Miroslav Plo
email: miroslav.ploscica@upjs.sk

Š Čica

Friedrich Wehrung
email: friedrich.wehrung01@unicaen.fr

SPECTRAL SUBSPACES OF SPECTRA OF ABELIAN LATTICE-ORDERED GROUPS IN SIZE ALEPH ONE

Keywords: 2010 Mathematics Subject Classification. 06D05, 06D20, 06D35, 06D50, 06F20, 46A55, 52A05, 52C35 Lattice-ordered, Abelian, group, vector lattice, ideal, homomorphic image, completely normal, distributive, lattice, countable, tree, relatively complete, joinirreducible, Heyting algebra, closed map, consonance, spectrum

may come

Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

Miroslav Ploščica, Friedrich

Introduction

A subset I, in a lattice-ordered group (in short ℓ-group) G, is an ℓ-ideal if it is an order-convex normal subgroup closed under the lattice operations. If I = G, we say in addition that I is prime if x ∧ y ∈ I implies that {x, y} ∩ I = ∅, whenever x, y ∈ G. In case G is Abelian, the ℓ-spectrum of G is defined as the set Spec G of all prime ℓ-ideals of G, endowed with the topology whose closed subsets are the {P ∈ Spec G | X ⊆ P } for X ⊆ G (often called the hull-kernel topology). Denote by G the class of all Abelian ℓ-groups.

The problem of the description of ℓ-spectra of all Abelian ℓ-groups (say the ℓ-spectrum problem) is stated, in the language of MV-algebras, in Mundici [START_REF] Mundici | Advanced Lukasiewicz Calculus and MV-Algebras[END_REF]Problem 2]. Now under Stone duality (cf. Grätzer [6, § II.5], Johnstone [8, § II.3], Rump and Yang [START_REF] Rump | The essential cover and the absolute cover of a schematic space[END_REF] for the case without top element, and Wehrung [ [4, Theorem 2], via a counterexample of cardinality ℵ 1 , that those properties are not sufficient to characterize Id c G. On the other hand, the second author proved in [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF] that every countable completely normal distributive 0-lattice belongs to Id c G. The categorical concept of condensate, initiated in the second author's work [START_REF] Gillibert | From Objects to Diagrams for Ranges of Functors[END_REF] with Pierre Gillibert, together with the main result of [START_REF]Cevian operations on distributive lattices[END_REF], enabled the second author to prove in [START_REF]From noncommutative diagrams to anti-elementary classes[END_REF] that Id c G is not the class of models of any class of L ∞λ sentences of lattice theory, for any infinite cardinal λ. Using further tools from infinitary logic, the second author extended those results in [START_REF]Projective classes as images of accessible functors, hal-03580184[END_REF] by proving that Id c G is not the complement of a projective class over L ∞∞ , thus verifying in particular that the additional property of all lattices Id c G coined by the first author in his proof of [START_REF] Ploščica | Cevian properties in ideal lattices of Abelian ℓ-groups[END_REF]Theorem 2.1] is still not sufficient to characterize Id c G.

Delzell and Madden observed in

As observed in the above-cited references, all those results extend to the class of all (lattice) homomorphic images of lattices Id c G. On the other hand, not every homomorphic image of a lattice of the form Id c G belongs to Id c G (cf. Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Example 10.6]). Recast in terms of spectra, via Stone duality, this means that not every spectral subspace of an ℓ-spectrum is an ℓ-spectrum.

Moreover, not every completely normal bounded distributive lattice is a homomorphic image of some Id c G: a counterexample of cardinality ℵ 2 is constructed in Wehrung [START_REF]Cevian operations on distributive lattices[END_REF].

In this paper we complete the picture above, by establishing that every completely normal distributive 0-lattice D, with at most ℵ 1 elements, is a homomorphic image of Id c G for some Abelian ℓ-group G. This also strengthens the first author's result, obtained in [START_REF] Ploščica | Cevian properties in ideal lattices of Abelian ℓ-groups[END_REF], that D is Cevian. In fact, we verify the slightly more general statement that G may be taken a vector lattice over any given countable totally ordered division ring k (cf. Theorem 7.4), modulo the obvious change in the definition of an ℓ-ideal (i.e., ℓ-ideals need to be closed under scalar multiplication by elements of k; see Wehrung [17, § 2.3] for more detail). Due to the results of [17, § 9], the countability assumption on k cannot be dispensed with.

Our argument will roughly follow the one from Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF], with the "Main Extension Lemma" [14, Lemma 4.2] strengthened from finite lattices to certain infinite lattices, and streamlined via the introduction of consonance kernels (cf. Definition 3.1), as Lemma 4.4. The proof of the "closure step" [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.2] fails in that more general context, so we get only "homomorphic image" as opposed to "isomorphic copy", of Id c G. This will also require a few known additional properties of finite distributive lattices and their homomorphisms, via Birkhoff duality (see in particular Lemma 2.4). Our final argument, given a completely normal distributive 0-lattice L, will start by expressing L as a directed union of an ascending ω 1 -sequence L = (L ξ | ξ < ω 1 ) of countable completely normal distributive 0-lattices, and then, with the help of Lemma 4.4, iteratively lift all subdiagrams (L ξ | ξ < α), with α < ω 1 , with respect to the functor Id c . That part of our argument turns out to be valid not only for the chain ω 1 but for any tree in which every element has countable height (cf. Theorem 7.3).

Basic concepts

2.1. Sets, posets, lattices. For any set X, Pow X denotes the powerset algebra of X. By "countable" we will mean "at most countable". For an element a in a partially ordered set (from now on poset ) P , we set P ↓ a def = {p ∈ P | p ≤ a} (or ↓a if P is understood). A subset A of P is a lower subset of P if P ↓ a ∈ A whenever a ∈ A. A poset P with bottom element is a tree if P ↓ a is well-ordered under the induced order whenever a ∈ P .

For a subset P in a poset Q and for x ∈ Q, x P (resp., x P ) denotes the least y ∈ P such that x ≤ y (resp., the largest y ∈ P such that y ≤ x) if it exists. We say that P is relatively complete in Q if x P and x P both exist for all x ∈ P . If P is a subalgebra of a Boolean algebra Q, it suffices to verify that x P exists whenever x ∈ Q (resp., x P exists whenever x ∈ Q).

Relative completeness has been used in a description of projective Boolean algebras. For the proof of the following (easy) assertion see Heindorf and Shapiro [7, Lemma 1.2.7].

Lemma 2.1. Let A, A ′ be subalgebras of a Boolean algebra B with A ′ finitely generated over A. If A is relatively complete in B, then so is A ′ .

For posets P and Q with respective top elements ⊤ P and ⊤ Q , a map f :

P → Q is top-faithful if f -1 {⊤ Q } = {⊤ P }.
For any poset P , P ⊔∞ denotes the poset obtained by adding an extra element, usually denoted by ∞, atop of P . For any map f : P → Q, we denote by f ⊔∞ : P ⊔∞ → Q ⊔∞ the unique extension of f sending ∞ to ∞. Such maps are exactly the top-faithful maps from P ⊔∞ to Q ⊔∞ .

We denote by Ji L (resp., Mi L) the set of all join-irreducible (resp., meet-irreducible) elements in a lattice L, endowed with the induced ordering. For any join-irreducible element p in a finite distributive lattice D, we denote by p * the unique lower cover of p in D, and by p † the largest element of D not above p; so p * = p ∧ p † . The assignment p → p † defines an order-isomorphism from Ji D onto Mi D.

As in Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF][START_REF]Real spectra and ℓ-spectra of algebras and vector lattices over countable fields[END_REF], two elements a and b in a 0-lattice (i.e., lattice with a bottom element) D are consonant if there exist u, v ∈ D such that a ≤ u ∨ b, b ≤ a ∨ v, and u ∧ v = 0. A subset X of D is consonant if any pair of elements in X is consonant. The lattice D is completely normal if it is consonant within itself.

We denote by Ji L (resp., Mi L) the set of all join-irreducible (resp., meet-irreducible) elements in a lattice L, endowed with the induced partial ordering. The assignment D → Ji D is part of Birkhoff duality between finite distributive lattices, with 0, 1-lattice homomorphisms, and finite posets, with isotone maps (cf. Grätzer [6, § II. 1.3]). The Birkhoff dual of a 0, 1-lattice homomorphism ϕ : D → E is the map Ji E → Ji D, q → q ϕ def = min {x ∈ D | q ≤ ϕ(x)}. For any distributive 0-lattice D, we denote by BR(D) the generalized Boolean algebra R-generated by D in the sense of Grätzer [6, § II.4] (aka the Boolean envelope of D). Equivalently, BR(D) is the universal generalized Boolean algebra of D. Up to isomorphism, BR(D) is the unique generalized Boolean algebra generated by D as a 0-sublattice. The assignment D → BR(D) canonically extends to a functor, which turns 0-lattice embeddings to embeddings of generalized Boolean algebras. For a 0-sublattice D of a distributive lattice E with 0, we will thus identify BR(D) with its canonical image in BR(E). If D is a finite distributive lattice and P def = Ji D, then the assingment x → P ↓ x defines an isomorphism from D onto the lattice Down P of all lower subsets of P . Since the universal Boolean algebra of Down P is the powerset lattice of P , with each {p} = (↓p) \ (↓p) * , it follows that the atoms of BR(D) are exactly the p ∧ ¬p * for p ∈ Ji D.

Lemma 2.2. The following statements hold, for any distributive 0-lattice D:

(1) For all a 1 , a 2 , b 1 , b 2 ∈ D, a 1 ∧ ¬b 1 ≤ a 2 ∧ ¬b 2 within BR(D) iff a 1 ≤ a 2 ∨ b 1 and a 1 ∧ b 2 ≤ b 1 within D. ( 2
Proof. ϕ(a) ≤ ϕ(b) ∨ c iff BR(ϕ)(a ∧ ¬b) ≤ c, iff BR(ϕ)(p ∧ ¬p * ) ≤ c whenever p ∈ Ji D such that p ≤ a and p b (we apply Lemma 2.2(2)). Now BR(ϕ)(p ∧ ¬p * ) ≤ c iff ϕ(p) ≤ ϕ(p * ) ∨ c.
For any elements x and y in a lattice E let x → E y denote the largest z ∈ E, if it exists, such that x ∧ z ≤ y (it is also called the pseudocomplement of x relative to y); so → E is the Heyting implication on E. If → E is defined on every pair of elements then we say that E is a generalized Heyting algebra. If, in addition, E has a bottom element, then we say that E is a Heyting algebra. Every Heyting algebra is a bounded distributive lattice, and every finite distributive lattice is a Heyting algebra 1 .

Dually, we denote by x E y the least z ∈ E, if it exists, such that x ≤ y ∨ z. It is the dual pseudocomplement of x relative to y.

A lattice homomorphism ϕ :

D → E is closed if whenever a 0 , a 1 ∈ D and b ∈ E, if ϕ(a 0 ) ≤ ϕ(a 1 ) ∨ b, then there exists x ∈ D such that a 0 ≤ a 1 ∨ x and ϕ(x) ≤ b.
If ϕ is an inclusion map we will say that D is a closed sublattice of E.

The following folklore lemma, whose easy proof we leave to the reader as an exercise, enables to read, on the Birkhoff dual, whether a given homomorphism, between finite distributive lattices, is a homomorphism of Heyting algebras or a closed homomorphism, respectively.

Lemma 2.4. The following statements hold, for any finite distributive lattices D and E and any 0, 1-lattice homomorphism ϕ : D → E:

(1) ϕ is a homomorphism of Heyting algebras iff for all p ∈ Ji D and all q ∈ Ji E, if p ≤ q ϕ , then there exists x ∈ Ji E such that x ≤ q and x ϕ = p. (2) ϕ is closed iff for all p ∈ Ji D and all q ∈ Ji E, if q ϕ ≤ p, then there exists

x ∈ Ji E such that q ≤ x and x ϕ = p.

2.2.

The lattices Bool(F, Ω), Op(F, Ω), and Op -(F, Ω). For more detail on this subsection we refer the reader to Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF][START_REF]Real spectra and ℓ-spectra of algebras and vector lattices over countable fields[END_REF]. For a right vector space E over a totally ordered division ring k, a map f :

E → k is an affine functional if f -f (0)
is a linear functional. Note that the affine functionals on E form a left vector space over k.

For functions f and g with common domain Ω and values in a poset T , we set

[[f ≤ g]] def = {x ∈ Ω | f (x) ≤ g(x)}; and similarly for [[f < g]], [[f = g]], [[f = g]],
and so on. Throughout this paper, f and g will always be restrictions, to a convex set Ω, of continuous affine functionals on a topological vector space E over a totally ordered division ring k. For a set F of maps from Ω to k, we will denote by Bool(F, Ω) the Boolean subalgebra of the powerset of Ω generated by all subsets [[f > 0]] and [[f < 0]] for f ∈ F. As in [START_REF]Real spectra and ℓ-spectra of algebras and vector lattices over countable fields[END_REF], we will also denote by Op -(F, Ω) the sublattice of Bool(F, Ω) generated by all [[f > 0]] and [[f < 0]] where f ∈ F, and then set Op(F, Ω) def = Op -(F, Ω) ∪ {Ω}. Evidently, Bool(F, Ω) is generated, as a Boolean algebra, by its 0-sublattice Op(F, Ω); so Bool(F, Ω) = BR Op(F, Ω) .

For any set I and any totally ordered division ring k, we will occasionally identify every element a = (a i | i ∈ I) ∈ k (I) with the corresponding (continuous) linear functional i∈I a i δ i (where δ i denotes the ith projection), thus justifying such notations as Bool(k (I) , k (I) ) and Op(k (I) , k (I) ); observe that in those notations, the first (resp., second) occurrence of k (I) is endowed with its structure of left (resp., right ) vector space over k. Moreover, in its second occurrence, k (I) is endowed with the coarsest topology making all canonical projections δ i continuous.

Denote by F ℓ (I, k) the free left 2 k-vector lattice on a set I. As observed in Baker [START_REF] Baker | Free vector lattices[END_REF], Bernau [START_REF] Bernau | Free abelian lattice groups[END_REF], Madden [9, Ch. III] (see also Wehrung [17,page 13] for a summary), F ℓ (I, k) canonically embeds into k k (I) . We sum up a few related facts. Lemma 2.5 (Folklore).

(1) F ℓ (I, k) is isomorphic to the sublattice of k k (I) generated by all linear functionals

i∈I a i δ i associated to elements a ∈ k (I) , via the assignment i → δ i . (2) The assignment x → [[x = 0]] defines an isomorphism from the lattice Id c F ℓ (I, k),
of all principal ℓ-ideals of the left k-vector lattice F ℓ (I, k), onto Op -(k (I) , k (I) ).

Consonance kernels

In this section we introduce a tool, the consonance kernels, expressing the consonance of the image a lattice homomorphism via its behavior on join-irreducible elements. e p ∧ e q = 0 , whenever p, q ∈ P are incomparable .

We then set x

⊘ e y def = {e p | p ∈ (P ↓ x) \ (P ↓ y)}, whenever x, y ∈ D. Lemma 3.2. In the context of Definition 3.1, f (x) = f (x ∧ y) ∨ (x ⊘ e y) whenever x, y ∈ D. Moreover, f is a lattice homomorphism. Proof. Setting c def = f (x ∧ y) ∨ (x ⊘ e y), it is obvious that c ≤ f (x).
In order to prove that f (x) ≤ c, it suffices to prove that f (p) ≤ c whenever p ∈ P ↓ x. By way of contradiction, let p be a minimal element of

P ↓x with f (p) c. Since p ≤ y implies f (p) ≤ f (x ∧ y) ≤ c, we get p ∈ (P ↓ x) \ (P ↓ y), so f (p) = f (p * ) ∨ e p . Since e p ≤ c, we get f (p * ) c. The case p * = 0 is impossible, because f (0) ≤ f (x ∧ y) ≤ c. Since f is a join-homomorphism, we get f (p * ) = {f (q) | q ∈ P ↓ p * }.
By the minimality assumption on p, we get f (q) ≤ c for every q ∈ P ↓ p * , hence f (p * ) ≤ c, a contradiction.

2 "Right" and "left" appear to have been unfortunately mixed up at various places in [START_REF]Real spectra and ℓ-spectra of algebras and vector lattices over countable fields[END_REF], particularly on pages 12 and 13. Since this is mostly a matter of choosing sides, that paper's results are unaffected. We nonetheless attempt to fix this here. Proof. Suppose first that the range of f is consonant in L. Since D is finite, there exists a finite 0-sublattice

K of L, containing f [D], such that the range of f is consonant in K. Setting e p def = f (p) K f (p * ) for each p ∈ Ji D, Condition (3.1) is obviously satisfied. Let p, q ∈ Ji D be incomparable. From p ∧ q ≤ p * we get e p = f (p) K f (p * ) ≤ f (p) K f (p ∧ q) = f (p) K (f (p) ∧ f (q)) = f (p) K f (q) ,
and, similarly, e q ≤ f (q) K f (p). Since f (p) and f (q) are consonant within K, we get (f (p) K f (q)) ∧ (f (q) K f (p)) = 0; whence e p ∧ e q = 0.

Let, conversely, (e p | p ∈ Ji D) be a consonance kernel for f and set

P def = Ji D. Let x, y ∈ D, set u def = x ⊘ e y and v def = y ⊘ e x. It follows from Lemma 3.2 that f (x) ≤ f (y) ∨ u and f (y) ≤ f (x) ∨ v.
Moreover, for all p ∈ (P ↓ x) \ (P ↓ y) and q ∈ (P ↓ y) \ (P ↓ x), p and q are incomparable, thus e p ∧ e q = 0; whence u ∧ v = 0. Therefore, the pair (u, v) witnesses the consonance of f (x) and f (y) in L.

An extension lemma for infinite distributive lattices

This section's main result, Lemma 4.4, states conditions under which a homomorphism f : D → L of distributive lattices can be extended to a homomorphism f : E → L in case E is generated over D by two disjoint elements a and b. One of its main improvements, over the original [14, Lemma 4.2] it stems from, is the possibility of D be infinite. Definition 4.1. A 0, 1-sublattice D of a bounded distributive lattice E is a semi-Heyting sublattice if for all x, y ∈ D, x → D y and x → E y both exist and are equal.

In particular, every semi-Heyting sublattice of E is a Heyting algebra (E itself may not be a Heyting algebra). Notation 4.2. Let D be a finite 0, 1-sublattice of a bounded distributive lattice E and let f : D → L be a 0-lattice homomorphism. We set

f e (a) def = {e p | p ∈ Ji D , p ≤ p * ∨ a}
for every consonance kernel e of f and every a ∈ E.

The following lemma arises from Wehrung [START_REF]Real spectra and ℓ-spectra of algebras and vector lattices over countable fields[END_REF]Remark 4.6]. We include a proof for convenience. Lemma 4.3. Let D be a finite semi-Heyting sublattice of a bounded distributive lattice E, let f : D → L be a 0-lattice homomorphism, and let a, b ∈ E such that a ∧ b = 0. Then any join-irreducible elements p and q in D such that p ≤ p * ∨ a and q ≤ q * ∨ b are incomparable. In particular, f e (a) ∧ f e (b) = 0 for any consonance kernel e for f . Proof. Suppose otherwise, say p ≤ q; thus p † ≤ q † . From a ∧ b = 0 we get p∧b ≤ (p * ∨a)∧b = p * ∧b ≤ p * , thus, by assumption, b ≤ p → E p * = p → D p * = p † . Since p † ≤ q † , we get b ≤ q † , so q ≤ q * ∨ b ≤ q † , a contradiction.

We are now reaching this section's main goal. In the next proof we use the following well known extension criterion. Let D and L be distributive lattices and X a generating subset of D. Then a map f : X → L can be extended to a (necessarily unique) homomorphism g : D → L if and only if

m i=1 x i ≤ n j=1 y j =⇒ m i=1 f (x i ) ≤ n j=1 f (y j ) (4.1)
for all m, n > 0 and all x 1 , . . . , x m , y 1 , . . . , y n ∈ X. Note. By the same token as the one used in the proof of Lemma 2.3, the condition that BR(f )(a B ) ≤ α is equivalent to saying that for all x, y ∈ D, x ≤ y ∨ a ⇒ f (x) ≤ f (y) ∨ α. By Lemma 2.3, if D is finite, then it suffices to restrict ourselves to the case where x = p ∈ Ji D and y = p * . Note that BR(f )(a B ) is an element of BR(L), usually not in L, so it cannot be taken as the lowest possible value of α a priori.

Proof. We start by proving (4.2). By (3), there is an expression of the form c B = i<n (¬u i ∨ v i ) (within B) where n < ω and all u i , v i ∈ D. For each i < n, c ≤ ¬u i ∨ v i within BR(E), thus u i ∧ c ≤ v i , and thus, since D is a semi-Heyting

sublattice of E, c ≤ u i → E v i = u i → D v i ; whence, setting w def = i<n (u i → D v i ), we get c ≤ w. For each i < n, w ≤ u i → D v i with w ∈ D, thus u i ∧ w ≤ v i , so w ≤ ¬u i ∨ v i within B, and so w ≤ c B . Since w ∈ D, it follows that w = c B = c D .
Now it is obvious that for every lattice homomorphism g : E → L extending f , the pair (α, β)

def = (g(a), g(b)) satisfies α ≤ f (a B ), β ≤ f (b B ), α ∧ β = 0, BR(f ) a B ≤ α, and BR(f ) b B ≤ β.
Let, conversely, (α, β) be such a pair.

We need to show the implication (4.1) for x i , y j ∈ D ∪ {a, b}. Since f is a lattice homomorphism, we can assume that exactly one x i and exactly one y j belong to D.

Since a ∧ b = 0, the inequality x ∧ a ≤ y ∨ b is equivalent to x ∧ a ≤ y. So, (4.1) boils down to the equality α ∧ β = 0 (which is assumed) and the following implications:

x ≤ y ∨ a ⇒ f (x) ≤ f (y) ∨ α ; (4.3) x ≤ y ∨ b ⇒ f (x) ≤ f (y) ∨ β ; (4.4) x ≤ y ∨ a ∨ b ⇒ f (x) ≤ f (y) ∨ α ∨ β ; (4.5) x ∧ a ≤ y ⇒ f (x) ∧ α ≤ f (y) ; (4.6) x ∧ b ≤ y ⇒ f (x) ∧ β ≤ f (y) . (4.7)
The implications (4.3) and (4.4) follow from BR(f ) a B ≤ α and BR(f ) b B ≤ β.

Owing to Condition (4), the implication (4.5) follows from the inequalities For

BR(f ) (a ∨ b) B = BR(f ) a B ∨ b B = BR(f )(a B ) ∨ BR(f )(b B ) ≤ α ∨ β . Suppose that x ∧ a ≤ y. Since D is a semi-Heyting sublattice of E, it follows that a ≤ x → E y = x → D y, thus, using (4.2), a D = a B ≤ x → D y. It follows that α ≤ f (a B ) ≤ f (x → D y), thus f (x) ∧ α ≤ f (x) ∧ f (x → D y) ≤ f (
every p ∈ Ji D ′ , p ≤ p * ∨ a (within E) implies that p ≤ p * ∨ a B (within D ′ ), thus, since p ∈ Ji D ′ , we get p ≤ a B , whence e p ≤ f (p) ≤ f (a B ). This proves that α ≤ f (a B ). Similarly, β ≤ f (b B ). Further, the equation α ∧ β = 0 follows from Lemma 4.3.
Let c ∈ {a, b} and let x, y ∈ D such that x ≤ y ∨ c, we need to prove that Using (4.9), we get

f (x) ≤ f (y) ∨ f ′ e (c). From x ∧ ¬y ≤ c (within BR(E)) it follows that x ∧ ¬y ≤ c B (within BR(D)). Set X def = {p ∈ Ji D ′ | p ∧ ¬p * ≤ c B } = {p ∈ Ji D ′ | p ≤ p * ∨ c}.
f (x) ∧ ¬f (y) = BR(f )(x ∧ ¬y) ≤ BR(f )(c B ) ≤ f ′ e (c) , so f (x) ≤ f (y) ∨ f ′ e (c).
5. Adjunctions between lattices Bool(F, k (I) )

Throughout this section k will be a totally ordered division ring. In this section we shall state a few properties of Boolean algebras of the form Bool(F, Ω), mostly related to relative completeness between such algebras.

The following observation is contained in the proof of Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.6].

Lemma 5.1. Let Ω be a convex subset in a right vector space E over k and let F∪{a} be a set of affine functionals on E.

Set A + def = [[a > 0]] and A -def = [[a < 0]]. Then for every U ∈ Bool(F, Ω), if U ⊆ A + ∪ A -, then there are U + , U -∈ Bool(F, Ω) such that U = U + ∪ U -whereas U + ⊆ A + and U -⊆ A -.
Proof. Since U is the union of finitely many cells, each of which being the intersection of finitely many sets of the form either [[±f > 0]] or [[±f ≥ 0]] where f ∈ F, it suffices to consider the case where U is such a cell. If U meets both A + and A -, pick x ∈ U ∩ A + and y ∈ U ∩ A -; so a(x) > 0 and a(y) < 0. Then λ def = (a(y)-a(x)) -1 a(y) belongs to the open interval ]0, 1[ and a(xλ+y(1-λ)) = 0, that is, xλ + y(1 -λ) / ∈ A + ∪ A -. On the other hand, since U is convex, xλ + y(1 -λ) ∈ U ; a contradiction since U ⊆ A + ∪ A -. Therefore, U is disjoint either from A + or from A -, thus it is contained either in A + or in A -. Corollary 5.2. In the context of Lemma 5.2, (A + ∪ A -) Bool(F,Ω) exists iff both (A + ) Bool(F,Ω) and (A -) Bool(F,Ω) exist, and then

(A + ∪ A -) Bool(F,Ω) = (A + ) Bool(F,Ω) ∪ (A -) Bool(F,Ω) .
In what follows we will identify every element f ∈ k (I) with the associated linear functional on k (I) , that is, x → i∈I f i x i . Moreover, whenever I ⊆ J we will identify k (I) with the subset of k (J) consisting of all vectors with support contained in I. Notation 5.3. For I ⊆ J, we define mappings

ε I,J : Pow k (I) → Pow k (J) , ρ ∨ J,I , ρ ∧ J,I : Pow k (J) → Pow k (I) , by ε I,J (X) = y ∈ k (J) | y↾ I ∈ X , ρ ∧ J,I (Y ) = x ∈ k (I) | (∀y ∈ k (J) )(y↾ I = x ⇒ y ∈ Y ) , ρ ∨ J,I (Y ) = {y↾ I | y ∈ Y } .
The following statements are immediate consequences of the definitions:

Extending a top-faithful map

In Lemmas 6.1 and 6.2 we fix a totally ordered division ring k. The following lemma takes care of the "domain step" required in the proof of Theorem 7.4. Lemma 6.1. Let I and J be sets, let L be a completely normal distributive 0lattice, let D be a finite subset of k (J) , and let e ∈ k (J) . Then every top-faithful 0-lattice homomorphism f : Op(k (I) ∪ D, k (J) ) → L ⊔∞ extends to a top-faithful lattice homomorphism g : Op(k

(I) ∪ D ∪ {e} , k (J) ) → L ⊔∞ (cf. Figure 6.1). L ⊔∞ Op(k (I) ∪ D, k (J) ) Op(k (I) ∪ D ∪ {e} , k (J) )) f ⊆ g Figure 6.1. A commutative triangle for Lemma 6.1 Proof. Set E def = D ∪ {e}, D def = Op(k (I) ∪ D, k (J) ), E def = Op(k (I) ∪ E, k (J) ), B def = BR(D) = Bool(k (I) ∪ D, k (J)
), and

C def = Bool(k (I) ∪ E, k (J) ).
f ′ def = f ↾ D ′ has a consonance kernel (e P | P ∈ Ji D ′ )
. By Lemma 4.4, f extends to a unique lattice homomorphism g : D → L such that g(x) = f ′ e (x) whenever x ∈ {a, b}. For any P ∈ Ji D ′ such that P ⊆ P * ∪ x, 0 / ∈ P * ∪ x, thus 0 / ∈ P , that is, P is not the top element of Op(k (I) , k (J) ). Since f is top-faithful, it follows that e P ≤ f (P ) < ∞; whence f ′ e (x) < ∞. It follows that g is top-faithful. The "surjectivity step" is much easily taken care of: Lemma 6.2. Let I and J be sets with I ⊂ J and J \I infinite, let L be a distributive 0-lattice, let D be a finite subset of k (J) , and let c ∈ L. Then every for every topfaithful 0-lattice homomorphism f : Op(k (I) ∪ D, k (J) ) → L ⊔∞ , there are e ∈ k (J) and a top-faithful lattice homomorphism g : Op(k

(I) ∪ D ∪ {e} , k (J) ) → L ⊔∞ such that g(e) = c.
Proof. Since D is finite and J \ I is infinite, there exists j ∈ J \ I not in the support of any element of D. Take e def = δ j , the jth canonical projection k (J) ։ k. By the argument of Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 8.3], Op(k (I) ∪D∪{δ j } , k (J) ) is the (internal) free amalgamated sum of Op(k

(I) ∪ D, k (J) ) and ∅, [[δ j > 0]], [[δ j < 0]], [[δ j = 0]], k (J)
within the category of bounded distributive lattices. Hence f extends to a unique lattice homomorphism g : Op(k (I) ∪ D ∪ {δ j } , k (J) ) → L such that g([[δ j > 0]]) = c and g([[δ j < 0]]) = 0. Since c < ∞ and f is top-faithful, it follows that g is also top-faithful.

Representing trees of countable lattices

In this section we will reach the paper's main goal, Theorem 7.4, which states that every completely normal distributive 0-lattice is a homomorphic image of some Id c F for some k-vector lattice F . In order to reach that result we will in fact prove (cf. Theorem 7.3) the apparently stronger statement that every diagram of countable completely normal distributive 0-lattices, indexed by a tree in which every element has countable height, can be represented in that fashion.

Towards that goal, our main technical tool is the following "one-step extension" theorem, which relies on the results of Section 6, together with the observation that for F ⊆ k (I) , Op(F, k (I) ) = Op -(F, k (I) ) ⊔ {∞} (where ∞ denotes here the full space k (I) ; so the top-faithful maps Op(F, k (I) ) → L ⊔∞ are exactly the g ⊔∞ where g : Op -(F, k (I) ) → L).

Theorem 7.1. Let k be a countable totally ordered division ring, let I and J be countable sets with I ⊂ J and J \ I infinite, let K and L be distributive 0-lattices with L countable completely normal, let ϕ : K → L be a 0-lattice homomorphism, and let f : Op -(k (I) , k (I) ) → K be a 0-lattice homomorphism. Then there exists a surjective lattice homomorphism g : Op

-(k (J) , k (J) ) ։ L such that g • ε I,J = ϕ • f .
The settings for Theorem 7.1 can be read on Figure 7.1. Its proof can be followed on

Given an extension g

n : Op -(k (I) ∪ D n , k (J) ) → L of g 0 def = ϕ • f , where D n ⊂ k (J) is finite, we extend the top-faithful extension g ⊔∞ n : Op(k (I) ∪ D n , k (J) ) → L ⊔∞ of g n to a top-faithful lattice homomorphism g ⊔∞ n+1 : Op(k (I) ∪ D n+1 , k (J) ) → L ⊔∞ , with D n ⊆ D n+1 , v ⌊n/2⌋ ∈ D n+1 if n
is even (via Lemma 6.1), and c ⌊n/2⌋ ∈ rng g n+1 if n is odd (via Lemma 6.2). The common extension g of all g n is as required.

By virtue of Lemma 2.5, Theorem 7.1 can be recast in terms of ℓ-ideal lattices of free vector lattices over k, as follows.

Theorem 7.2. Let k be a countable totally ordered division ring, let I and J be countable sets with I ⊂ J and J \ I infinite, let K and L be distributive 0lattices with L countable completely normal, let ϕ : K → L be a 0-lattice homomorphism, and let f : Id c F ℓ (I, k) → K be a 0-lattice homomorphism. Denote by η I,J : Id c F ℓ (I, k) ֒→ Id c F ℓ (J, k) the canonical embedding. Then there exists a surjective lattice homomorphism g : Id c F ℓ (J, k) ։ L such that g • η I,J = ϕ • f . By using the functoriality of the assignment I → Id c F ℓ (I, k), Theorem 7.2 can further be extended to diagrams indexed by trees, as follows.

Theorem 7.3. Let k be a countable totally ordered division ring, let T be a tree in which every element has countable height, and let

L def = (L s , ϕ s,t | s ≤ t in T ) be a commutative T -indexed diagram of distributive 0-lattices such that L t is countable completely normal whenever t ∈ T \ {⊥}. Let I ⊥ ⊆ {⊥} × ω and set I t def = (T ↓ t) × ω whenever t ∈ T \ {⊥}. Set I def = (I s , η Is,It | s ≤ t in T ). Then every 0-lattice homomorphism χ ⊥ : Id c F ℓ (I ⊥ , k) → L ⊥ extends to a natural trans- formation χ : Id c F ℓ ( I, k)
. → L such that χ t is a surjective lattice homomorphism whenever t ∈ T \ {⊥}.

Proof. The proof can be partly followed on Figure 7.3. By Zorn's Lemma, there This shows that our conclusion holds at T ′ ∪ {t}, in contradiction with the maximality assumption on T ′ . This leads us to the following positive solution of the problem stated at the end of Wehrung [START_REF]Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF]. By applying Stone duality for distributive 0-lattices, we obtain the following formulation in terms of spectra.

L s L <t L t Id c F ℓ (I s , k) Id c F ℓ (I <t , k) Id c F ℓ (I t , k)
Corollary 7.6. Every completely normal generalized spectral space with at most ℵ 1 compact open sets embeds, as a spectral subspace, into the ℓ-spectrum of an Abelian ℓ-group. Corollary 7.5 also strengthens Ploščica [START_REF] Ploščica | Cevian properties in ideal lattices of Abelian ℓ-groups[END_REF]Theorem 3.2], which states that every completely normal distributive 0-lattice of cardinality at most ℵ 1 is Cevian; that is, it carries a binary operation (x, y) → x y such that x ≤ y ∨ (x y), (x y) ∧ (y x) = 0, and x z ≤ (x y) ∨ (y z) for all x, y, z. Indeed, Id c G is Cevian for any Abelian ℓ-group G, and any homomorphic image of a Cevian lattice is Cevian (cf. Wehrung [15, § 5]).

Problem. Let D be a completely normal distributive 0-lattice such that for all a, b ∈ D there exists a sequence (c n | n < ω) from D such that for all x ∈ D, a ≤ b ∨ x iff there exists n < ω such that c n ≤ x (in [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF] we say that D has countably based differences). If card D = ℵ 1 , does D ∼ = Id c G for some Abelian ℓ-group G?

The cases where card D ≤ ℵ 0 and card D ≥ ℵ 2 are settled in Wehrung [START_REF] Wehrung | Spectral spaces of countable Abelian lattice-ordered groups[END_REF][START_REF]Cevian operations on distributive lattices[END_REF], in the positive and the negative, respectively (the counterexample constructed in [START_REF]Cevian operations on distributive lattices[END_REF] is not even Cevian, thus it is not a homomorphic image of any Id c G). A Cevian counterexample (of size continuum plus) is constructed in Ploščica [START_REF] Ploščica | Cevian properties in ideal lattices of Abelian ℓ-groups[END_REF].

  17, § 2.2] for a summary), for any G ∈ G, Spec G corresponds to the lattice Id c G of all principal ℓ-ideals of G; that is, Id c G = { a | a ∈ G + } where each a def = {x ∈ G | (∃n ∈ N)(|x| ≤ na)}. This enables us to restate the ℓ-spectrum problem as the description problem of the class Id c G def = {D | (∃G ∈ G)(D ∼ = Id c G)}. All such lattices are clearly distributive with smallest element (usually denoted by 0). They are also completely normal (cf. Bigard, Keimel, and Wolfenstein [3, Ch. 10]), that is, they satisfy the statement (∀a, b)(∃x, y)(a ∨ b = a ∨ y = x ∨ b and x ∧ y = 0) .

)Lemma 2 . 3 .

 23 If D is finite, then a∧¬b = {p ∧ ¬p * | p ∈ Ji D , p ≤ a , p b} within BR(D), whenever a, b ∈ D. Let D and L be distributive 0-lattices with D finite, let ϕ : D → L be a 0-lattice homomorphism, let a, b ∈ D, and let c ∈ L. Then ϕ(a) ≤ ϕ(b) ∨ c iff ϕ(p) ≤ ϕ(p * ) ∨ c whenever p ∈ Ji D with p ≤ a and p b.

Definition 3 . 1 .

 31 Let D and L be distributive lattices, with D finite and L with a zero element, and let f : D → L be a join-homomorphism. Set P def = Ji D. A consonance kernel for f is a family (e p | p ∈ P ) of elements of L such that f (p) = f (p * ) ∨ e p , whenever p ∈ P ; (3.1)

  Now let x, y ∈ D. By the result of the paragraph above,f (x) = f (x∧y)∨(x ⊘ e y) and f (y) = f (x ∧ y) ∨ (y ⊘ e x). Due to (3.2), (x ⊘ e y) ∧ (y ⊘ e x) = 0; whence f (x) ∧ f (y) = f (x ∧ y).Lemma 3.3. Let D and L be distributive lattices, with D finite and L with a zero element. Then a lattice homomorphism f : D → L has a consonance kernel iff the range of f is consonant in L.

Lemma 4 . 4 ( 3 )

 443 Main Extension Lemma). Let D be a semi-Heyting sublattice of a bounded distributive lattice E and let a, b ∈ E. Setting B def = BR(D), we assume the following: (1) E is generated, as a lattice, by D ∪ {a, b}. (2) a ∧ b = 0. (All elements a B , b B , (a ∨ b) B , a B , and b B are defined.(4) (a ∨ b) B = a B ∨ b B . Then c B ∈ D whenever c ∈ {a, b, a ∨ b} .(4.2)Further, for every 0-lattice homomorphism f : D → L and all α, β ∈ L, the following conditions are equivalent.(i) (α, β) = (g(a), g(b)) for some lattice homomorphism g :E → L extending f ; (ii) α ≤ f (a B ), β ≤ f (b B ), α ∧ β = 0, BR(f ) a B ≤ α,and BR(f ) b B ≤ β. Moreover, for any finite semi-Heyting sublattice D ′ of D such that {a B , b B } ⊆ BR(D ′ ) and a B , b B ⊆ D ′ , and any consonance kernel e of f ′ def = f ↾ D ′ , (f ′ e (a), f ′ e (b)) is a pair satisfying (ii).

  y). The implication (4.6) follows. The proof of (4.7) is similar.For the remainder of the proof, let D ′ be a finite semi-Heyting sublattice of D such that {a B , b B } ⊆ BR(D ′ ) and a B , b B ⊆ D ′ (cf.

Figure 4 . 1 )Figure 4 . 1 .

 4141 Figure 4.1. Illustrating the proof of Lemma 4.4

By ( 3 )

 3 and since BR(D ′ ) is a finite Boolean algebra with atoms p∧¬p * for p ∈ Ji D ′ , c B = {p ∧ ¬p * | p ∈ X} within B . (4.8) By the definition of X, f (p) = f (p * ) ∨ e p ≤ f (p * ) ∨ f ′ e (c) whenever p ∈ X , so f (p) ∧ ¬f (p * ) ≤ f ′ e (c) within BR(L), whenever p ∈ X; whence, using (4.8), BR(f )(c B ) = {f (p) ∧ ¬f (p * ) | p ∈ X} ≤ f ′ e (c) within BR(L) . (4.9)

By Proposition 5. 5 ,B

 5 is relatively complete in C. In particular, setting a def = [[e > 0]] and b def = [[e < 0]], the elements a B , b B , a B , b B , and (a ∨ b) B are all defined. By Corollary 5.2, (a ∨ b) B = a B ∨ b B . Let D ′ be a finite subset of k (I) ∪ D such that a B , b B , a B , and b B all belong to B ′ def = Bool(D ′ , k (J) ). By Wehrung [14, Lemma 5.4] (see also Wehrung [17, Lemma 4.1] for the more general form of that statement), D is a Heyting subalgebra of E and D ′ def = Op(D ′ , k (J) ) is a Heyting subalgebra of D. Since L is completely normal and f [D ′ ] is finite, it follows from Lemma 3.3 that

Figure 7 Figure 7 . 1 .Figure 7 . 2 . 1 Proof.

 771721 Figure 7.1. A commutative diagram for Theorem 7.1

Theorem 7 . 4 .Corollary 7 . 5 .

 7475 Let k be a countable totally ordered division ring. Then every completely normal distributive 0-lattice L with at most ℵ 1 elements is a surjective homomorphic image of Id c F for some vector lattice F over k.Proof. Write L as the directed union of an ascending ω 1 -sequence L = (L ξ | ξ < ω 1 ) of countable completely normal distributive 0-lattices, with L 0 = {0}. Theorem 7.3, applied to the well-ordered chain ω 1 , yields anω 1 -indexed commutative diagram F = (F ξ , f ξ,η | ξ ≤ η < ω 1 )of k-vector lattices together with a natural transformation χ : Id c F . → L all of whose components are surjective lattice homomorphisms. Letting F def = lim -→ F , the universal property of the colimit yields a surjective homomorphism from Id c F onto L.Due to Wehrung [17, Corollary 9.5], Theorem 7.4 cannot be generalized to uncountable totally ordered division rings k. On the other hand, setting k as any countable Archimedean totally ordered field (for example the rationals), Id c F is identical to the ℓ-ideal lattice of the underlying ℓ-group of F . Hence, Every completely normal distributive 0-lattice L with at most ℵ 1 elements is a surjective homomorphic image of Id c F for some Abelian ℓ-group F .

  ′ where s ≤ s ′ < t and limiting maps ϕ s,<t : L s → L <t for s < t) and the set Id c F ℓ (I t , k)and ϕ <t,t : L <t → L t , such that η I<t,It • η Is,I<t = η Is,It and ϕ <t,t • ϕ s,<t = ϕ s,t I<t,It = ϕ <t,t • χ <t . Therefore, for each s < t, χ t • η Is,It = χ t • η I<t,It • η Is,I<t = ϕ <t,t • χ <t • η Is,I<t = ϕ <t,t • ϕ s,<t • χ s = ϕ s,t • χ s .

				ϕs,t
			ϕs,<t	ϕ<t,t
			χs	lim -→	χ<t
			ηI s ,I <t lim -→	ηI <t ,I t
				ηI s ,I t
	ϕ s,s I <t	def =	def = lim -→s<t {I s | s < t}. The universal property of the colimit ensures the existence L s (with transition maps
	of unique 0-lattice homomorphisms
			η I<t,It : Id c F ℓ (I <t , k) = lim -→
	whenever s < t. Further, the natural transformation (χ s | s < t) induces a unique
	0-lattice homomorphism
			χ

χt Figure 7.3. Illustrating the proof of Theorem 7.3

exists a maximal lower subset T ′ of T , containing {⊥}, on which the conclusion of Theorem 7.3 holds. Suppose, by way of contradiction, that T ′ = T and let t be a minimal element of T \ T ′ ; so T ′ ∪ {t} is also a lower subset of T . Since the height of t is countable, so are the lattice L <t s<t

Id c F ℓ (I s , k) → <t : Id c F ℓ (I <t , k) → L <t such that χ <t • η Is,I<t = ϕ s,<t

• χ s whenever s < t. By Theorem 7.2, there exists a surjective lattice homomorphism χ t : Id c F ℓ (I t , k) ։ L t such that χ t • η

Strictly speaking we should set the Heyting implication → apart from the lattice signature, thus for example stating that "every finite distributive lattice expands to a unique Heyting algebra". The shorter formulation, which we shall keep for the sake of simplicity, reflects a standard abuse of terminology that should not create any confusion here.

• ε I,J is an embedding of Boolean algebras, ρ ∧ J,I is a meet-homomorphism, and ρ ∨ J,I is a join-homomorphism. Moreover, ρ ∧ J,I and ρ ∨ J,I are right and left adjoint to ε I,J , respectively.

• ρ ∧ J,I and ρ ∨ J,I are conjugate, that is,

Lemma 5.4. Let I and J be sets with I ⊆ J. The following statements hold:

Proof. Ad (1) and ( 2) are both trivial. In order to prove (3), it suffices, since ρ ∧ J,I and ρ ∨ J,I are conjugate, to establish the result for ρ ∨ J,I . For every X ∈ Bool(k

The set V of all (m + n)-tuples of elements of k of the form (a ′′ 0 (z), . . . , a ′′ m-1 (z), b ′′ 0 (z), . . . , b ′′ n-1 (z)) is a vector subspace of k m+n . Hence, an element x ∈ k (I) belongs to ρ ∨ J,I (Y ) iff there exists u ∈ V such that a ′ i (x) + u i ≥ 0 whenever i < m and b ′ j (x) + u m+j > 0 whenever j < n. Since membership in V , of any (m + n)-tuple of elements of k, can be expressed by a finite set of linear equations, the statement that a given x ∈ k (I) belongs to ρ ∨ J,I (Y ) can be expressed by a sentence, over the first-order language L def = {<, 0, -, +} ∪ {•λ | λ ∈ k} of ordered Abelian groups augmented with right scalar multiplications by elements of k, in (a ′ 0 (x), . . . , a ′ m-1 (x), b ′ 0 (x), . . . , b ′ n-1 (x)). Now every L-sentence is equivalent, over all nonzero totally ordered right k-vector spaces, to a quantifier-free L-sentence (cf. van den Dries [13, Corollary I.7.8]). Therefore, ρ ∨ J,I (Y ) belongs to Bool(F, k (I) ) for a finite set F of linear combinations of the a ′ i and the b ′ j .

Proposition 5.5. Let I and J be sets with I ⊆ J and let D be a finite subset of k (J) . Then Bool(k (I) ∪ D, k (J) ) is relatively complete in Bool(k (J) , k (J) ).

Proof. We first prove that Bool(k