st x (χ)) As usual, we process the lines one by one, and instantiate the minimal instantiations of P and Q into st x (χ). First compute st x (χ) = ∃s(xRs ∧ P s) ∨ ∃t(xRt ∧ Qt) In the first line of (18) we get σ(P ) = λu.y ≤ u and σ(Q) = λu.u = u. Then [σ(P )/P, σ(Q)/Q] st x (χ) holds, because its first disjoint is valid. The second line of (18) yields a similar situation. So both these give rise to first-order formulae that are always valid. The third line of (18) gives σ(P ) = λu.z ≤ u and σ(Q) = λu.z ′ ≤ u. Instantiating this we get that a frame (X, ≤) validates (17) if and only if for all x, y, z, z ′ ∈ X such that xRy and z ∧ z ′ ≤ y we have either • there exists a v such that xRv and z ≤ v; or • there exists a v ′ such that xRv ′ and z ′ ≤ v ′ ; or • there exist v, v ′ , w, w ′ such that z ≤ v, z ′ ≤ v ′ , w ∧ w ′ ≤ x, wRv and w ′ Rv ′ .
The first and third item can be depicted as follows: The second item can be depicted in a similar way as the first one. ⊳

Sahlqvist Canonicity
As we have seen, for each modal L-space X we have two flavours of "underlying frame," principal and non-principal modal L-frames. This gives rise to two different notions of persistence. We stress that our approach is along the same lines as classical Sahlqvist canonicity results [36,6].
In this section we show that every modal consequence pair is p-persistent, and that all Sahlqvist consequence pairs are d-persistent. We begin by working towards p-persistence. As we will see, this simply follows from an extension of Lemma 3.37. Next suppose ϕ = ψ. The inclusion V ( ψ) ⊆ V < · U U ( ψ) is clear because of monotonicity of the diamond. For the converse, assume x ∈ V < · U U ( ψ). We need to find some y such that xRy and y ∈ V (ψ) = V < · U U (ψ). So we want to find an element in Since all sets in this intersection are closed, it suffices to show that it has the finite intersection property. Given a finite subcollection R [x], U 1 , . . . , U n . Then V <· (U 1 ∩ · · · ∩ U n ), so x ∈ (U 1 ∩ · · · ∩ U n )( ψ) and hence there exists a state y ′ such that xRy ′ and y ′ ∈ (U 1 ∩ · · · ∩ U n )(ψ). This witnesses the finite intersection property.
Since every principal valuation is closed, it follows immediately that every consequence pair which is valid on a modal L-space X is also valid in the underlying principal L-frame πX.
4.41 Theorem. Every modal consequence pair is p-persistent.
Proof. The principal valuations for πX are precisely the closed valuations of X. Let V be such a closed valuation. If X ϕ ψ then U (ϕ) ⊆ U (ψ) for every clopen valuation U for X. This implies V (ϕ) = and therefore πX ϕ ψ.
As an immediate consequence we obtain (soundness and) completeness of L (Γ) with respect to the class of principal modal L-frames that validate all modal consequence pairs in Γ.
4.42 Theorem. Let Γ be a set of modal consequence pairs and write PMLFrm(Γ) for the class of principal modal L-frames that validate all consequence pairs in Γ. Then for all ϕ, ψ ∈ L we have ϕ ⊢ Γ ψ iff ϕ PMLFrm(Γ) ψ.
Moreover, we obtain a modal counterpart of Baker and Hales' theorem: 4.43 Theorem. Every variety of modal lattices is closed under filter extensions.
Proof. It suffices to show that filter extensions preserve inequalities. When writing inequalities, we may use the set of proposition letters as variables, so that an inequality is of the form ϕ ≤ ψ, where ϕ and ψ are L -formulae. Moreover, a modal lattice = (A, , ) satisfies the inequality ϕ ≤ ψ if and only if the modal consequence pair ϕ ψ is valid on . So it suffices to show that filter extensions preserve validity of modal consequence pairs. So suppose ϕ ψ. Then the dual modal L-space * validates ϕ ψ, and by Theorem 4.41 we have π * ϕ ψ. This implies (π * ) † ϕ ψ, and since this is exactly the filter extension of we have proven the theorem.
Next we focus on d-persistence. While we do not have an explicit example of a modal consequence pair that is not d-persistent, we have the following conjecture. We expect that a proof can be given inspired by [20]. The following lemma plays a key role in the Sahlqvist canonicity theorem.
Since we have z ∈ [R](X \ a i ) iff R[z] ⊆ (X \ a i ) for all z ∈ X, this implies R[c] ⊆ (X \ a 1 ) ∪ · · · ∪ (X \ a n ). Then a 1 ∩ · · · ∩ a n is a clopen neighbourhood of y disjoint from R[c], as required.
(3) As a consequence of Definition 4.1(4) and item (1) R n [x] is closed under meets for all natural numbers n. This implies that ↑R n [x] is a filter. It follows from items (2), Lemma 2.10 and Lemma 4.16 that it is also closed.
Proof. The proof is essentially the same as that of Theorem 3.38. Using Lemma 4.45 one can show that minimal valuations are closed.
As a corollary we obtain completeness with respect to classes of modal L-fames for extensions of L with Sahlqvist consequence pairs. 4.47 Theorem. Let Γ be a set of Sahlqvist consequence pairs and write MLFrm(Γ) for the class of modal L-frames that validate all consequence pairs in Γ. Then for all ϕ, ψ ∈ L we have ϕ ⊢ Γ ψ iff ϕ MLFrm(Γ) ψ.

4.48
Remark. Similarly to Remark 3.41 we point out that our notion of d-persistence corresponds algebraically not to canonical extensions, as in the setting of distributive modal logics, but to F 2 -completions. The reason for this is that our d-persistence allows us to move from valuations of clopen filters to valuations of all filters. Algebraically this corresponds exactly to F 2 -completion, while canonical extensions correspond to taking saturated filters (Theorem 2.26). ⊳

Conclusion
We have given a new duality for bounded (not necessarily distributive) lattices which more closely resembles Stone-type dualities than existing dualities. It arises as a restriction of a known duality for the category of bounded meet-semilattices given by Hofmann, Mislove and Stralka [27]. The relation between our duality and the duality by Hofmann, Mislove and Stralka is similar to the relation between Esakia duality and Priestley duality. It can also be seen as a Stone-type analogue of Jipsen and Moshier's spectral duality for lattices [34]. One of the advantages of the duality for bounded lattices presented in this paper is that it more closely resembles known dualities used in (modal) logic. Consequently, it allows us to use similar tools and techniques. To showcase this, we proved Sahlqvist correspondence and canonicity results along the lines of [6].
Furthermore, we extended the duality to a duality for a modal extension based on positive (non-distributive) logic with a and . While these are interpreted using a relation in the way as in normal modal logic over a classical base, the changed interpretation of joins in our frame caused to no longer be join-preserving. This interesting phenomenon has also been observed in the context of modal intuitionistic logic [31].
There are many intriguing avenues for further research, some of which we list below.
Algebraic characterisation of the double filter completion. As noted at the end of Section 2.4, it is not yet know if and how we can characterise the double filter extension algebraically. While it follows from the definition that it is compact, it seems likely that the double filter completion is only "half dense." Finite model property. While it is easy to derive the finite model property for positive (nondistributive) logic, the same result for the modal extension presented in this paper appears to be non-trivial.
Relation to ortho(modular) lattices. Ortholattices and orthomodular lattices provide other interesting classes of (not necessarily distributive) lattices with operators. However, in ortholattices the orthocomplement is turning joins into meets. Duality for these structures has been discussed by Goldblatt [18,19] and Bimbo [5]. In [13, Chapter 6] a duality developed in this paper is extended to modal operators that turn joins into meets. These are called ∇-algebras there. Recently modal ortholattices have been studied in [28]. We leave it an an interesting open problem to see whether the Sahlqvist correspondence and canonicity results of this paper can be extended to ∇-algebras. It is also open whether these technique could be extended to orthomodular lattices [30]. This is especially interesting as orthomodalur lattices provide algebraic structures of quantum logic [12] and therefore these methods could be relevant in the study of quantum logic.

Different modalities.
Yet another question is what other modal extensions of positive (nondistributive) logic we can define. In particular, it would be interesting to define a form of neighbourhood semantics based on the L-frames used in this paper and investigate the behaviour of the resulting modalities.