Friedrich Wehrung
email: friedrich.wehrung01@unicaen.fr

Klaus Keimel

REAL SPECTRA AND ℓ-SPECTRA OF ALGEBRAS AND VECTOR LATTICES OVER COUNTABLE FIELDS

Keywords: 2010 Mathematics Subject Classification. 14P10, 12D15, 13J30, 46A55, 06D05, 06F20 Brumfiel spectrum, real spectrum, ℓ-spectrum, flat triangulation, completely normal, lattice

In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Combining those methods with Baro's Normal Triangulation Theorem, we obtain the following result:

Theorem. For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.

The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

Introduction

The same way the usual Zariski spectrum of a commutative unital ring A topologizes the collection of all surjective homomorphisms from A onto a domain, the real spectrum of A topologizes the collection of all surjective homomorphisms from A onto a totally ordered domain. However, while the class of all topological spaces that arise as Zariski spectra of commutative unital rings has been well understood since Hochster [START_REF] Hochster | Prime ideal structure in commutative rings[END_REF] (we get exactly the so-called spectral spaces), the corresponding problem for real spectra has been open for decades, and dates back at least to Problem 12 in Klaus Keimel's 1995 survey paper [START_REF] Keimel | Some trends in lattice-ordered groups and rings, Lattice theory and its applications[END_REF]. Observe that the order structure of real spectra (without the topology) got completely elucidated in Dickmann, Gluschankof, and Lucas [START_REF] Dickmann | The order structure of the real spectrum of commutative rings[END_REF].

Building on Delzell and Madden's construction of a completely normal spectral space which cannot arise as a real spectrum [START_REF] Delzell | A completely normal spectral space that is not a real spectrum[END_REF], Mellor and Tressl [START_REF] Mellor | Non-axiomatizability of real spectra in L ∞λ[END_REF] proved that the class R of all Stone duals of real spectra (which is thus a class of bounded distributive lattices) is not closed under L ∞λ -elementary equivalence, for any infinite cardinal λ. As usual, L ∞λ denotes the extension of first-order logic allowing conjunctions and disjunctions of arbitrary length and universal or existential quantifiers over strings of variables of length less than λ. There is not much room for improvement of Mellor and Tressl's solution beyond L ∞λ : an infinite distributive lattice D belongs to R iff it carries a ring structure R and an isomorphism from the Stone dual of the real spectrum of R onto D; thus R is the class of lattice reducts of the class of all models of some L ω1ω sentence in a larger first-order language, and thus it is a so-called projective class (cf. the argument concluding the Introduction in Wehrung [START_REF]From non-commutative diagrams to anti-elementary classes[END_REF]).

Mellor and Tressl's solution involves (necessarily) the construction of lattices of arbitrarily large cardinality. The main result of the present paper is thus a positive counterpart of Mellor and Tressl's negative solution (cf. Corollary 8.7):

Theorem A. Let k be a countable formally real field. Every second countable completely normal spectral space arises as the real spectrum of some commutative unital k-algebra.

Our road to Theorem A involves an extension of the tools introduced, in the author's paper [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF], for proving an analogous result for the corresponding problem on Abelian ℓ-groups. Those tools are a blend of lattice theory and semilinear geometry, and they were originally focused on finite lattices of polyhedral cones in real vector spaces. One of the main underlying lattice-theoretical concepts of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF] is the one of consonance (cf. Section 2.1). We will need to introduce here the lattice-theoretical concept of CN-purity (cf. Definition 7.8) and relate it to Baro's Normal Triangulation Theorem from [START_REF] Baro | Normal triangulations in o-minimal structures[END_REF] (cf. Proposition 7.9).

The main result of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF] is a precursor of Theorem A for ℓ-spectra of Abelian ℓgroups. The concept of ℓ-spectrum can in turn be extended to (right) vector lattices over totally ordered division rings k, by allowing the ℓ-ideals to be closed under the action of k. A byproduct of our study is the following version of Theorem A for vector lattices, which also extends the main result of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF] to those structures (cf. Corollary 6.3).

Theorem B. Let k be a countable totally ordered division ring. Every second countable completely normal spectral space arises as the ℓ-spectrum of some right k-vector lattice.

Analogues of Mellor and Tressl's result, for spectra of various classes of ℓ-groups (including all Abelian ones), also answering related questions from Iberkleid et al. [START_REF] Iberkleid | Conrad frames[END_REF], were recently obtained by the author in [START_REF]Cevian operations on distributive lattices[END_REF][START_REF]From non-commutative diagrams to anti-elementary classes[END_REF].

We also prove that the countability assumption on k, in both Theorems A and B, cannot be dispensed with (cf. Corollary 9.7):

Theorem C. There exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring or the real spectrum of any commutative unital algebra over an uncountable field.

Theorem C follows from a more general result, Theorem 9.3, that states that if a homomorphic preimage of a distributive lattice D can be represented via a vector lattice on a directed partially ordered division ring of cardinality greater than the one of D, then every indecomposable element of D is join-irreducible.

We also prove (cf. Corollary 10.7) that a completely normal generalized spectral space X can be represented via vector lattices over arbitrary totally ordered division rings iff the Stone dual of X is a generalized dual Heyting algebra. The real spectrum analogue of that result fails (cf. Example 10.9).

Basic concepts

2.1. Lattices. A standard reference on lattice theory is Grätzer [START_REF] Grätzer | Lattice Theory: Foundation[END_REF]. A nonzero element p in a lattice L with zero is [START_REF] Baker | Free vector lattices[END_REF] indecomposable if (p = x ∨ y and x ∧ y = 0) ⇒ (x = 0 or y = 0), (2) join-irreducible if p = x ∨ y ⇒ (p = x or p = y),

(3) join-prime if p ≤ x ∨ y ⇒ (p ≤ x or p ≤ y), for all x, y ∈ L. In particular, (3)⇒( 2)⇒(1), and (2)⇔(3) if L is distributive. We will denote by Ji L the set of all join-irreducible elements in L. If L is finite distributive, then for every p ∈ Ji L there exists a largest p † ∈ L such that p p † . Furthermore, p ∧ p † = p * , the unique lower cover of p in L, and the assignment p → p † is order-preserving.

We say that a distributive lattice D is a generalized dual Heyting algebra if for all a, b ∈ D there exists a smallest x ∈ D, then denoted a D b, such that a ≤ b ∨ x.

We denote by → D the Heyting implication in any Heyting algebra D. That is, for a, b ∈ D, a → D b is the largest x ∈ D such that a ∧ x ≤ b.

Two elements a and b in a distributive lattice D with zero are consonant (cf. Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Definition 2.2]) if there are x, y ∈ D such that a ≤ b ∨ x, b ≤ a ∨ y, and x∧y = 0. A lattice map f : D → E is consonant if f (x) and f (y) are consonant whenever x, y ∈ D. We say that D is completely normal if every pair of elements in D is consonant.

For join-semilattices A and B, a join-homomorphism f : A → B is closed (cf. Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Definition 2.4]) if for all a 0 , a 1 ∈ A and b ∈ B, if f (a 0 ) ≤ f (a 1 ) ∨ b, then there exists x ∈ A such that a 0 ≤ a 1 ∨ x and f (x) ≤ b.

Stone duality for distributive lattices.

A topological space X is generalized spectral if it is sober and the set • K(X) of all compact open subsets of X is a basis of the topology of X, closed under binary intersection. If, in addition, X is compact, we say that X is a spectral space. A map, between generalized spectral spaces, is spectral if the inverse image of any compact open subset is compact open. Borrowing notation from Rump and Yang [START_REF] Rump | The essential cover and the absolute cover of a schematic space[END_REF], let us denote by DL 0 the category of all distributive 0-lattices with 0-lattice homomorphisms with cofinal1 range, and by GSp the category of all generalized spectral spaces with spectral maps. Stone duality, originating in Stone [START_REF] Marshall | Topological representations of distributive lattices and Brouwerian logics[END_REF], states an equivalence of categories between DL 0 and the opposite category of GSp. That duality sends every distributive lattice D with zero to its spectrum, which is the space Spec D of all prime ideals of D with the hull-kernel topology; in the other direction, it sends every generalized spectral space X to the lattice • K(X). The unit and counit of the duality are given by

ε D : D ∼ = → • K(Spec D) , a → {P ∈ Spec D | a / ∈ P } , whenever D ∈ DL 0 ; η X : X ∼ = → Spec • K(X) , x → V ∈ • K(X) | x / ∈ V , whenever X ∈ GSp .
For more detail about Stone duality, see Grätzer [15, § II.5] (where the functorial aspects are not treated), Johnstone [18, § II.3] (where the more standard case of bounded distributive lattices, and spectral spaces, is handled), and Rump and Yang [27, page 63] for the general case (in that paper generalized spectral spaces are called schematic spaces).

A generalized spectral space X is completely normal if for any points x and y in the closure of a singleton {z}, either x belongs to the closure of {y} or y belongs to the closure of {x}. It follows from Monteiro [25, Théorème V.3.1] that X is a completely normal space iff • K(X) is a completely normal lattice.

2.3.

Vector lattices, ℓ-spectra. Denote by G + the positive cone of any partially ordered group G. A division ring k, endowed with a translation-invariant partial ordering, is a partially ordered division ring if k ++ def = k + \ {0} is a (nonempty) multiplicative subsemigroup of k, closed under ξ → ξ -1 . It is only in Sections 9 and 10 that the partially ordered division rings in question will not necessarily be totally ordered.

Let k be a partially ordered division ring. A right vector space V over k, endowed with a translation-invariant lattice order, is a vector lattice if

V + k + ⊆ V + . Then xλ ∨ yλ = (x ∨ y)λ and xλ ∧ yλ = (x ∧ y)λ , for all x, y ∈ V and λ ∈ k + . (2.1)
Define ℓ-k-homomorphisms as the maps preserving both the ℓ-group structure and the vector space structure. As customary in module theory, the structure of right vector space of V over k will be emphasized by writing V k instead of just V. A linear subspace (not only an additive subgroup) of V is an ℓ-ideal of V k if it is both order-convex and closed under x → |x|. The lattice Id ℓ V k of all ℓ-ideals of V k is algebraic (i.e., it is complete and every element is a join of compact elements; see Grätzer [START_REF] Grätzer | Lattice Theory: Foundation[END_REF]). Now assume that k is directed (equivalently, k = k + + (-k + )). Then the ℓ-ideals of V k are exactly the ℓ-ideals of the underlying Abelian ℓ-group of V which are closed under multiplication by positive scalars. Further, the usual argument, involving (2.1), shows that the collection Id ℓ c V k of all finitely generated (equivalently, principal) ℓ-ideals of V k is a 0-sublattice of Id ℓ V k . The elements of Id ℓ c V k are those of the form

x ℓ k def = {y ∈ V | (∃λ ∈ k + )(|y| ≤ |x|λ)} for x ∈ V. The lattice Id ℓ c V k is a homomorphic image of the lattice of all convex additive ℓ-subgroups of V (via x ℓ → x ℓ k ), thus it is completely normal (cf. Madden [20, § I.2]
). The ℓ-spectrum of V k , denoted by Spec ℓ V k , can be defined as the Stone dual of the lattice Id ℓ c V k . It is a completely normal generalized spectral space.

2.4. f -rings, k-algebras, Brumfiel spectrum, and real spectrum. Here we shall only set up some basic facts and notation over the structures in question, referring the reader to Delzell and Madden [START_REF]Lattice-ordered rings and semialgebraic geometry. I, Real analytic and algebraic geometry[END_REF], Johnstone [18, Chapter 5], Keimel [START_REF] Keimel | Some trends in lattice-ordered groups and rings, Lattice theory and its applications[END_REF], Coste and Roy [START_REF] Coste | La topologie du spectre réel, Ordered fields and real algebraic geometry[END_REF], Dickmann [START_REF] Dickmann | Applications of model theory to real algebraic geometry. A survey[END_REF]Chapter 6], or Wehrung [34, § 4] for more information. Recall (cf. Bigard et al. [START_REF] Bigard | Groupes et Anneaux Réticulés[END_REF]) that an f-ring is a (not necessarily unital) ring A endowed with a translation-invariant lattice order for which A + • A + ⊆ A + and x ∧ y = 0 implies that x ∧ yz = x ∧ zy = 0 whenever x, y ∈ A and z ∈ A + . A subset I of A is an ℓ-ideal if it is simultaneously an ℓ-ideal of the underlying ℓgroup of A and a two-sided ideal of the underlying ring of A. The lattice Id r A of all radical 2 ℓ-ideals of a commutative f -ring A is algebraic, and the collection Id r c A of all its finitely generated (equivalently, principal) members is a completely normal 2 An ideal I of a commutative ring A is radical if x n ∈ I implies that x ∈ I whenever x ∈ A and n is a positive integer. The Stone dual of Id r c A is the Brumfiel spectrum Spec B A of A. A ring A, endowed with an additional structure of right vector space over a field k, is a k-algebra if (xy) • λ = (x • λ)y = x(y • λ) whenever x, y ∈ A and λ ∈ k. If A is unital then this amounts to a unital ring homomorphism from k to A with central range (viz. λ → 1 A • λ). If, in addition, k is a partially ordered field and A is an f -ring with A + • k + ⊆ A + , we say that A is an f-k-algebra (cf. Delzell and Madden [11, § 6] for the unital case with k totally ordered).

Note 2.1. For a commutative f -k-algebra A, the meaning of x r , for x ∈ A, is unambiguous with respect to the action of k, for whenever x ∈ A and λ ∈ k,

(x • λ) 2 = (x • λ 2 )x ∈ x r whence x • λ ∈ x r .
Let A be a commutative unital ring. A subset of A is a cone if it is both an additive and a multiplicative submonoid of A, containing all squares in A. A cone P of A is prime if A = P ∪(-P ) and P ∩(-P ) is a prime ideal of A. The real spectrum of A is the collection Spec r A of all prime cones of A, endowed with the topology generated by all subsets of the form {P ∈ Spec r A | a / ∈ P } for a ∈ A.

2.5. Semi-algebraic sets and maps. For more background, we refer the reader to Bochnak et al [START_REF] Bochnak | Real Algebraic Geometry[END_REF], van den Dries [START_REF] Van Den Dries | Tame Topology and o-Minimal Structures[END_REF]. A field k is • formally real if -1 is not a sum of squares in k;

• real-closed if it is formally real, every polynomial of odd degree with coefficients in k has a root in k, and every element of k is either x 2 or -x 2 for some x ∈ k. For a real-closed field k and a nonnegative integer d, a subset of k d is semialgebraic if it is a Boolean combination of subsets of the form x ∈ k d | f (x) ≥ 0 for polynomials f ∈ k[x 1 , . . . , x d ]. By the Tarski-Seidenberg quantifier elimination theorem, a subset of k d is semi-algebraic iff it is first-order definable (with parameters) in the language of ring theory.

For semi-algebraic subsets A ⊆ k m and B ⊆ k n , a map f : A → B is semialgebraic if its graph is a semi-algebraic subset of k m+n . 2.6. A final bit of notation. The operators of closure and interior, relative to a subspace Ω in a given topological space, will be denoted by cl Ω and int Ω , respectively. If Ω is the ambient space we will omit the subscript Ω in that notation.

For any functions f , g, any subset Ω of the intersection of the domains of f and g, and any binary relation ✁,

we will set [[f ✁ g]] Ω def = {x ∈ Ω | f (x) ✁ g(x)}.
Throughout the paper "countable" will mean "at most countable", and ω will denote the set of all nonnegative integers, also identified with the first infinite ordinal. We will denote by card X the cardinality of any set X.

Semilinear triangulation over totally ordered fields

Results from this section will be required in Sections 7 and 8. Our main sources will be Coppel [START_REF] Andrew | Foundations of Convex Geometry[END_REF], Rourke and Sanderson [START_REF] Rourke | Introduction to Piecewise-Linear Topology[END_REF], Schrijver [START_REF] Schrijver | Theory of Linear and Integer Programming[END_REF], van den Dries [START_REF] Van Den Dries | Tame Topology and o-Minimal Structures[END_REF].

First some standard terminology. Let V be a vector space over a totally ordered field k. A map f :

V → k is an affine functional if f -f (0) is a linear functional. A closed (resp., open) affine half-space of V is a subset of V of the form {x ∈ V | f (x) ≥ 0} (resp., {x ∈ V | f (x) > 0}) for some nonconstant affine func- tional f on V. A subset C of V is • semilinear if it is a Boolean combination of half-spaces,
• a polytope if if it is the convex hull of a finite subset X of V. Proposition 3.1. Let V be finite-dimensional. Then the polytopes of V are exactly the bounded intersections of finite collections of closed affine half-spaces of V. Proposition 3.1 is usually stated in the literature over real vector spaces (see, e.g., Ziegler [START_REF] Günter | Lectures on Polytopes[END_REF]Theorem 1.1]). Abstracting away some properties of the convex hull operator in V yields linear geometries, for which a generalization of Proposition 3.1, contained in Propositions IV.19 and V.60 in Coppel [START_REF] Andrew | Foundations of Convex Geometry[END_REF], is valid; that result implies that Proposition 3.1 extends to vector spaces over totally ordered division rings. On the other hand, the proof provided in Schrijver [28, Corollary 7.1(c)] carries over almost word for word to the latter context.

For affinely independent elements a 0 , . . . , a n ∈ V, we define

• the simplex [a 0 , . . . , a n ] def = { n i=0 a i λ i | all λ i ≥ 0 , i λ i = 1}, • the open simplex a 0 , . . . , a n def = { n i=0 a i λ i | all λ i > 0 , i λ i = 1}
, which is the relative interior (not to be confused with the topological interior) of [a 0 , . . . , a n ].

A face of a simplex (resp., open simplex) as above is the simplex (resp., open simplex) spanned by a nonempty subset of {a 0 , . . . , a n }. A set K of simplexes is a simplicial complex if any face of a member of K belongs to K and the nonempty intersection of any two members of K belongs to K. The geometric realization of K is its union |K| Proposition 3.2. For all semilinear subsets S, S 1 , . . . , S k of V, with S closed bounded and S 1 , . . . , S k ⊆ S, there exists a simplicial complex K such that S = |K| and each S i is partitioned by K.

Proof. By the result of Exercise (2.14)2, Chapter 8 in van den Dries [START_REF] Van Den Dries | Tame Topology and o-Minimal Structures[END_REF] (stated there over any totally ordered field), there exists a "complex" K 0 , with union S, such that every S i is a union of members of K 0 . By "complex" it is meant there that K 0 is a finite set of open simplexes and for all

C 1 , C 2 ∈ K 0 , if cl C 1 ∩ cl C 2 = ∅, then cl C 1 ∩ cl C 2 = cl C for a common face C (not necessarily in K 0 ) of C 1 and C 2 .
The "closure" K 1 of K 0 , defined on page 121 of van den Dries [START_REF] Van Den Dries | Tame Topology and o-Minimal Structures[END_REF] as the set of all faces of all open simplexes in K 0 , is also a complex, closed in the sense that every face of every member of K 1 belongs to K 1 . It follows that

K def = {cl C | C ∈ K 1 } is a simplicial complex. Since S is closed, |K| = |K 1 | = S.
Since every S i is a union of members of K 0 and every member of K 0 is the relative interior or a member of K (namely its closure), every S i is partitioned by K.

Alternatively, a proof of Proposition 3.2 can also be obtained by adapting the one of Rourke and Sanderson [26, Addendum 2.12] from the reals to arbitrary totally ordered division rings (once again commutativity is not needed). Numerous changes are required in both definitions and arguments. The details, although not hard, are quite tedious.

Lattices of relatively open semilinear sets

Throughout this section we shall fix a totally ordered division ring k endowed with its interval topology, together with a topological right k-vector space V. For any set F of affine functionals on V and any subset Ω of V, we define

• Σ F,Ω def = {[[f > 0]] Ω | f ∈ F} ∪ {[[f < 0]] Ω | f ∈ F}, • Σ F,Ω def = {[[f ≥ 0]] Ω | f ∈ F} ∪ {[[f ≤ 0]] Ω | f ∈ F},
• Bool(F, Ω), the Boolean subalgebra of the powerset of Ω generated by Σ F,Ω (equivalenly, by Σ F,Ω ), • Clos(F, Ω) (resp., Op(F, Ω)), the 0, 1-sublattice of Bool(F, Ω) consisting of all closed (resp., open) members of Bool(F, Ω).

The notation Bool(F, Ω), Clos(F, Ω), Op(F, Ω) thus differs from the one of the author's paper [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF] in two aspects: F is now a set of affine functionals (as opposed to a set of hyperplanes), and the second parameter Ω stands for the ambient space. Aside from those differences, the corresponding material will be very similar to the one introduced in [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF].

A subset of Ω is

• basic, with respect to (F, Ω), if it is the intersection of a finite collection of sets of the form [[f > 0]] Ω , [[f < 0]] Ω , or [[f = 0]] Ω , • basic open, with respect to (F, Ω), if it is the intersection of a finite collec- tion of sets of the form [[f > 0]] Ω or [[f < 0]] Ω .
In particular, Ω is basic open. In case the subset Ω is understood from the context, we will often use the shorthand

f 0 = f 0 def = [[f = 0]] Ω ;
(4.1)

f + def = [[f > 0]] Ω ; f -def = [[f < 0]] Ω ; f + def = [[f ≥ 0]] Ω ; f -def = [[f ≤ 0]] Ω (4.2)
for every f : V → k.

The following lemma originates, in a more restrictive context, in Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 5.4]. We present an alternative, streamlined proof. Lemma 4.1. Let Ω be a convex subset of V and let F be a set of continuous affine functionals on V. Then Bool(F, Ω) is closed under both topological interior and closure operators relative to Ω, and Op(F, Ω) is a Heyting subalgebra of the Heyting algebra of all relative open subsets of Ω. Furthermore, the 0-sublattice Op

-(F, Ω) of Op(F, Ω) generated by Σ F,Ω satisfies Op(F, Ω) = Op -(F, Ω) ∪ {Ω}.
Proof. The first part of Lemma 4.1 boils down to establishing that whenever A is nonempty basic in Bool(F, Ω), its closure cl Ω (A) belongs to Bool(F, Ω) (for then, Bool(F, Ω) is closed under topological closure). There are a finite subset F 0 of F and an assignment ε : F 0 → {0, +, -} such that (applying Notation (4.1), (4.2))

A = f ∈F0 f ε(f ) . The closure of A relative to Ω is contained in B def = f ∈F0 f ε(f ) .
Pick a ∈ A. Since every member of F 0 is an affine functional and since Ω is convex,

for every b ∈ B the half-open segment [a, b) is contained in A; thus b ∈ cl Ω (A).
This completes the proof that B = cl Ω (A).

For the last part of Lemma 4.1, it suffices to observe that every proper basic open subset of Ω belongs, by definition, to Op -(F, Ω). Notation 4.2. For every U ∈ Op(F, Ω), we set

F U def = {f ∈ F | U ∩ [[f = 0]] Ω = ∅}. The set ∇U def = ([[f = 0]] Ω | f ∈ F U ) is a closed affine subspace of V.
The following result is an analogue of Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.4]. Since [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 5.1] fails to relativize to arbitrary convex subsets of V, we need to modify the original proof of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.4], taking the opportunity to slightly amplify its original statement. Lemma 4.3. Let Ω be a convex subset of V and let F be a finite set of continuous affine functionals on V. Then a member P of Op(F, Ω) is join-irreducible in Op(F, Ω) iff it is basic open and P ∩ ∇P = ∅. Moreover, for each such P , (1) the lower cover P * of P in Op(F, Ω) is equal to P \ ∇P ;

(2) there exists a unique ε : F \ F P → {+, -} (the "sign function" of P ) such that

(cf. (4.2)) P = f ∈F\FP f ε(f ) , and then cl Ω (P ) = f ∈F\FP f ε(f ) ;
(3) the largest member

P † of Op(F, Ω) not containing P is the complement in Ω of the set cl Ω (P ∩ ∇P ) = cl Ω (P ) ∩ ∇P = f ∈F\FP f ε(f ) ∩ ∇P . (4.3)
Proof. We begin with a claim. Suppose, conversely, that P is basic open and

P ∩ ∇P = ∅. For each f ∈ F \ F P , P is contained in [[f = 0]] Ω = f + ∪ f -(cf. (4.2)), thus, since P is nonempty and convex, P ⊆ f ε(f ) for a unique ε(f ) ∈ {+, -}; whence P ⊆ f ∈F\FP f ε(f ) . Since P is basic open, there are X ⊆ F and η : X → {+, -} such that P = f ∈X f η(f ) . For each f ∈ X, P is contained in f η(f ) thus it is disjoint from f 0 , so f ∈ F \ F P and ε(f ) = η(f ); whence X ⊆ F \ F P , and so P ⊇ f ∈F\FP f ε(f ) . Therefore P = f ∈F\FP f ε(f ) . (4.4)
Now the relative closure of P is obviously contained in

P def = f ∈F\FP f ε(f ) . Con-
versely, picking p ∈ P , any x ∈ P satisfies that the half-open segment [p, x) is contained in P (this is again because Ω is convex and every member of F is an affine functional); thus x ∈ cl Ω (P ). This completes the proof that P = cl Ω (P ), and thus of (2) above.

In order to prove that P is join-irreducible and that (1) holds, it suffices to prove that every join-irreducible element Q of Op(F, Ω) properly contained in P is contained in P \ ∇P . Applying (2), together with the Claim above, to Q, we get a map η :

F \ F Q → {+, -} such that Q = f ∈F\FQ f η(f ) . From Q P and (4.4) it thus follows that F Q F P and η extends ε. Pick f ∈ F P \ F Q . Then Q ⊆ f η(f ) (because f / ∈ F Q ) and ∇P ⊆ f 0 (because f ∈ F P ); whence Q ∩ ∇P = ∅, as required.
The join-irreducibility of P , together with (1), follows.

Since the second equation in (4.3) follows from (2), establishing (4.3) reduces to proving the containment f ∈F\FP f ε(f ) ∩ ∇P ⊆ cl Ω (P ∩ ∇P ). Letting x be any element of the left hand side of that containment, and picking p ∈ P ∩∇P , the halfopen segment [p, x) is contained in P ∩ ∇P ; whence x ∈ cl Ω (P ∩ ∇P ), as required. This completes the proof of (4.3). The statement that P † is the complement in Ω of cl Ω (P ∩ ∇P ) is then proved the same way as at the end of the proof of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.4].

The proof of the following easy corollary is, mutatis mutandis, identical to the one of Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Proposition 6.5], so we omit it.

Corollary 4.4. Let Ω be a convex subset of V, let F be a finite set of continuous affine functionals on V, and let P, Q ∈ Ji Op(F, Ω). If P Q, then ∇Q ∇P .

Although the following observation will not be needed in the remainder of the paper, it might find further uses so we record it here. Proposition 4.5. Let F be a finite set of continuous affine functionals on V and let P , Q be join-irreducible members of Op(F, Ω).

If P ∩ Q = ∅, then P ∩ Q is join-irreducible in Op(F, Ω) and F P ∩Q = F P ∩ F Q . In particular, Ji Op(F, Ω) ∪ {∅} is a meet-semilattice under intersection.
Proof. Pick x ∈ P ∩ ∇P and y ∈ Q ∩ ∇Q, and set z def = 1 2 (x + y). For each f ∈ F, we shall determine the position of z with respect to [[f = 0]]. Denote by ε and η the sign functions of P and of Q, respectively. We separate cases.

• Case 1.

f ∈ ∁F P ∩ ∁F Q . Then P ⊆ f ε(f ) and Q ⊆ f η(f ) . Since P ∩ Q = ∅ it follows that ε(f ) = η(f ), whence x ∈ P ⊆ f ε(f ) and y ∈ Q ⊆ f ε(f ) , so z ∈ f ε(f ) = f η(f ) as well. • Case 2. f ∈ ∁F P ∩ F Q . Then ∇Q ⊆ f 0 (thus f (y) = 0) and P ⊆ f ε(f ) (thus ε(f )f (x) > 0). It follows that ε(f )f (z) > 0, that is, z ∈ f ε(f ) . • Case 3. f ∈ F P ∩ ∁F Q . In a similar fashion as in Case 2, we get z ∈ f η(f ) . • Case 4. f ∈ F P ∩ F Q .
Then ∇P and ∇Q are both contained in f 0 ; whence z ∈ f 0 . Using Lemma 4.3, we get from Cases 1-4 above that

z ∈ P ∩ Q ∩ f 0 | f ∈ F P ∩ F Q . (4.5)
In particular, F P ∩ F Q is contained in F P ∩Q . The converse containment being trivial, we get F P ∩Q = F P ∩ F Q . By Lemma 4.3 together with (4.5), it follows that

P ∩ Q is join-irreducible.
The following Lemma 4.7 extends the author's [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.6] from the linear case to the relativization of the affine case to a convex set. Its proof involves a lattice-theoretical homomorphism extension result established in [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 4.2]. Let us first state a remark about the latter result.

Remark 4.6. In the original statement of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 4.2], the author had overlooked the fact that in the statement of that lemma, Condition (5) follows from Conditions (2) and (3); thus it is redundant. Let us verify this, thus assuming that D is a Heyting subalgebra of a finite distributive lattice E (endowed with its unique structure of Heyting algebra) and a, b ∈ E with a ∧ b = 0. We need to verify that for all join-irreducible p, q ∈ D, if p ≤ p * ∨ a and q ≤ q * ∨ b, then p and q are incomparable. Suppose, to the contrary, that p and q are comparable, say p ≤ q. We get

p ∧ b ≤ (p * ∨ a) ∧ b = (p * ∧ b) ∨ (a ∧ b) ≤ p * , thus, using the assumption that D is a Heyting subalgebra of E, we get b ≤ (p → E p * ) = (p → D p * ) = p † . Now p ≤ q implies p † ≤ q † , so b ≤ q † . But then, q ≤ q * ∨ b ≤ q † , a contradiction. Lemma 4.7.
Let Ω be a convex subset of V, let F ∪ {g} be a finite set of continuous affine functionals on V, and let L be a generalized dual Heyting algebra. Every consonant 0-lattice homomorphism ϕ : Op(F, Ω) → L extends to a unique lattice homomorphism ψ : Op(F∪{g} , Ω) → L such that for every

G ∈ {[[g > 0]] Ω , [[g < 0]] Ω }, ψ(G) = ϕ * (G) def = (ϕ(P ) L ϕ(P * ) | P ∈ Ji Op(F, Ω), P ⊆ P * ∪ G) .
Proof. We verify that Conditions (1)-( 4 Let Ω be a convex subset of V, let F be a finite set of continuous affine functionals on V, and let a, b ∈ F. We set F λ def = F ∪ {a -λb} for evey λ ∈ k. Then for all large enough positive λ ∈ k, the following statement holds: for every generalized dual Heyting algebra L, every consonant 0-lattice homomorphism ϕ : Op(F, Ω) → L can be extended to a lattice homomorphism ψ :

Op(F λ , Ω) → L such that ψ [[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω = ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) . (4.6)
Proof. The proof goes along the same lines as the one of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.1], with a trivialization of the proof of the corresponding Claim, as follows.

Let us state how large λ should be. Set

M def = {U ∈ Op(F, Ω) | [[b > 0]] Ω ⊆ U } and pick x U ∈ [[b > 0]] Ω \ U for every U ∈ M. Let λ be a strict upper bound in k of {0} ∪ a(x U )b(x U ) -1 | U ∈ M . We claim that [[a < λb]] Ω ⊆ U ⇒ [[b > 0]] Ω ⊆ U , whenever U ∈ Op(F, Ω) . Indeed, suppose otherwise. Then U ∈ M by definition, thus λ > a(x U )b(x U ) -1
by assumption on λ, and thus (as b(x U ) > 0) a(x U ) < λb(x U ), and therefore

x U ∈ [[a < λb]] Ω \ U , a contradiction.
This completes the proof of our claim. The remainder of the proof goes exactly like the part of the proof of [35, Lemma 7.1] following its claim, with m replaced by λ and [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.6] by its extension given in Lemma 4.7. Due to the crucial character of Lemma 4.8, we include a proof for convenience.

Let L be a generalized dual Heyting algebra and let ϕ : Op(F, Ω) → L be a consonant 0-lattice homomorphism. We consider the extension ψ of ϕ, to a homomorphism from Op(F λ , Ω) to L, given by Lemma 4.7, with a -λb in place of g. In particular,

ψ([[a > λb]] Ω ) = (ϕ(P ) L ϕ(P * ) | P ∈ Ji Op(F, Ω) , P ⊆ P * ∪ [[a > λb]] Ω ) . (4.7)
We claim that the following inequality holds:

ϕ([[a > 0]] Ω ) ∧ ψ([[a > λb]] Ω ) ≤ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) . (4.8)
Since L is distributive, this amounts to proving the following statement:

ϕ([[a > 0]] Ω ) ∧ ϕ(P ) L ϕ(P * ) ≤ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) ,
for every P ∈ Ji Op(F, Ω) such that

P ⊆ P * ∪ [[a > λb]] Ω . (4.9)
Let P as in the premise of (4.9); so

P ∩ ∇P ⊆ [[a > λb]] Ω . By Lemma 4.3, we get cl Ω (P ) ∩ ∇P = cl Ω (P ∩ ∇P ) ⊆ [[a ≥ λb]] Ω , that is, [[a < λb]] Ω ⊆ P † . From the assumption on λ we thus get [[b > 0]] Ω ⊆ P † , that is, P ⊆ [[b > 0]] Ω . Since [[b > 0]] Ω ∈ Op(F, Ω), it follows that P ∩[[b > 0]] Ω ⊆ P * . Now suppose that P ⊆ [[a > 0]] Ω . Since P ∩ [[b > 0]] Ω ⊆ P * , the inequalities P ⊆ P * ∪ [[a > 0]] Ω and P ∩ [[b > 0]] Ω ⊆ P * both hold, thus also ϕ(P ) ≤ ϕ(P * ) ∨ ϕ([[a > 0]] Ω ) and ϕ(P ) ∧ ϕ([[b > 0]] Ω ) ≤ ϕ(P * ) .
By Lemmas 3.5 and 3.6 in [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF], we get ϕ(P

) L ϕ(P * ) ≤ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ),
which implies (4.9) right away.

It remains to handle the case where

P ⊆ [[a > 0]] Ω . From λ > 0 it follows that [[a > λb]] Ω ⊆ [[a > 0]] Ω ∪ [[b < 0]] Ω , thus P ⊆ P * ∪ [[a > 0]] Ω ∪ [[b < 0]] Ω ,
]] Ω )∧ϕ([[b < 0]] Ω ) = 0 and ϕ([[a > 0]] Ω ) ≤ ϕ([[b > 0]] Ω ) ∨ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) , ϕ([[a > 0]] Ω ) ∧ ϕ(P ) L ϕ(P * ) ≤ ϕ([[a > 0]] Ω ) ∧ ϕ([[b < 0]] Ω ) ≤ ϕ([[b > 0]] Ω ) ∧ ϕ([[b < 0]] Ω ) ∨ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) ∧ ϕ([[b < 0]] Ω ) = ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) ∧ ϕ([[b < 0]] Ω ) ≤ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) ,
thus completing the proof of (4.9) in the general case, and therefore of (4.8).

Now

[[a > 0]] Ω ⊆ [[b > 0]] Ω ∪ ([[a > 0]] Ω ∩ [[a > λb]] Ω ), that is, [[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω ⊆ [[a > 0]] Ω ∩ [[a > λb]] Ω . It follows that ψ([[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω ) ≤ ψ([[a > 0]] Ω ∩ [[a > λb]] Ω ) = ϕ([[a > 0]] Ω ) ∧ ψ([[a > λb]] Ω ) ≤ ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) . Since ϕ([[a > 0]] Ω ) ≤ ϕ([[b > 0]] Ω ) ∨ ψ([[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω ), the converse inequality ϕ([[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) ≤ ψ([[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω )
holds, and therefore ϕ(

[[a > 0]] Ω ) L ϕ([[b > 0]] Ω ) = ψ([[a > 0]] Ω Op(F λ ,Ω) [[b > 0]] Ω ).
An iteration, of finite length 4 • (card F) 2 , of Lemma 4.8, involving a feature of consonance stated in Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 3.9], leads to the following extension, to our context, of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.2]. As Lemma 4.9 is, mutatis mutandis, obtained from Lemma 4.8 the same way as [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.2] is obtained from [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.1], we omit its proof. Lemma 4.9. Let Ω be a convex subset of V, let F be a finite set of continuous affine functionals on V, let L be a completely normal distributive lattice with zero, let ϕ : Op(F, Ω) → L be a 0-lattice homomorphism, and let U, V ∈ Op -(F, Ω) and γ ∈ L such that ϕ(U ) ≤ ϕ(V ) ∨ γ. Then there exists a finite set G, containing F, of linear combinations of F, together with W ∈ Op -(G, Ω) and a lattice homomorphism ψ : Op(G, Ω) → L extending ϕ, such that U ⊆ V ∪ W and ψ(W ) ≤ γ.

Enlarging the range of a homomorphism: semilinear case

Throughout this short section k will be a totally ordered division ring endowed with its interval topology and I will be a set. We endow the right k-vector space k (I) , with basis I, with the topology induced by the product topology on k I where k is given the discrete topology. This section will be devoted to stating the modifications required for adapting to our context the author's results from [35, § 8] regarding the enlargement of the range of a homomorphism from some Op(F, Ω) to a bounded distributive lattice L.

Denote by F I the set of all linear combinations of the form f = i∈S δ i ξ i + λ, for a finite subset S of I and elements ξ i ∈ k \ {0} and λ ∈ k. In particular, f is a continuous affine functional on k (I) . Write supp(f ) The proofs of the following lemmas are, mutatis mutandis, identical to those of Lemmas 8.2 and 8.3 of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF], respectively, and we thus omit them.

Lemma 5.1. Let F ⊆ F I , let S def = supp(F), and let i ∈ I \ supp(F). We denote by ψ : Op(F, k (I) ) → Op(F ∪ {δ i } , k (I) ) the inclusion map and ϕ : Op(F, k (I) ) → Op(F, k (I) ) * J 2 the diagonal map, respectively, and we set

ε(X, Y, Z) def = (X ∩ [[δ i > 0]] k (I) ) ∪ (Y ∩ [[δ i < 0]] k (I) ) ∪ Z for all (X, Y, Z) ∈ Op(F, k (I) ) * J 2 .
Then ε is an isomorphism and ψ = εϕ. Lemma 5.2. Let F ⊆ F I and let i ∈ I \ supp(F). Let L be a bounded distributive lattice and let a, b ∈ L such that a ∧ b = 0. Then every 0, 1-lattice homomorphism ϕ : Op(F, k (I) ) → L extends to a unique lattice homomorphism

ψ : Op(F ∪ {δ i } , k (I) ) → L such that ψ([[δ i > 0]] k (I) ) = a and ψ([[δ i < 0]] k (I) ) = b.

ℓ-spectra of countable vector lattices

The author proved in [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Theorem 9.1] that every second countable completely normal generalized spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. In this section we shall show how to extend that result to ℓ-spectra of right vector lattices over arbitrary countable totally ordered division rings. As unproblematic as that extension may seem, the auxiliary results of Sections 4-5 will also play a crucial role in the proof of the extension of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Theorem 9.1] to f -rings stated in Theorem 8.4.

For a totally ordered division ring k and any set I, denote by F ℓ (I, k) the free right k-vector lattice on I, and by O pl (k (I) ) the sublattice of the powerset of k (I) generated by all subsets [[f > 0]] k (I) where f : k (I) → k is a (finite) linear combination of coordinate projections. Madden's arguments in [20, Ch. III] (in particular Proposition III.1.3 and Lemma III.5.2 from that thesis), themselves extending results from Baker [START_REF] Baker | Free vector lattices[END_REF] and Bernau [START_REF] Bernau | Free abelian lattice groups[END_REF], from subfields of the reals to any totally ordered field, do not require the commutativity of k. Hence, Proposition 6.1. For any totally ordered division ring k and any set I, the assignment

f ℓ k → [[f = 0]] k (I) defines a lattice isomorphism from Id ℓ c F ℓ (I, k) k onto O pl (k (I) ).
Theorem 6.2. Let k be a countable totally ordered division ring. Then every countable, completely normal, distributive lattice L with zero is isomorphic to Id ℓ c V k for some right k-vector lattice V.

Outline of proof. Our proof follows exactly the same lines as the one of Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Theorem 9.1], thus we will just show the required changes:

• The additive subgroup Λ generated by all coordinate projection maps R (ω) ։ R is to be replaced by the subspace Λ k consisting of all linear combinations of coordinate projections k (ω) ։ k. • The notation Op(H), used in [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF] with H a set of (linear) hyperplanes, is replaced here by Op(A, k (ω) ), for a finite subset A of Λ k .

• The step n = 3m involves Lemma 5.2 (linear case, Ω def = k (ω) ) instead of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 8.3].

• The step n = 3m + 1 involves Lemma 4.7 (linear case, Ω def = k (ω) ) instead of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 6.6].

• The step n = 3m + 2 involves Lemma 4.9 (linear case, Ω def = k (ω) ) instead of [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Lemma 7.2].

• The final step, invoking Baker's representation of the ℓ-ideals of the free Abelian ℓ-group on a countably infinite set, is to be replaced by Proposition 6.1.

By applying Stone duality, we thus obtain: Corollary 6.3. Let k be a countable totally ordered division ring. Then every second countable, completely normal generalized spectral space is homeomorphic to the ℓ-spectrum of some right k-vector lattice.

Extending homomorphisms of lattices of open semi-algebraic sets

Now we move to the context of semi-algebraic sets and real spectra. Throughout this section we shall fix a real-closed field k endowed with its interval topology, together with a positive integer d. We shall endow k d with its natural structure of topological vector space over k, and establish a few extension theorems for lattice embeddings between finite lattices of semi-algebraic relatively open subsets of finite powers of the unit interval of k. Those finite lattices will resemble our previously introduced lattices Op(A, Ω), modulo a semi-algebraic homeomorphism whose existence will be ensured by a strong form of triangulation, due to Baro, stated in Proposition 7.3. Definition 7.1. Given semi-algebraic subsets S, S 1 , . . . , S l of k d with all S i ⊆ S, a triangulation of (S; S 1 , . . . , S l ) is a pair (K, τ ) where K is a simplicial complex in k d and τ : S → |K| is a semi-algebraic homeomorphism such that each τ [S i ] is partitioned by K. The Triangulation Theorem for semi-algebraic sets (cf. Bochnak et al. [5, Theorem 9.2.1]) states that every finite sequence (S; S 1 , . . . , S l ), where S and all S i are semi-algebraic, S is closed bounded, and each S i ⊆ S, has a triangulation. Baro strengthens this result in [START_REF] Baro | Normal triangulations in o-minimal structures[END_REF] with his "Normal Triangulation Theorem" , valid in any o-minimal structure over a real-closed field. Baro's [2, Theorem 1.4], together with the remark on top of [2, p. 277], can be formulated as follows.

Proposition 7.3. Let K be a simplicial complex of k d and let S 1 , . . . , S l be semialgebraic subsets of |K|. Then there exists a triangulation (L, ψ) of (|K|; S 1 , . . . , S l ) such that L is a subdivision of K and ψ[S] = S for every open simplex S of K. Observe that the definition of a flatly triangulable lattice involves not only the algebraic structure of the lattice, but also the way it is embedded as a lattice of semialgebraic open sets. Our next lemma states that the polytope Ω of Definition 7.5 can be taken as the geometric realization of a simplicial complex partitioning all elements of Op(A, Ω). Proof. Let Ω, A, τ witness the flat triangulability of D. By Proposition 3.2, there exists a simplicial complex K such that |K| = Ω and every member of Bool(A, Ω) is partitioned by K. Adjoining to A the constant function with value 1 ensures that Op(A, Ω) = Op -(A, Ω).

Lemma 7.7. Every finite subset X of O sa ([-1, 1] d ) is contained in a flatly trian- gulable sublattice of O sa ([-1, 1] d ).
Proof. By the Triangulation Theorem for semi-algebraic sets (cf. Bochnak et al.

[5, Theorem 9.2.1]), there exists a triangulation (K, τ ) of ([-1, 1] d ; (X | X ∈ X)). Now by Proposition 3.1, there exists a finite set A of nonconstant affine functionals on k d such that every simplex in K belongs to Clos(A, k d ). For every X ∈ X, τ [X] is partitioned by K, thus it belongs to Bool(A, k d ); it is also a relative open subset of |K|, thus it in fact belongs to Op(A, |K|). Therefore, X is contained in the flatly triangulable lattice τ -1 Op(A, |K|).

The following lattice-theoretical concept will be repeatedly used throughout the remainder of the paper. Definition 7.8. Let A be a 0-sublattice of a distributive lattice B with zero. We say that A is CN-pure in B if every 0-lattice homomorphism from A to a completely normal distributive lattice L with zero extends to a lattice homomorphism from B to L.

Most of our interest in flat triangulability arises from the following result. We conclude this section with a variant of Lemma 5.1 tailored to the semialgebraic case.

Lemma 7.10. Let I ∪ {j} be a finite set, with j / ∈ I, and let D be a 0, 1-sublattice

of O sa ([-1, 1] I ). We set ε(X, Y, Z) def = (X ×(0, 1])∪(Y ×[-1, 0))∪(Z ×[-1, 1]) whenever (X, Y, Z) ∈ D * J 2 .
Then ε defines a 0, 1-lattice embedding from D * J 2 into O sa ([-1, 1] I∪{j} ). Furthermore, if D is flatly triangulable, then so is the range of ε.

Proof. The set T def = ε(X, Y, Z) determines the triple (X, Y, Z) ∈ D * J 2 , because X = t ∈ [-1, 1] I | (t, 1) ∈ T , Y = t ∈ [-1, 1] I | (t, -1) ∈ T , Z = t ∈ [-1, 1] I | (t, 0) ∈ T .
It follows that ε is a 0, 1-lattice embedding. Denote by D the range of ε.

Now suppose that D is flatly triangulable. Let Ω be a polytope, let A be a finite set of affine functionals on k I , and let τ : [-1, 1] I → Ω be a semi-algebraic homeomorphism such that τ D = Op(A, Ω). Then Ω × [-1, 1] is a polytope of k I∪{j} and τ :

[-1, 1] I∪{j} = [-1, 1] I × [-1, 1] → Ω × [-1, 1], (x, t) → (τ (x), t)
is a semi-algebraic homeomorphism. Furthermore, denoting by p : k I∪{j} ։ k the jth coordinate projection, defining a : (x, t) → a(x) for a ∈ A, and setting

A def = {a | a ∈ A}, it is not hard to verify that τ D = Op A ∪ {p} , Ω × [-1, 1] .

Real spectra of countable algebras

Now we are about to reach our main result, Theorem 8.4. We will state that result in the language of f -rings, radical ℓ-ideals, and Brumfiel spectra. Translations to real spectra will follow right away. Definition 8.1. For any real-closed field k and any set I, we set

[-1, 1] (I) def = x ∈ k (I) | -1 ≤ x i ≤ 1 for every i ∈ I , the cube of dimension I in k .
We endow the k-algebra k [-1,1] (I) , of all maps [-1, 1] (I) → k, with its componentwise ordering and we denote by k I the f -subring of k [-1,1] (I) generated by the constant map with value 1 together with all projection maps ([-1, 1] The following easy lemma was already stated in Wehrung [START_REF] Wehrung | Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF]Lemma 4.9].

(I) → k, p → p(i)) for i ∈ I. A subset X of [-1, 1] (I) is semi-algebraic if there are a finite subset J of I and a semi-algebraic subset Y of [-1, 1] (J) such that X = p ∈ [-1, 1] (I) | p↾ J ∈ Y - from now
Lemma 8.3. Let A be a commutative f-ring, let L be a distributive lattice with zero, and let ϕ : Id r c A ։ L be a closed surjective lattice homomorphism. Then the subset

I def = {x ∈ A | ϕ( x r ) = 0} is a radical ℓ-ideal of A and there is a unique isomorphism ψ : Id r c (A/I) → L such that ψ( x + I r ) = ϕ( x r ) for every x ∈ A.
Theorem 8.4. Let k be a countable totally ordered field. Then every countable, completely normal, bounded distributive lattice L is isomorphic to Id r c A for some commutative unital f-k-algebra A.

Proof. If k denotes the real closure of k, then any f -k-algebra is also a f -k-algebra. Since Id r c A is not affected by the change of field (cf. Note 2.1), we may thus replace k by its real closure, and thus assume that k is real-closed. Now we follow the lines of the proof of Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Theorem 9.1], replacing the original auxiliary results by those obtained in the present paper's Sections 4-7.

We fix a generating subset {a m | m < ω} of L and a sequence ((S m , A m ) | m < ω), where each S m is a finite subset of ω and each

A m ∈ O sa ([-1, 1] (Sm) ), such that A 0 = ∅ and O sa ([-1, 1] (ω) ) = A m × [-1, 1] (ω\Sm) | m < ω . Further, we fix an enumeration ((X k , Y k , γ k ) | k < ω) of all triples (X, Y, γ) where X, Y ∈ O sa ([-1, 1] (ω) ) and γ ∈ L.
We shall construct inductively a sequence ((

I n , D n , ϕ n ) | n < ω)
, where for every n < ω, I n is a finite subset of ω, D n is a flatly triangulable 0, 1-sublattice of O sa ([-1, 1] (In) ), and ϕ n : D n → L is a 0, 1-lattice homomorphism, subjected to the following conditions:

(1) every a m belongs to the range of some ϕ n , (2) for all m there exists n such that S m ⊆ I n and

A m × [-1, 1] (In\Sm) ∈ D n , (3) for all k ≤ n, if X k = X × [-1, 1] (ω\In) and Y k = Y × [-1, 1] (ω\In) for some X, Y ∈ D n such that ϕ n (X) ≤ ϕ n (Y ) ∨ γ k , then there exists W ∈ D n+1 such that X ×[-1, 1] (In+1\In) ⊆ Y × [-1, 1] (In+1\In) ∪W and ϕ n+1 (W ) ≤ γ k , ( 4
) for all n < ω and every

U ∈ D n , U × [-1, 1] (In+1\In) ∈ D n+1 and the equal- ity ϕ n (U ) = ϕ n+1 (U × [-1, 1] (In+1\In) ) holds.
For n = 0 we just set

I 0 def = {0} (thus [-1, 1] (I0) = [-1, 1] {0} ∼ = [-1, 1]), D 0 def = {∅, [-1, 1]}, and ϕ 0 : D 0 → {0 L , 1 L } is the unique isomorphism.
Suppose (I n , D n , ϕ n ) already constructed.

If n = 3m for an integer m, denote by k the least element of ω\I n , set I n+1 def = I n ∪ {k}, and let ε :

D n * J 2 ֒→ O sa ([-1, 1] (In∪{k})
) be the 0, 1-lattice embedding defined in Lemma 7.10. By that lemma, the range D n+1 of ε is flatly triangulable and there exists a unique 0, 1-lattice homomorphism ϕ n+1 : D n+1 → L extending ϕ n in the sense of (4) above whereas sending [-1, 1] (In) × (0, 1] to a m and [-1, 1] (In) × [-1, 0) to 0. This will take care of (1) above.

If n = 3m + 1 for an integer m, we set

I n+1 def = I n ∪ S m . Setting D ′ n def = U × [-1, 1] (Sm\In) | U ∈ D n , it follows from Lemma 7.7 that D ′ n ∪ A m × [-1, 1] (In\Sm) is contained in a flatly triangulable sublattice D n+1 of O sa ([-1, 1] (In+1) ). Now D ′ n
is easily seen to be flatly triangulable. Hence, by Proposition 7.9, the 0, 1-lattice homomorphism

D ′ n → L, U × [-1, 1] (Sm\In) → ϕ n (U )
extends to a lattice homomorphism ϕ n+1 : D n+1 → L. This will take care of (2) above.

Let finally n = 3m + 2 for some integer m. Now set I n+1 def = I n . Since D n is flatly triangulable, there are a polytope Ω, a finite set A of affine functionals on k In , and a semi-algebraic homeomorphism τ : [-1, 1] (In) → Ω such that τ D n = Op(A, Ω). We may add to A the constant function with value 1 and thus assume that Op(A, Ω) = Op -(A, Ω). By a finite iteration of Lemma 4.9, initially applied to the homomorphism ϕ n τ -1 : Op(A, Ω) → L, there exists a finite set B of linear combinations of A, containing A (so Op(B, Ω) = Op -(B, Ω)), together with a homomorphism ψ :

Op(B, Ω) → L extending ϕ n τ -1 , such that for every k ≤ n, if X k = X × [-1, 1] (ω\In) and Y k = Y × [-1, 1] (ω\In) for some X, Y ∈ D n with ϕ n (X) ≤ ϕ n (Y ) ∨ γ k , there exists W ∈ Op(B, Ω) such that τ [X] ⊆ τ [Y ] ∪ W and ψ(W ) ≤ γ k . The lattice D n+1 def = τ -1 Op(B, Ω
) is flatly triangulable and contains D n and the homomorphism ϕ n+1 def = ψτ extends ϕ n . This will take care of (3) above.

The sequence ((

I n , D n , ϕ n ) | n < ω) satisfies Conditions (1)-(4). For each n < ω, E n def = U × [-1, 1] (ω\In) | U ∈ D n is an isomorphic copy of D n in O sa ([-1, 1] (ω) )
and the assignment U × [-1, 1] (ω\In) → ϕ n (U ) defines a 0, 1-lattice homomorphism ψ n : E n → L. It follows from Condition (4) that both sequences (E n | n < ω) and (ψ n | n < ω) are ascending; thus the union ψ of all ψ n is a 0, 1-lattice homomor-

phism from E def = (E n | n < ω) to L. Moreover, it follows from Condition (1) that ψ is surjective, from Condition (2) that E = O sa ([-1, 1] (ω)
), and from Condition (3) that ψ is a closed map. By Proposition 8.2, ψ thus induces a closed surjective lattice homomorphism ψ ′ : Id r c k ω ։ L. Now by Lemma 8.3, ψ ′ in turn induces an isomorphism Id r c A → L where A is the quotient of k ω by a suitable radical ℓ-ideal. Corollary 8.5. Let k be a countable totally ordered field. Then every countable, completely normal distributive lattice L with zero is isomorphic to Id r c A for some commutative f-k-algebra A.

Proof. Observe that L is an ideal of the bounded distributive lattice L ∞ def = L∪{∞} for a new top element ∞. Since L ∞ is a countable completely normal bounded distributive lattice, it follows from Theorem 8.4 that L ∞ ∼ = Id r c A for some commutative unital f -k-algebra A. Therefore, Corollary 8.6. Let k be a countable totally ordered field. Every second countable, completely normal generalized spectral space X is homeomorphic to the Brumfiel spectrum of some commutative f-k-algebra A. If, in addition, X is compact (i.e., it is a spectral space), then A can be taken unital.

J def = {x ∈ A | x r = ∞} is a radical ℓ-ideal of A,
In Wehrung [34, § 4], we observe, invoking results by Delzell and Madden [START_REF]Lattice-ordered rings and semialgebraic geometry. I, Real analytic and algebraic geometry[END_REF], Schwartz and Madden [START_REF] Schwartz | Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings[END_REF], and Schwartz [START_REF] Schwartz | The basic theory of real closed spaces[END_REF], that the class of real spectra of all commutative unital rings (resp., so-called real-closed rings) and the class of Brumfiel spectra of all commutative unital f-rings coincide. The corresponding constructions carry over to algebras over totally ordered fields. This is used in our next result.

Corollary 8.7. Let k be a countable formally real field. Every second countable completely normal spectral space is homeomorphic to the real spectrum of some countable commutative unital k-algebra.

Proof. Since k is formally real, it has a compatible total ordering, that we shall fix through the remainder of the proof. By Corollary 8.6, the given space is homeomorphic to the Brumfiel spectrum of some commutative unital f -k-algebra A. Now we argue as in the proof of [START_REF] Wehrung | Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF]Corollary 4.17], reducing the Brumfiel spectrum to the real spectrum. In some more detail: first observe that the assignment P → A + + P defines a homeomorphism from Spec B A onto Spec r (A, A + ) def = {Q ∈ Spec r A | A + ⊆ Q}. Now Spec r (A, A + ) is a closed subspace of Spec r A, which is homeomorphic to the Zariski spectrum Spec C r (A) of the real closure C r (A) of A (cf. Schwartz [START_REF] Schwartz | The basic theory of real closed spaces[END_REF]); thus Spec r (A, A + ) ∼ = Spec C r (A)/I for some radical ideal I of C r (A). Now the ring C r (A)/I is real-closed, thus Spec C r (A)/I ∼ = Spec r C r (A)/I . Therefore, Spec B A ∼ = Spec r C r (A)/I . Now since A is a kalgebra, so is C r (A)/I.

The case of uncountable partially ordered division rings

This section will be mostly devoted to the discussion of an example, showing that the countability assumption on k cannot be dispensed with in our positive representability results, most notably Theorems 6.2 and 8.4. This will be done by isolating an additional property satisfied by all homomorphic images of lattices that can be represented over division rings of larger cardinality. Definition 9.1. Let k be a partially ordered division ring, let V be a right kvector lattice, and let D be a distributive lattice with zero. A map µ :

V → D is a premeasure if µ↾ V + is a 0-lattice homomorphism, µ(x) = µ(|x|), and µ(xλ) ≤ µ(x) whenever (x, λ) ∈ V × k. If, in addition, µ -1 {0 D } = {0 V },
then we say that µ is a measure.

Lemma 9.2. The following statements hold, for any premeasure µ : V → D:

(1) µ(x + y) ≤ µ(x) ∨ µ(y) for all x, y ∈ V, with equality holding if x, y ∈ V + ;

(2) µ(xλ) = µ(x) for all x ∈ V and all λ ∈ k \ {0};

(3) the subset I def = µ -1 {0} is an ℓ-ideal of V k and the assignment x + I → µ(x) defines a measure from (V/I) k to D.

Proof. Ad (1). If x, y ∈ V + , then, since x ∨ y ≤ x + y ≤ 2(x ∨ y) within V + and µ(2(x ∨ y)) = µ((x ∨ y) • 2) ≤ µ(x ∨ y) we get µ(x + y) = µ(x ∨ y) = µ(x) ∨ µ(y). In the general case, µ(x Proof of Claim. By Claim 4 and since 0 < λ -1 , we get λ -1 / ∈ r(x). By Claim 1, it follows that either eλ -1 ≤ x or x ≤ eλ -1 . In the first case, µ(x) ≥ µ(eλ -1 ) = µ(e) = e, in contradiction with 0 ∈ r(x). Therefore the second case holds.

+ y) = µ(|x + y|) ≤ µ(|x| + |y|) = µ(|x|) ∨ µ(|y|) = µ(x) ∨ µ(y). Ad (2). Just observe that µ(x) = µ(xλλ -1 ) ≤ µ(xλ) ≤ µ(x). Ad (
Claim 5.

Now we can finish the proof of Theorem 9.3. Since {a, b} ⊆ E and µ(a) and µ(b) are both distinct from e, we get 0 ∈ r(a) and 0 ∈ r(b). By Claim 5, it follows that 3a ≤ e and 3b ≤ e. Adding those inequalities together yields 3e ≤ 2e, so e = 0, and so e = 0, a contradiction.

Throughout the remainder of this section we shall denote by [0, 1] the unit interval of the rational ordered field Q. The lattice D of all finite unions of relatively open intervals of [0, 1] (with rational endpoints) is obviously bounded and distributive. Proof.

X def = {z ∈ [0, 1] | d(z, B ′ ) < d(z, A ′ )} and Y def = {z ∈ [0, 1] | d(z, A ′ ) < d(z, B ′ )} both belong to D. Moreover, A ⊆ B ∪ X, B ⊆ A ∪ Y ,
The assumption that k is directed ensures that x ∧ y ℓ k = x ℓ k ∧ y ℓ k for all x, y ∈ V + (use (2.1)); whence Id ℓ c V k is a sublattice of Id ℓ V k . Now suppose that the conclusion of Corollary 9.5 fails. By precomposing the given homomorphism with the map x → x ℓ k , we obtain a surjective premeasure µ : V ։ D. The top element of D, namely [0, 1], is indecomposable in D, but not join-irreducible (e.g., [0, 1] = [0, 1) ∪ (0, 1]). Apply Theorem 9.3. Corollary 9.6. Let k be a directed partially ordered field. If k is uncountable, then there are no commutative f-k-algebra A and no lattice homomorphism from Id r c A onto D.

Proof. Denote by V the underlying right k-vector lattice of A. As in Wehrung [START_REF] Wehrung | Real spectrum versus ℓ-spectrum via Brumfiel spectrum[END_REF]Proposition 5.6] and using Note 2.1, observe that the assignment x ℓ k → x r defines a surjective lattice homomorphism from Id ℓ c V k onto Id r c A. Apply Corollary 9.5. By Proposition 9.4, Spec D is a completely normal spectral space. By applying Stone duality to our previous results, we obtain: Corollary 9.7. The space Spec D cannot be embedded, as a spectral subspace, into either following kind of spectral space:

(1) Spec ℓ V k , for any uncountable directed partially ordered division ring k and any right k-vector lattice V; (2) Spec r A, for any uncountable field k and any commutative unital k-algebra A.

Proof. Ad [START_REF] Baker | Free vector lattices[END_REF]. Apply Stone duality to Corollary 9.5.

Ad [START_REF] Baro | Normal triangulations in o-minimal structures[END_REF]. Suppose that Spec D embeds, as a spectral subspace, into Spec r A for a commutative unital algebra A over an uncountable field k. Since Spec r A is nonempty, k is formally real, which means that it can be partially ordered via the positive cone k + consisting of all sums of squares in k. Observe that this ordering makes k directed, for

x = x + 1 2 2 - x -1 2 2 for all x ∈ k .
Since Spec r A is homeomorphic to the Brumfiel spectrum of the universal commutative unital f -ring A of A (cf. Delzell and Madden [START_REF]Lattice-ordered rings and semialgebraic geometry. I, Real analytic and algebraic geometry[END_REF]Proposition 3.3]), Spec D embeds as a spectral subspace into Spec B A. Applying Stone duality, it follows that D is a homomorphic image of Id r c A. Since every square of k belongs to A + and by definition of the ordering on k, the canonical map from k to A is orderpreserving, A is also an f -k-algebra; in contradiction with Corollary 9.6.

Let us point that analogues of Corollary 9.5, established in a completely different, module-theoretical context, can be found in Goodearl and Menal's paper [START_REF] Goodearl | Stable range one for rings with many units[END_REF]. A key point is [14, Lemma 2.1], which can be viewed as an analogue of Claim 3 in the proof of Theorem 9.3.

Representability over arbitrary totally ordered division ring

It is well known that every completely normal generalized dual Heyting algebra is isomorphic to the principal ℓ-ideal lattice of some Abelian ℓ-group (cf. Cignoil et al. ). In this section we shall amplify that result, in that process noticeably modifying the presentation of the above works, enabling us to extend it to vector lattices over arbitrary totally ordered division rings and to state it in functorial terms. Functoriality will be achieved by a construction via generators and relations (cf. Definition 10.1).

Following the notation of Iberkleid et al. Lemma 10.2. The assignment D → k⌊D⌋ can be extended to a functor, from the category of all distributive lattices with zero and closed homomorphisms, to right k-vector lattices with ℓ-k-homomorphisms.

: D → V + is a pre-scale if a ≪ b implies that ν(a)λ ≤ ν(b) whenever a, b ∈ D and λ ∈ k. If, in addition, ν -1 {0 V } = {0 D },
Proof. Any closed 0-lattice homomorphism f : D → E, between distributive lattices with zero, preserves the relations defining the construction k⌊ -⌋. This means that there exists a unique ℓ-k-homomorphism k⌊f ⌋ : k⌊D⌋ → k⌊E⌋ sending each a to f (a).

We will be interested only in the case where D is a completely normal generalized dual Heyting algebra and k is a totally ordered division ring. As we will see shortly, in that case everything boils down to the case where D is finite, that is, the poset P def = Ji D is a root system (i.e., every principal upper subset of P is a chain). For x ∈ k P , denote by Γ(x) the set of all maximal elements of the support {p ∈ P | x(p) = 0} of x. Letting x ≤ y hold iff x(p) < y(p) for every p ∈ Γ(y -x), the Hahn power (k P , +, 0, ≤) is a right k-vector lattice. The argument of the proof in Iberkleid et al. Our next lemma shows that (k P , ν D ) is the universal (pre-)scale on D.

Lemma 10.4. Let D be a finite completely normal dual Heyting algebra, set P def = Ji D, let k be a totally ordered division ring, and let V be a right k-vector lattice. Then for every pre-scale ν : D → V + , there exists a unique ℓ-k-homomorphism ϕ :

k P → V such that ν = ϕ • ν D . Moreover, if ν is a scale, then ϕ is an ℓ-k- embedding. Proof. Necessarily, ϕ : k P → V, x → (ν(p)x(p) | p ∈ P ). Let x ∈ k P .
Since P is a root system, the lower subsets P ↓ p, for p ∈ Γ(x), are pairwise disjoint, thus the elements x p def = x↾ P ↓p (extended by 0 outside P ↓ p) generate pairwise orthogonal ℓ-ideals in k P and x = (x p | p ∈ Γ(x)). Now each x p is either positive or negative according to whether x(p) > 0 or x(p) < 0, respectively.

In particular, setting Γ + (x)

def = {p ∈ Γ(x) | x(p) > 0}, we get x ∨ 0 = x p | p ∈ Γ + (x) and ϕ(x) ∨ 0 = ϕ(x p ) | p ∈ Γ + (x) .
Therefore, ϕ is an ℓ-k-homomorphism. Now suppose that ν is a scale. We must prove that ϕ(x) ≥ 0 implies that x ≥ 0, for every x ∈ k P . Using the decomposition x = (x p | p ∈ Γ(x)) above, it suffices to consider the case where the support of x has a largest element p. In that case, ν(p) > 0, together with the assumption that ν is a pre-scale, entails that ϕ(x) is either positive or negative according to whether x(p) > 0 or x(p) < 0, that is, x > 0 or x < 0, respectively; so we are done.

Theorem 10.5. Let k be a totally ordered division ring. Then for every completely normal generalized dual Heyting algebra D, the assignment a → a ℓ k defines an isomorphism from D onto Id ℓ c k⌊D⌋ k . This homomorphism is natural in D with respect to closed 0-lattice homomorphisms.

Proof. Owing to Maksimova [START_REF] Maksimova | Pretabular superintuitionistic logics[END_REF] (where the result is stated in terms of Heyting algebras), the variety CNdHA of all completely normal generalized dual Heyting algebras (in the signature (∨, ∧, )) is generated by all finite chains. Now for every positive integer n, every totally ordered member of CNdHA generated by an nelement chain has at most n + 1 elements. Using McKenzie et al. Lemma 4.98], it follows that every n-generated member of CNdHA has at most (n + 1) (n+1) n elements; so every member of CNdHA is the directed union of its finite subalgebras. This reduces the verification of Theorem 10.5 to the finite case. By Stone duality, we obtain: Corollary 10.7. A completely normal generalized spectral space X is the ℓ-spectrum of a right vector lattice over any totally ordered division ring iff the Stone dual of X is a generalized dual Heyting algebra.

We shall now relate Theorem 10.6 to the results of Section 9. Theorem 9.3 is not the first occurrence of the property "indecomposable ⇒ joinirreducible" in the literature. In particular, a slight modification of the argument of Martínez [22, Although Theorem 9.3 and Proposition 10.8 may raise the hope that the representability of a distributive lattice D, as the range of premeasures on vector lattices over arbitrary totally ordered fields, implies that D is a generalized dual Heyting algebra, the following example shows that this is not the case; and a bit more.

Example 10.9. A countable, bounded, completely normal distributive lattice D (necessarily not a generalized dual Heyting algebra) such that:

(1) For every totally ordered division ring k, there are a right k-vector lattice V such that Spec D embeds into Spec ℓ V k as a spectral subspace. (2) For every formally real field k, there exists a commutative unital k-algebra A such that Spec r A ∼ = Spec D. (3) There exists a totally ordered field k such that there is no right k-vector lattice V with Spec ℓ V k ∼ = Spec D.

Proof. Let D be the lattice denoted by D ω in Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Example 10.5]. That is, D consists of all pairs (X, k), where X is a subset of ω that is either finite or cofinite, k ∈ {0, 1, 2}, k = 0 implies that X is finite, and k > 0 implies that X is cofinite. The ordering on D is defined componentwise. Setting 3 def = {0, 1, 2}, the sublattice E of 3 ω , consisting of all eventually constant sequences, is a dual Heyting algebra. For each x ∈ E, the subset supp(x) def = {n < ω | x(n) = 0} is either finite or cofinite in ω and x(∞), defined as x(n) for large enough n, belongs to 3. The map π : E → D, x → (supp(x), x(∞)) is a surjective lattice homomorphism (cf. Wehrung [START_REF]Spectral spaces of countable Abelian lattice-ordered groups[END_REF]Example 10.6]).

Let k be a totally ordered division ring and consider the lexicographical product K of k by itself, endowed with its canonical structure of (totally ordered) right k-vector lattice. The map ρ : K → 3 defined by the rule ρ(x, y) where the "condensate" R is defined as the k-algebra of all eventually constant sequences of elements of K with limit in A.

Finally, there is no least x ∈ D such that (ω, 2) ≤ (ω, 1) ∨ x; whence D is not a generalized dual Heyting algebra. Using Corollary 10.7, Item (3) follows.

This suggests the following problem.

Problem. Characterize the spectral spaces that can be represented as real spectra of algebras over any formally real field.

  sublattice. The elements of that lattice are those of the form x r def = y ∈ A | |y| n ≤ |x|z for some z ∈ A + and n a positive integer for x ∈ A .

  def = K = {S | S ∈ K}. The relative interiors of the simplexes in K form a partition of |K|; the unions of elements of that partition are the sets partitioned by K. A simplicial complex L is a subdivision of K if every open simplex of K is partitioned by L and every open simplex of L is contained in an open simplex of K; of course in that case |K| = |L|.

Claim.

  If P is join-irreducible, then it is basic open and P ∩ ∇P = ∅. Proof of Claim. Since every element of Op(F, Ω) is a finite union of basic open sets, every join-irreducible element of Op(F, Ω) is basic open, thus convex. Now suppose that P ∩ ∇P = ∅, that is, P ⊆ f ∈FP [[f = 0]] Ω . Since P is join-irreducible in the distributive lattice Op(F, Ω), P ⊆ [[f = 0]] Ω for some f ∈ F P , in contradiction with the definition of F P .Claim.

  and thus, since P is join-prime in Op(F, Ω), we get P ⊆ [[b < 0]] Ω . It follows that ϕ(P ) L ϕ(P * ) ≤ ϕ([[b < 0]] Ω ), thus, using the relations ϕ([[b > 0

def= 1 Figure 5 . 1 .

 151 Figure 5.1. The lattice J 2

Note 7 . 2 .

 72 Triangulations are often defined by replacing τ by τ -1 in Definition 7.1.

Remark 7 . 4 .Definition 7 . 5 .

 7475 Shiota establishes in [31, Proposition 3.9] a version of Proposition 7.3 for k = R. It is conceivable that a suitable application of Tarski-Seidenberg's transfer principle, similar to the argument of Coste [8], might turn Shiota's result into the general form of Proposition 7.3. The author did not investigate this. For any semi-algebraic subset Ω of k d , we denote by O sa (Ω) the lattice of all semi-algebraic subsets of Ω that are open with respect to the relative topology on Ω. A (necessarily finite) sublattice D of O sa ([-1, 1] d ) is flatly triangulable if there are a polytope Ω of k d , a semi-algebraic homeomorphism τ : [-1, 1] d → Ω, and a finite set A of affine functionals on k d such that τ D def = {τ [U ] | U ∈ D} is equal to Op(A, Ω).

Lemma 7 . 6 .

 76 Let D be a flatly triangulable sublattice of O sa ([-1, 1] d ). Then there are a simplicial complex K of k d with |K| convex, a semi-algebraic homeomorphism τ : [-1, 1] d → |K|, and a finite set A of affine functionals on k d such that τ D = Op(A, |K|) = Op -(A, |K|) and every member of Bool(A, |K|) is partitioned by K.

Proposition 7 . 9 .Figure 7 . 1 .

 7971 Figure 7.1. A commutative diagram of finite distributive lattices

Proposition 8 . 2 .

 82 on let us write this as X = Y × [-1, 1] (I\J) . Accordingly, we extend the notation O sa (Ω) to the infinite-dimensional case (cf. Definition 7.5), and we denote by O sa (Ω) the lattice of all semi-algebraic relatively open subsets (with respect to the topology induced by the product topology on [-1, 1] (I) ) of a semi-algebraic subset Ω of [-1, 1] (I) . In particular, k I is a commutative unital f -ring. For all x ∈ k I , the cozero set of x, COZ(x) def = p ∈ [-1, 1] (I) | x(p) = 0 is a semi-algebraic relatively open subset of [-1, 1] (I) . For any real-closed field k and any set I, the assignment a r → COZ(a) defines a lattice isomorphism from Id r c k I onto O sa ([-1, 1] (I) ). Proof. Since every element of k I or O sa ([-1, 1] (I) ), respectively, depends only on a finite set of coordinates in I, it is easy to reduce the problem to the case where I is finite. Now that case is covered by Delzell and Madden [11, Proposition 6.2].

  thus a commutative f -k-algebra, and Id r c J ∼ = L. Since Id r c A is the Stone dual of the Brumfiel spectrum of A, we thus obtain the following topological version of Theorem 8.4 and Corollary 8.5.

3 ) is straightforward. Theorem 9 . 3 .Claim 1 . 2 . 3 . 3 . 4 . 4 . 5 .

 3931233445 Let D be a distributive lattice with zero. If there are a partially ordered division ring k with card D < card k + , a right k-vector lattice V, and a surjective premeasure µ : V k ։ D, then every indecomposable element of D is joinirreducible.Proof. It follows from Lemma 9.2(3) that µ can be taken a measure. Suppose that D has an indecomposable element e which is not join-irreducible. Since µ is surjective, there are a, b ∈ V + such that e = µ(a) ∨ µ(b) whereas µ(a) < e and µ(b) < e. Setting e def = a + b, we get µ(e) = e.Set x ydef = x -(x ∧ y), for all x, y ∈ V. Let x, y ∈ V be incomparable. Then µ(x -y) = e.Proof of Claim. From (x y) ∧ (y x) = 0 we get µ(x y) ∧ µ(y x) = 0. Since x and y are incomparable and µ -1 {0} = {0}, µ(x y) and µ(y x) are both nonzero. Since e is indecomposable in D, it follows that µ(x y) ∨ µ(y x) = e, so µ(x -y) = µ(|x -y|) = µ(x y) ∨ µ(y x) = e.Claim 1.Until the end of the proof of Theorem 9.3 we will setE def = {x ∈ V | µ(x) ≤ e}.Observe that E is an ℓ-ideal of V k containing e as an element. Setr(x) def = {λ ∈ k | µ(x -eλ) = e} = {λ ∈ k | µ(x -eλ) < e} , for all x ∈ E .Claim 2. For every x ∈ E, the set r(x) is order-convex in k.Proof of Claim. Let α < γ < β in k with {α, β} ⊆ r(x). If γ / ∈ r(x), then, by Claim 1, either x ≤ eγ or eγ ≤ x. In the first case, eβ -x ≥ e(β -γ) ≥ 0 with β -γ ∈ k ++ , thus, using Lemma 9.2, µ(eβ -x) ≥ µ(e(β -γ)) = µ(e) = e, in contradiction with β ∈ r(x). In the second case,x -eα ≥ e(γ -α) ≥ 0 with γ -α ∈ k ++ , thus µ(x -eα) ≥ µ(e(γ -α)) = µ(e) = e, in contradiction with α ∈ r(x). Claim Claim For all x ∈ E and all distinct α, β ∈ k, µ(x -eα) ∨ µ(x -eβ) = e.Proof of Claim. From e(α -β) = (x -eβ) -(x -eα) it follows, using Lemma 9.2, that e = µ(e) = µ(e(α -β)) ≤ µ(x -eα) ∨ µ(x -eβ). The converse inequality follows from x ∈ E. Claim Claim For every x ∈ E, any two distinct elements of r(x) are incomparable. Proof of Claim. Suppose, to the contrary, that there are α < β in r(x). By Claim 2, the interval [α, β] is contained in r(x). By Claim 3, it follows that the assignment γ → µ(x -eγ) defines a one-to-one map from [α, β] into D; whence card[α, β] ≤ card D. Since ξ → α + (1 + ξ) -1 (β -α) defines a one-to-one mapping from k + into (α, β], it follows that card k + ≤ card D, a contradiction. Claim Claim Let x ∈ E. If 0 ∈ r(x), then xλ ≤ e for every λ ∈ k ++ .

Proposition 9 . 4 .

 94 The lattice D is completely normal. Proof. Denote by d : (x, y) → |x -y| the canonical distance function on [0, 1]. For A, B ∈ D, with respective complements A ′ and B ′ , the subsets X and Y defined by

3 . 9 . 5 .

 395 and X ∩ Y = ∅. Now we can start reaping the consequences of Theorem 9.Corollary Let k be a directed partially ordered division ring. If k is uncountable, then there are no right vector lattice V over k and no surjective lattice homomorphism from Id ℓ c V k onto D.

[ 6 ,

 6 Theorem 3.3], Iberkleid et al. [17, Theorem 3.1.1]

  [START_REF] Iberkleid | Conrad frames[END_REF], for any elements a and b in a distributive lattice D, let a ≪ b hold if a ≤ b and for all x ∈ D, b ≤ a ∨ x implies that b ≤ x. Equivalently, a ≤ b and b D a = b. Definition 10.1. Let D be a distributive lattice with zero. For any totally ordered division ring k and any right k-vector lattice V, a 0-lattice homomorphism ν

  we say that ν is a scale. Given D and k, we denote by k⌊D⌋ the (image of the) universal pre-scale on D. In more detail, k⌊D⌋ is thus the right k-vector lattice defined by generators a, for a ∈ D, and relations 0 = 0, a ∨ b = a ∨ b, and a ∧ b = a ∧ b, for a, b ∈ D, together with aλ ≤ b whenever a ≪ b in D and λ ∈ k.

[ 17 ,

 17 Theorem 3.1.1(3)] yields the following. Lemma 10.3. Let D be finite. The map ν D : D → (k P ) + , sending every a to the characteristic function of the set of all maximal elements of P ↓a def = {p ∈ P | p ≤ a}, is a scale on D.

  Now if D is finite, all required proofs are straightforward except perhaps the one of the surjectivity of a → a ℓ k . However, the decomposition x = (x p | p ∈ Γ(x)) introduced in the proof of Lemma 10.4 reduces the problem to elements x whose support has a largest element p, in which casex ℓ k = p ℓ k .Finally, the required naturality statement follows from Lemma 10.2.Theorem 10.6. For any completely normal distributive lattice D with zero, the following are equivalent: (i) There are a partially ordered division ring k and a right k-vector lattice V, such that every subset of k with at most card D elements has an upper bound, and D ∼ = Id ℓ c V k . (ii) For every totally ordered division ring k, there exists a right k-vector lattice V such that D ∼ = Id ℓ c V k . (iii) D is a generalized dual Heyting algebra. Proof. (i)⇒(iii). Let D = Id ℓ c V k and set, as in the proof of Theorem 9.3, x y def = x -(x ∧ y) for all x, y ∈ V. For all a ∈ D pick a ∈ V + such that a = a ℓ k . For all a, b ∈ D, the set a bλ ℓ k | λ ∈ k ++ is contained in D, thus there exists a subset S a,b of k ++ , with at most card D elements, such that a bλ ℓ k | λ ∈ k ++ = a bλ ℓ k | λ ∈ S a,b . The set S def = (S a,b | a, b ∈ D) has thus at most (card D) 3 elements, and a bλ ℓ k | λ ∈ k ++ = a bλ ℓ k | λ ∈ S whenever a, b ∈ D . The set S has an upper bound ξ in k ++ (this is obvious if D is finite, and follows from our assumption otherwise). Hence a bξ ℓ k ⊆ a bλ ℓ k for all a, b ∈ D and all λ ∈ k ++ . We claim that for all a, b ∈ D, a D b is defined and equal to a bξ ℓ k . Indeed, a ℓ k ⊆ b ℓ k ∨ a bξ ℓ k . Further, for all x ∈ D such that a ≤ b ∨ x, there exists λ ∈ k ++ such that a ≤ (b + x)λ, which implies that a bλ ≤ xλ, whence a bξ ℓ k ⊆ a bλ ℓ k ⊆ x ℓ k = x, thus completing the proof of our claim. (iii)⇒(ii) follows from Theorem 10.5. (ii)⇒(i). There are totally ordered fields as in (i).

  Lemma 2.2.3] yields the following: Proposition 10.8. Let D be a completely normal generalized dual Heyting algebra. Then every indecomposable element of D is join-irreducible. Proof. Let e ∈ D and let a, b ∈ D such that e = a ∨ b. From the complete normality of D we get the disjoint join e = a ′ ∨ b ′ with a ′ def = a D (b D a) and b ′ def = (b D a) D a = b D a. If e is indecomposable, then e ∈ {a ′ , b ′ }, thus, since a ′ ≤ a and b ′ ≤ b, we get e ∈ {a, b}. Therefore, e is join-irreducible.

0 ,

 0 if x = y = 0 , 1 , if x = 0 and y = 0 , 2 , if x = 0 , for all (x, y) ∈ K is a surjective measure from K k onto 3. Denoting by V the right k-vector lattice consisting of all eventually constant sequences of elements of K, the assignment((x n , y n ) | n < ω) → (ρ(x n , y n ) | n < ω) defines a surjective measure ρ : V k ։ E. Therefore, π • ρ is a surjective measure from V k onto D,so it defines a surjective homomorphism from Id ℓ c V k onto D. By Stone duality, (1) follows. Now let us prove (2). We may assume that k is a real-closed field. Replacing the real field R by k in the argument of the proof, relative to R[[G]], of Dickmann et al. [13, Proposition 1.1], then taking G a divisible totally ordered Abelian group with exactly two nonzero Archimedean classes, we get a totally ordered real-closed k-algebra A, without zero divisors, such that the real spectrum of A is the twoelement chain. Denoting by K any real-closed field extension of A, it follows that the map ε : 3 ։ 2, defined by ε(0) = 0 and ε(1) = ε(2) = 1, is represented, with respect to the Stone dual of the functor Spec r , by the inclusion map from A into K. Therefore, mimicking part of the argument of [34, Theorem 5.4], Spec D ∼ = Spec r R

A subset X in a poset P is cofinal if every element of P lies below some element of X.


"Agro innovation, food quality and safety" Reviving perspectives on innovation in Agri-Food Systems (AFS)

For two main raisons :

1. First, because The AFS are characterized by specificities at different levels :

• Depends on natural unstable environment  High level of risk for investment.

• Biological process of production  different from industrial products.

• Strong cultural/patrimonial aspects in both production and consumption sides.

• Wide variety of stakeholders between research and farm, in global value chains and territories.

Secondly, because Innovation in AFS are driven by news global challenges

• Food and nutritional security. We have the traditional need of increasing quantity of food production at global level. But also to response the new demand for sanitary, quality, nutrition and health. • Social and territorial issues. in the rural areas, AFS is number 1 job supplier in the world.

• Climate change. AFS contribute to the emission of Greenhouse Gas. At the same time they contribute to mitigation of the Climate change. • Ecological transition, ….calling for new perspective to both "economic actors" and "researchers" who analyse and gouvern the evolution of agriculture and food.

Methodologic framework and outline

• The main insight from a research program carried out by the joint research unit "Innovation". between 3 organizations (disciplines : Economy -Sociology -Management -Agronomy -Geography)

• Initiate a collective action in coordination for writing one book. This book using reviews and analyses of case studies undertaken and in many countries in Europe and the Global South: Africa, south America, and Asia.

• This book's contributors, all share a systemic approach to innovation The main contributions of this book in four points:

•
• To explain why Innovation in AFS is the subject of increasing social and political debates

• To present how Innovation in AFS is analyzed by interdisciplinary research

• To analyze the debates regarding different models and the possible coexistence of various agri-food models.

• To propose new methods for supporting innovation 1 -Why Innovation in AFS is the subject of increasing social and political debates ?

• No common understanding of ethical considerations that define what is a "good" or "bad" innovation (ex GMO)

• Where some see : a sign of progress a way of reducing the use of pesticides or of improving food production • Others see : only a strategy to increase the profits of multinational firms and reduce the farmers' autonomy Given that an innovation cannot be considered good -just because it is an innovation we need to develop methodologic impact and risk assessment

• Different objectives that go beyond the economic aspects and open up to the challenges of sustainable development

• During the 20 century, innovation was presented as the only factor of economic growth • Today innovation should answer several goals responding to the major challenges that confront societies and public policies: food security, biodiversity, fight against pollution

• Who are the real beneficiaries of innovation : enterprises or population ?

• The literature in innovation process is focus on the performance of enterprise.

• But in reaction, is the necessity to address the innovation concerning the publics goods or the commons in favor of populations that are economically or socially the most vulnerable.

• The models of innovation : the way to innovate, with more open and collaborative methods (contestation of 'top-down')

• The idea is instead to involve a growing number of heterogeneous stake holder in the development of innovations.

• To create more democratic regime of innovation based on collective experimentation by the creation of local platforms and territorial 'living lab

In the "Innovations studies", these are mainly economists and policists who work on the governance of innovation. They take into account technical, organizational and institutional dimensions, at different levels, national, regional, local.

In the "Science and Technology

Studies", There are mainly sociologists, who work on the network of stakeholders. They underline the sociotechnical dimensions of innovation and the influence of learning networks.

In the "Transition studies", sociologists and economists analyse structural conditions which explain the mechanism of transition between technological systems. They use multi-levels perspective.

3 -these innovations refer to different agricultural and food production models

• The agro-industrial model is intensive in capital and industrial inputs : it increases concentration of farms and enterprises.

• Other existing models use ecological inputs and other indicators for performance :

-Social and environmental externality -Proximity between supply and consumers.

-Food diversity -Social inclusiveness

The question is to understand the modalities of coexistence of these different models (space, time)

You have situation where the relation is confrontation.

But you also have situations where you have good complementary.

Coexistence of different of these models depends both as a feature of globalization and as a sectoral specificity :

• while innovations can be part of each of these models, • they can also result from their combinations and interactions

Conclusion This book presents new perspectives on innovation in AFS, it proposes a set of new methods

Co-design approaches using modeling and experimentation with farmers Evolution of advice and consulting based on NTIC New systemic methods to asses impact and risk (Impress)

  www.quae.com/produit/1540/9782759229604/i nnovation-and-development-in-agricultural-and-foodsystems


Full 3D printed hexapod robots with energy efficient joints

Ilya Brodoline, Stéphane Viollet, Julien Serres

To cite this version: