Model-completion of scaled lattices^{∗}

Luck Darnière^{†}

Abstract

It is known from Grzegorczyk’s paper [Grz51] that the lattice of real semi-algebraic closed subsets of ℝ^{n} is undecidable for every integer n ≥ 2. More generally, if X is any definable set over a real or algebraically closed field K, then the lattice L(X) of all definable subsets of X closed in X is undecidable whenever dim X ≥ 2. Nevertheless, we investigate in this paper the model theory of the class SC_{def}(K, d) of all such lattices L(X) with dim X ≤ d and K as above or a henselian valued field of characteristic zero.

We show that the universal theory of SC_{def}(K, d), in a natural expansion by definition of the lattice language, is the same for every such field K. We give a finite axiomatization of it and prove that it is locally finite and admits a model-completion, which turns to be decidable as well as all its completions. We expect L(ℚ_{p}^{d}) to be a model of (a little variant of) this model-completion. This leads us to a new conjecture in p-adic semialgebraic geometry which, combined with the results of this paper, would give decidability (via a natural recursive axiomatization) and elimination of quantifiers for the complete theory of L(ℝ_{p}^{d}), uniformly in p.

1 Introduction

In this paper we study the model-theory of a class of lattices coming from the following examples.

Example 1.1 Let K be a henselian valued field of characteristic zero, a real closed field or an algebraically closed field. There exists a good notion of dimension for definables sets A over K (see [vdD89] for the henselian case, and any book of real or complex algebraic geometry for the other cases). For any positive integer i let:

C^{i}(A) = \overline{\{a ∈ A / dim(A, a) = i\}}

∗This paper is a revised version of [Dar04], submitted to APAL the 5th of April 2005 and eventually rejected the 24th of March 2006.

Keywords: model-theory, quantifier elimination, scaled lattice, Heyting algebra, p-adic.

MSC classes: 03C10, 06D20, 06D99.

†Département de mathématiques, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers cedex 01 (France)

where dim(A, a) is the maximal dimension of definable neighborhood of a in A, and the overline stands for the topological closure in A. This is a definable subset of A which we call the i-pure^{1}component of A. Given a definable set X over K of dimension d, let L_{def}(X) be the lattice of all definable subsets of X closed in X, enriched^{2} with the unary functions (C^{i})_{i≤d} and the binary function:

A − B = \overline{A \ B}

Eventually let SC_{def}(K, d) denote the class of lattices L_{def}(X) for all definable sets over K of dimension at most d.

Example 1.2 Let K be any infinite field, X a constructible subset over K (that is a boolean combination of Zariski closed subsets of K^{n} for some positive integer n) and L_{Zar}(X) be the lattice of all constructible subsets of X which are Zariski closed in X enriched with the following structure. For any A, B in L_{Zar}(X) let A − B be as in the above example and C^{i}(A) be the union of the irreductible components of A of Krull dimension i (in the usual sense for topological spaces).

Eventually let SC_{Zar}(K, d) denote the class of lattices L_{Zar}(X) for all definable sets over K of dimension at most d. Of course SC_{Zar}(K, d) = SC_{def}(K, d) when K is algebraically closed.

It is known from an argument of [Grz51] that the complete theory of L_{def}(K^{n}) is undecidable for every real closed field K and every integer n ≥ 2, and the argument can easily be adapted to algebraically closed fields K. This paper gives some reason to believe that the complete theory of L_{def}(K^{n}) is decidable for every p-adically closed field K and every n. It is organized as follows.

We first give in Section 2 a finite list of universal axioms of a theory T_{d} in a language L_{SC_{d}} extending the language of lattices, the model of which we call d-subscaled lattices. The examples given above are all d-subscalled lattices. After some preliminar techinal results in Section 3 we prove in Section 4 that every finiteley generated d-subscaled lattice is finite. Combining this result with a linear representation for finite d-subscaled lattices and with the model-theoretic compactness theorem, we then prove in Section 5 that T_{d} is precisely the universal theory of SC_{Zar}(K, d). In particular this theory is finitely axiomatizable and, remarkably enough, does not depend on K. Eventually a detailed study of finitely generated extensions of finite d-subscaled lattices, achieved in Sections 6 and 7 allows us to exhibit in Section 8 a model-completion T̄_{d} for T_{d}, having a finite axiomatisation. Moreover we show that T̄_{d} has finitely many completions, each of which is finitely axiomatizable and ℵ_{0}-categorical. It follows that T̄_{d} is decidable.

It is difficult to find a model of T̄_{d} coming from geometry because such models are atomless. We present in the last section a similar model completion and decidability result for a theory T̄_{d}^{∗} authorizing atoms. These results lead us to the following conjecture (or question):

1If C^{i}(A) is non-empty it has pure dimension i, that is the local dimension dim(A, a) = i at every point of A.

2The additionnal functions are definable in the lattice structure of L_{def}(X).

Conjecture 1.3 Let K be a p-adically closed field and A be an infinite definable subset of K^{n} which is open in its closure. Let (B_{k})_{k≤q} be a finite collection of closed definable subsets of Ā \ A. Then there exists a collection (A_{k})_{k≤q} of non-empty definable subsets of A clopen in A such that:

∀k ≤ q, \overline{A_{k}} = A_{k} ∪ B_{k}

If this conjecture is true then it follows immediatly that L_{def}(K^{n}) is a model of T_{n}^{∗} hence has a decidable complete theory (not depending on p).

Remark 1.4 Since 0-subscaled lattices are exactly non-trivial boolean algebras (with the L_{SC_{0}}-structure and the boolean structure being quantifier-free bi-definable) our model-completion result for subscaled lattices is a generalisation to arbitrary finite dimension d of the well known theorem on the modelcompletion of boolean algebras.

Remark 1.5 The duals of d-subscaled lattices form an elementary class of Heyting algebras so this paper may also be considered as a contribution to the model-theory of Heyting algebras. However the usual geometric objects whose study motivated this paper are closed sets (points, curves, surfaces, and so on). From this point of view the lattice L_{Zar}(K^{d}) is a more natural object to consider than its dual, the Heyting algebra of open algebraic sets. This is the reason why we had to present our results in this settings and not in terms of Heyting algebra.

2 Notation and definitions

The set of all positive integers is denoted by ℕ, and ℕ^{∗} = ℕ \ {0}. If N is an unbounded subset of ℕ (resp. the empty subset) we set max N = ∞ (resp. max N = −1). The symbols ⊆ and ⊂ denote respectively the inclusion and the strict inclusion.

2.1 Lattices

Let L_{lat} = {0, 1, ∨, ∧} be the language of lattices. An upper semi-lattice is an L_{up}-substructure of a lattice, with L_{up} = {0, ∨}. As usually b ≤ a is an abreviation for a ∨ b = a and similarly for b < a, b ≥ a and b > a. Iterated ∨ and ∧ operations are denoted by ∨∨_{i∈I} a_{i} and ∧∧_{i∈I} a_{i} respectively. If the index set I is empty then of course ∨∨_{i∈I} a_{i} = 0 and ∧∧_{i∈I} a_{i} = 1. The logical connectives ‘or’, ‘and’ and their iterated forms will be denoted by ⋀, ⋁, ⋀ ; ⋀ and ⋁ ; ⋁ respectively. We consider the following relation, definable in any lattice:

b ≪ a ⟺ b < a and ∀c (c < a ⇒ b ∨ c < a)
⟺ b ≤ a ≠ 0 and ∀c (c < a ⇒ b ∨ c < a)

The spectrum of a distributive lattice L is the set Spec(L) of all prime filters of L, endowed with the so-called Zarisky topology, defined by taking as a basis of closed sets the family:

P (a) = {p ∈ Spec(L) / a ∈ p}

for a ranging over L. Stone-Priestley’s duality asserts that a ↦ P (a) is an isomorphism between L and the lattice of closed subsets of Spec(L) whose complement in Spec(L) is compact. We call a lattice noetherian if it is isomorphic to the lattice of closed sets of a noetherian topological space. By Stone-Priestley’s duality a lattice L is noetherian if and only if its spectrum is a noetherian topological space. In such a lattice every filter is principal and every element a writes uniquely as the join of its ∨-irreducible components, which are the (finitely many) maximal elements in the set of non-zero ∨-irreducible elements of L smaller than a. We denote by I(L) the set of all non-zero ∨-irreducible elements of L.

We define the lattice dimension of an element a in a lattice L as the least upper bound (in ℕ ∪ {−1, ∞}) of the set of positive integers n such that:

∃p_{0} ⊂ ⋯ ⊂ p_{n} ∈ P (a)

This is nothing but the ordinary topological dimension (defined by chains of irreducible closed subsets) of the spectral space P (a). We denote this dimension by dim_{L} a. By construction dim_{L} a = −1 if and only if a = 0_{L}. The index L is necessary since dim_{L} a is not preserved by L_{lat}-embeddings, nevertheless we omit it whenever the ambiant lattice is clear from the context and we often call the lattice dimension simply the dimension. We let the lattice dimension of L be the lattice dimension of 1_{L} in L.

Fact 2.2 If L = L(X) is any of the lattices of the introduction, then for any A ∈ L(X), dim_{L(X)} A is exactly the usual dimension of A as a definable (or constructible) set over K.

Fact 2.4 below is a key argument in the proof of this result, which is non-obvious because the definition of dim_{L(X)} A lies on prime filters of closed definable subsets of A, an object which do not have a natural geometric meaning.

2.3 TC-lattices

Let L_{TC} = L_{lat} ∪ {−} with ‘−’ a binary function symbol. A topologically complemented lattice, or TC-lattice for short, is an L_{TC}-structure which is a lattice and in which the relative topological complement a − b is defined as the least element c such that a ≤ b ∨ c (or equivalently P (a − b) is the topolological closure of the relative complement P (a)\P (b), so the name). This is clearly the dual of a Heyting algebra with a − b in the TC-lattice becoming b → a in its dual. So we know from the theory of Heyting algebra (see for example[Joh82]) that every TC-lattice is distributive, and that the class of all TC-lattices is a variety (in the sense of universal algebra). Observe that in TC-lattices the ≪ relation is quantifier-free definable since:

a − b = a ⟺ b ∧ a ≪ a or a = 0

So it will be preserved by L_{TC}-embeddings. On the other hand the lattice dimension will not be preserved in general by L_{TC}-embeddings of TC-lattices.

We will use the following rules, the proof of which are elementary exercises (using either dual properties, if known, of Heyting algebras or, more directly, Stone-Priestley’s duality).

TC_{1} : a = (a ∧ b) ∨ (a − b).

In particular if a is ∨-irreducible then b < a ⟹ b ≪ a.

TC_{2} : (a_{1} ∨ a_{2}) − b = (a_{1} − b) ∨ (a_{2} − b).

TC_{3} : (a − b) − b = a − b.

So either a − b = 0 or (a − b) ∧ b ≪ a − b ≤ a.

TC_{4} : More generally a − (b_{1} ∨ b_{2}) = (a − b_{1}) − b_{2}.

So if a − b_{1} = a then a − (b_{1} ∨ b_{2}) = a − b_{2}.

Fact 2.4 For any TC-lattice L and any a ∈ L, dim_{L} a is exactly the least upper bound of the set of positive integers n such that there exists a_{0}, …, a_{n} ∈ L such that:

0 ≠ a_{0} ≪ a_{1} ≪ ⋯ ≪ a_{n} ≤ a

The proof is a good exercise that we leave to the reader. The fact that dim_{L} a is at least equal to the above least upper bound is true in any lattice. Equality holds in TC-lattices because they are min-compact (that is for any a ∈ L the set of elements of P (a) which are minimal with respect to the inclusion, is compact).

2.5 (Sub)scaled lattices.

For a given positive integer d let L_{SC_{d}} = L_{TC} ∪ {C^{i}}_{0≤i≤d} where the C^{i}’s are unary function symbols. With the examples of the introduction in mind, define the sc-dimension of an element a of an L_{SC_{d}}-structure L as:

sc-dim a = min{l ≤ d / a = ∨∨_{0≤i≤l} C^{i}(a)} ∈ ℕ ∪ {−1, ∞}

The sc-dimension of L, denoted sc-dim(L), is by definition the sc-dimension of 1_{L}. In general the lattice dimension of an element is not preserved by L_{lat}-embeddings neither by L_{TC}-lattices. On the other hand the sc-dimension of an element is obviously preserved by L_{SC_{d}}-embeddings.

A d-subscaled lattice is then an L_{SC_{d}}-structure which is a topologically complemented lattice and which satisfies the following list of axioms:

SC_{1} : ∨∨_{0≤i≤d} C^{i}(a) = a

SC_{2} : ∀I ⊆ {0, …, d}, ∀k:

C^{k}(∨∨_{i∈I} C^{i}(a)) = \{ \begin{matrix}0 & if k ∉ I \\ C^{k}(a) & if k ∈ I \end{matrix}

SC_{3} : ∀k ≥ max(sc-dim(a), sc-dim(b)), C^{k}(a ∨ b) = C^{k}(a) ∨ C^{k}(b)

SC_{4} : ∀i ≠ j, sc-dim (C^{i}(a) ∧ C^{j}(b)) < min(i, j)

SC_{5} : ∀k ≥ sc-dim(b), C^{k}(a) − b = C^{k}(a) − C^{k}(b)

In particular, by SC_{3}: sc-dim b < a ⟹ C^{k}(a) − b = C^{k}(a).

SC_{6} : If b ≪ a then sc-dim b < sc-dim a.

Axioms SC_{1} to SC_{5} are easily seen to be equivalent to a finite set of equations in L_{SC_{d}}. On the other hand SC_{6} is expressible by a universal formula in L_{SC_{d}} but not by an equation (indeed the class of d-subscaled lattices is not preserved by L_{SC_{d}}-projections, hence is not a variety). We call d-scaled lattices the d-subscaled lattices satisfying the following additional property:

SC_{0} : sc-dim a = dim a

This is an elementary class, which is not preserved by L_{SC_{d}}-substructures. By Fact 2.2 all the examples of L_{SC_{d}}-structures given in the introduction are d-scaled lattices in which the lattice dimension and the sc-dimension coincide with the usual geometric dimension. As the terminology suggests, we will see that d-subscaled lattices are precisely the L_{SC_{d}}-substructures of d-scaled lattices.

The finite language L_{SC_{d}} allows to write finite axiomatisations. When this is not essential we consider the language L_{SC} = L_{TC} ∪ {C^{i}}_{i∈ℕ} and define subscaled lattices (resp. scaled lattices) as those L_{SC}-structures whose L_{SC_{d}}-reduct is a d-subscaled lattice (resp. a d-scaled lattice) for every d large enough. This is not an elementary class, but for any fixed integer d the class of (sub)scaled lattices of sc-dimension at most d (resp. exactly d) is elementary. Any d-(sub)scaled lattice expands uniquely to a (sub)scaled lattice of dimension at most d by realizing C^{i} as the constant map x ↦ 0 for every i > d. Conversely every (sub)scaled lattice L is of that kind for every d ≥ sc-dim(L).

2.6 Basic properties

The next additionnal properties follow easily from the axioms of subscaled lattices.

SC_{7} : sc-dim a = max{k / C^{k}(a) ≠ 0} In particular ∀k, sc-dim C^{k}(a) = k ⟺ C^{k}(a) ≠ 0

SC_{8} : sc-dim a ∨ b = max(sc-dim a, sc-dim b)

In particular b ≤ a ⇒ sc-dim b ≤ sc-dim a.

SC_{9} : ∀k, C^{k}(a) = ∨∨{b / b ≤ ∨∨_{0≤i≤k} C^{i}(a) and C^{k}(b) = b}

SC_{10} : sc-dim a ≤ dim a

SC_{11} : ∀I ⊆ {0, …, d}, a − ∨∨_{i∈I} C^{i}(a) = ∨∨_{i∉I} C^{i}(a)

In particular sc-dim(a − C^{d}(a)) < d.

SC_{12} : ∀k, C^{k}(a) − b = C^{k}(C^{k}(a) − b)

SC_{13} : ∀k, C^{k}(a) = a ⟺ ∀b (a − b ≠ 0 ⇒ sc-dim a − b = k)

Proof: (Sketch) SC_{7} follows from SC_{1} and SC_{2}; SC_{8} from SC_{2}, SC_{3} and SC_{7}; SC_{9} from SC_{2} and SC_{3}; eventually SC_{10} is equivalent to SC_{6} modulo the other axioms. Only the three last properties require a little effort.

SC_{11}: For every l ∈ I, C^{l}(a) ≤ ∨∨_{i∈I} C^{i}(a) hence C^{l}(a) − ∨∨_{i∈I} C^{i}(a) = 0. On the other hand for every l ∉ I and every i ∈ I, C^{l}(a) − C^{i}(a) = C^{l}(a) by SC_{4} and SC_{5}. So C^{l}(a) − ∨∨_{i∈I} C^{i}(a) = C^{l}(a) by TC_{4}. Eventually by SC_{1} and TC_{2}:

a − ∨∨_{i∈I} C^{i}(a) = ∨∨_{l≤d}(C^{l}(a) − ∨∨_{i>k} C^{i}(a)) = ∨∨_{l∉I} C^{l}(a)

In particular if I = {d} then a−C^{d}(a) = ∨∨_{k<d} C^{k}(a) hence sc-dim(a−C^{d}(a)) < d.

SC_{12}: By TC_{1}, C^{k}(a) = (C^{k}(a)−b)∨(C^{k}(a)∧b), and by SC_{8} both C^{k}(a)−b and C^{k}(a) ∧ b have sc-dimension at most k, so by SC_{3}:

C^{k}(C^{k}(a)) = C^{k}(C^{k}(a) − b) ∨ C^{k}(C^{k}(a) ∧ b)

By SC_{1}, C^{k}(C^{k}(a) ∧ b) ≤ C^{k}(a) ∧ b ≤ b, and by SC_{2}, C^{k}(a) = C^{k}(C^{k}(a)) so:

C^{k}(a) ≤ C^{k}(C^{k}(a) − b) ∨ b

It follows that C^{k}(a) − b ≤ C^{k}(C^{k}(a) − b), and equality holds by SC_{1}.

SC_{13}: If a = C^{k}(a) then a − b = C^{k}(a − b) for every b by SC_{12}. If moreover a − b ≠ 0 then sc-dim(a − b) = k by SC_{7}. Conversely assume that a ≠ C^{k}(a) (hence a ≠ 0). If sc-dim a ≠ k then a − 0 = a ≠ 0 hence sc-dim a − 0 ≠ k. If sc-dim a = k, let b = C^{k}(a). Then by assumption a − b ≠ 0, and by SC_{11}, a − b = ∨∨_{i<k} C^{i}(a) hence sc-dim a − b < k.

2.7 Miscellanies

Given an integer k we say that an element a of a distributive lattice L is k-pure in L if and only if^{3}:

∀b ∈ L (a − b ≠ 0 ⇒ dim_{L} a − b = k)

3Remember Example 1.1 and Footnote 1.

Then either a = 0 or dim_{L} a = k. In the latter case we say that a has pure dimension k in L.

Similarly in any d-subscaled lattice we say that an element a is k-sc-pure if and only if:

∀b ∈ L (a − b ≠ 0 ⇒ sc-dim a − b = k)

By SC_{13}, a is k-sc-pure if and only if a = C^{k}(a). Then by SC_{7}, either a = 0 or sc-dim a = k. In the latter case we say that a has pure sc-dimension k. For any a, the element C^{k}(a) is called the k-sc-pure component of a, and simply its k-pure component if L is a scaled lattice.

Proposition 2.8 The L_{SC_{d}}-structure of a d-scaled lattice L is uniformly definable in the L_{lat}-structure of L. In particular it is uniquely determined by this L_{lat}-structure.

Proof: Clearly the TC-structure is an extension by definition of the lattice structure of L. For every positive integer k the class of k-pure elements is uniformly definable, using the definablility of ≪. Then so is the function C^{k} for every k, by decreasing induction on k. Indeed by SC_{9} and SC_{11}, C^{k}(a) is the largest k-pure element c such that c ≤ a − ∨∨_{i>k} C^{i}(a).

Our study of (sub)scaled lattices is motivated by the examples given in the introduction. Although they are less natural, the following examples which will be needed further in this paper.

Example 2.9 In an arbitrary noetherian lattice L an L_{SC}-structure can be defined as follows. For every a, b ∈ L, if C(a) denotes the set of all ∨-irreducible components of a, let:

a − b = ∨∨{c ∈ C(a) / c ≰ b}
(∀k) C^{k}(a) = ∨∨{c ∈ C(a) / dim_{L} c = k}

This L_{TC}-structure (resp. L_{SC}-structure) is the only one (by Proposition 2.8) which turns L into a TC-lattice (resp. a scaled lattice). On the other hand, for any strictly increasing map D: I(L) → ℕ and any a, b ∈ L define a − b as above and:

(∀k) C_{D}^{k}(a) = ∨∨{c ∈ C(a) / D(c) = k}

This L_{SC}-structure turns L into a subscaled lattice which is not a scaled lattice (except if D coincides with the map dim_{L}). We will use without further mention the following obvious fact:

Fact 2.10 Every noetherian (hence in particular every finite) subscaled lattice is of the above kind.

Eventually the following notation will be convenient in induction arguments. If L is any of our languages L_{lat}, L_{TC}, L_{SC_{d}} or L_{SC} we let L^{∗} = L \ {1}. Given an L-structure L whose reduct to L_{lat} is a lattice, for any a ∈ L we denote by:

L(a) = {b ∈ L / b ≤ a}

L(a) is a typical example of L^{∗}-substructure of L.

3 Embeddinds of subscaled lattices

We need a reasonably easy criterion for an L_{lat}-embedding of subscaled lattices to be an L_{SC}-embedding. The special case of a noetherian embedded lattice, presented in the next proposition, is sufficient for this paper. However combining the model-theoretic compactness theorem with the local finiteness Theorem 4.1, one can easily derive from Proposition 3.1 that an L_{lat}-embedding φ : L → L′ between arbitrary subscaled lattices is an L_{SC}-embedding if and only if it preserves the sc-dimension and sc-purity, that is for every a ∈ L and every k ∈ ℕ:

C^{k}(a) = a ⟹ C^{k}(φ(a)) = φ(a)

Proposition 3.1 Let L_{0} be a noetherian subscaled lattice and φ : L_{0} → L an L_{lat}-embedding such that for every a ∈ I(L_{0}), φ(a) is sc-pure and has the same sc-dimension as a. Then φ is an L_{SC}-embedding.

Remark 3.2 Clearly the same statement remains true with L_{lat} and L_{SC} replaced respectively by L_{lat}^{∗} and L_{SC}^{∗} (or L_{SC_{d}}^{∗}). We will freely use these variants.

Proof: Let d = sc-dim L, and for any a ∈ L_{0}:

(a_{0}, …, a_{d}) = (φ(C^{0}(a)), …, φ(C^{d}(a)))

For every positive integer k, C^{k}(a) is k-sc-pure by SC_{13}, hence each ∨-irreducible component c of C^{k}(a) in L_{0} has pure sc-dimension k. By assumption each such φ(c) then has pure sc-dimension k. The join of finitely many elements of pure sc-dimension k is easily seen to be k-sc-pure by definition and by TC_{2}, so we have proved:

∀k, a_{k} is k-sc-pure.  (1)

Moreover for any k ≠ l, sc-dim(C^{k}(a) ∧ C^{l}(a)) < min(k, l) by SC_{4}. It follows that each ∨-irreducible component c of C^{k}(a) ∧ C^{l}(a) has sc-dimension stricly less than min(k, l), hence so does φ(c) by assumption. By SC_{8} we conclude that:

∀k ≠ l, sc-dim(a_{k} ∧ a_{l}) < min(k, l)  (2)

For every k > d, φ(C^{k}(a)) = φ(0) = 0. Each ∨-irreducible component of a has sc-dimension at most d hence so does φ(a) by assumption, so by SC_{8}, sc-dim(φ(a)) ≤ d. It follows that C^{k}(φ(a)) = 0 = C^{k}(φ(a)).

For every k ≤ d let:

b_{k} = ∨∨_{0≤l≤k} a_{l}  and  c_{k} = ∨∨_{0≤l≤k} C^{l}(φ(a))

By SC_{11}, c_{k} − C^{k}(φ(a)) = c_{k−1}. Moreover the proof of SC_{11} proves as well that b_{k} − a_{k} = ∨∨_{l≤k}(a_{l} − a_{k}) = b_{k−1}, thanks to (1) and (2) above.

Now assume that b_{k} = c_{k} for some k ≤ d. Then C^{k}(φ(a)) = C^{k}(c_{k}) by SC_{2}, so C^{k}(φ(a)) = C^{k}(b_{k}) by assumption. It follows by SC_{5} that:

C^{k}(φ(a)) − a_{k} = C^{k}(b_{k}) − C^{k}(a_{k}) = C^{k}(b_{k} − a_{k}) = C^{k}(b_{k−1})

SC_{8} implies that sc-dim(b_{k−1}) ≤ k − 1 so C^{k}(b_{k−1}) = 0 by SC_{2}. It follows that C^{k}(φ(a)) − a_{k} = 0 that is C^{k}(φ(a)) ≤ a_{k}. On the other hand a_{k} ≤ ∨∨_{l≤k} C^{k}(φ(a)) and C^{k}(a_{k}) = a_{k} by (1) so a_{k} ≤ C^{k}(φ(a)) by SC_{9}. We have proved:

b_{k} = c_{k} ⟹ a_{k} = C^{k}(φ(a))

Then b_{k−1} = b_{k} − a_{k} = c_{k} − C^{k}(φ(a)) = c_{k−1}. Since b_{d} = c_{d} = a it follows by decreasing induction that b_{k} = c_{k} for every k, that is φ(C^{k}(a)) = C^{k}(φ(a)). Incidentally, since φ is injective, this implies by SC_{7} that for every a ∈ L_{0}:

sc-dim a = sc-dim φ(a)  (3)

Now let a, b ∈ L_{0}, and a′, b′ be their images by φ. We have to show that φ(a− b) = a′ − b′. By TC_{2}, replacing if necessary a by its ∨-irreducible components, we may assume w.l.o.g. that a itself is ∨-irreducible in L_{0}. This implies that a = C^{k}(a) for some k. It then remains two possibilities for a − b:

• If b ≥ a then a − b = 0 and b′ ≥ a′ hence φ(a − b) = 0 = a′ − b′.

• Otherwise c = b ∧ a < a hence c ≪ a by TC_{1}, so sc-dim c < sc-dim a by SC_{6}. Let c′ = φ(c) = b′ ∧ a′, by assumption a′ = C^{k}(a) and by (3) sc-dim(c′) < sc-dim(a′) = k, hence by SC_{5} a′ − c′ = a′. We conclude that a − b = a and a′ − b′ = a′ hence φ(a − b) = φ(a) = a′ = a′ − b′.

We have proved that φ preserves − and the C^{k}’s operations, so φ is an L_{SC}-embedding.

Corollary 3.3 Let L_{0} be a noetherian sublattice of a subscaled lattice L. If for any b < a ∈ I(L), a is sc-pure in L and sc-dim b < sc-dim a in L, then L_{0} is an L_{SC}-substructure of L_{0}.

Proof: The assumptions imply that the map D : a ↦ sc-dim a is a stricly inreasing map from I(L_{0}) to ℕ. Endow L_{0} with the structure of subscaled lattice determined by D as in Example 2.9. Proposition 3.1 then applies to the inclusion map φ from L_{0} to L.

4 Local finiteness

We prove in this section that every finitely generated subscaled lattice is finite. This result is far non-obvious, due to the lack of any known normal form for terms in L_{SC}. It contrasts with the situation in TC-lattices, which can be both infinite and generated by a single element. Our main ingredient, which explains this difference, is the uniform bound given a priori for the sc-dimension of any element in a given subscaled lattice.

Theorem 4.1 Any subscaled lattice L of sc-dimension d generated by n elements is finite. More precisely, the cardinality of I(L) is then bounded by the fonction μ(n, d) defined by:

\begin{aligned}μ(n, −1) & = & 0 & (∀n) \\ μ(n, d) & = & 2^{n} + μ(2^{n+1}, d − 1) & (∀n, ∀d ≥ 0) \end{aligned}

Proof: If d = −1 the only subscaled lattice of dimension −1 is the one-element lattice {0}, so the result is trivial.

Assume the d ≥ 0 and that the result is proved for every d′ < d and every positive integer n. Let L be a subscaled lattice of sc-dimension d generated by elements x_{1}, …, x_{n}. Let Ω_{n} be the family of all subsets of {1, …, n} (so Ω_{0} = {∅}). For every I ∈ Ω_{n} let I^{c} = Ω_{n} \ I and:

y_{I} = (∧∧_{i∈I} x_{i}) − ( ∨∨_{i∈I^{c}} x_{i}), z_{I} = C^{d}(y_{I})

The family of all Y_{I} = ⋂_{i∈I} P (x_{i}) ∩ ⋂_{i∈I^{c}} P (x_{i})^{c} is a partition of Spec(L). Indeed the Y_{i}’s are the atoms of the boolean algebra generated in the power set P(Spec(L)) by the P (x_{i})’s. Moreover each P (y_{I}) is the topological closure \overline{Y}_{I} of Y_{I} in Spec(L) hence for every x ∈ L:

P (x) = ⋃_{I∈Ω_{n}} P (x) ∩ Y_{I} ⊆ ⋃_{I∈Ω_{n}} P (x) ∩ \overline{Y}_{I} = P ( ∨∨_{I∈Ω_{n}} x ∧ y_{I})

So x ≤ ∨∨_{I∈Ω_{n}}(x ∧ y_{I}) by Stone-Priestley’s duality. The reverse inequality being

obvious we have proved:

∀x ∈ L, x ≤ ∨∨_{I∈Ω_{n}}(x ∧ y_{I})  (4)

In particular SC_{3} also gives:

C^{d}(1) = C^{d}( ∨∨_{I∈Ω_{n}} y_{I}) = ∨∨_{I∈Ω_{n}} z_{I}  (5)

For every I ≠ J ∈ Ω_{n}, if for example I ⊈ J choose any i ∈ I \ J and observe that y_{I} ≤ x_{i} and y_{J} ≤ 1 − x_{i} so y_{I} ∧ y_{J} ≪ 1 − x_{i} by TC_{3}. By SC_{6} and the d-sc-purity of the z_{I}’s it follows that:

sc-dim z_{I} ∧ z_{J} < d hence z_{I} − z_{J} = z_{I}  (6)

It follows from SC_{8}, SC_{11} and (6) above, that the element:

a = (1 − C^{d}(1)) ∨ ( ∨∨_{I∈Ω_{n}}(y_{I} − z_{I})) ∨ ( ∨∨_{I≠J∈Ω_{n}}(z_{I} ∧ z_{J}))

has sc-dimension strictly smaller than d. So the induction hypothesis applies to the L_{SC}-substructure L_{0}^{−} of L(a) generated by the (y_{I} − z_{I})’s and the (z_{I} ∧ a)’s: L_{0}^{−} is finite, with at most μ(2|Ω_{n}|, d − 1) non-zero ∨-irreducible elements. Eventually let L_{1} be the upper semi-lattice generated in L by L_{0}^{−} ∪ {z_{I}}_{I∈Ω_{n}}. By construction L_{1} is finite and I(L_{1}) ⊆ I(L_{0}) ∪ {z_{I}}_{I∈Ω_{n}}, so |I(L_{1})| ≤ 2^{n} + μ(2^{n+1}, d − 1) = μ(n, d). It is then sufficient to show that L_{1} = L.

By (5), 1 = C^{d}(1) ∨ a = ∨∨_{I∈Ω_{n}} z_{I} ∨ a ∈ L_{1}. For every I ∈ Ω_{n} and every b ∈ L_{0}^{−}, z_{I} ∧ b = (z_{I} ∧ a) ∧ b ∈ L_{0}^{−}. Eventually for every I ≠ J ∈ Ω_{n}, z_{I} ∧z_{j} = (z_{I} ∧a)∧(z_{J} ∧a) ∈ L_{0}^{−}. So by the distributivity law, L_{1} is a sublattice of L.

Since I(L_{1}) ⊆ I(L_{0}^{−}) ∪ {z_{I}}_{I∈Ω_{n}} it is immediate that for any b′ < b in I(L_{1}), sc-dim b′ < sc-dim b. So L_{1} is an L_{SC}-substructure of L by Corollary 3.3. Moreover each y_{I} = (y_{I} − z_{I}) ∨ z_{I} ∈ L_{1} and for every i ≤ n, (4) gives:

x_{i} = ∨∨_{I∈Ω_{n}} x_{i} ∧ y_{I} ≤ ∨∨_{{I∈Ωn \atop i∈I}} y_{I} ≤ x_{i}

So equality holds, hence each x_{i} ∈ L_{1} and eventually L = L_{1}.

Corollary 4.2 For every n, d there are finitely many non-isomorphic subscaled lattices of sc-dimension d generated by n elements.

Proof: Any such subscaled lattice L is finite, with |I(L)| ≤ μ(n, d) by Theorem 4.1. Clearly there are finitely many non-isomorphic lattices such that |I(L)| ≤ μ(n, d) and each of them admits finitely many non-isomorphic L_{SC_{d}}-structures. The conclusion follows.

5 Linear representation

In this section we prove that the theory of d-subscaled lattices is the universal theory of various natural classes of L_{SC_{d}}-structures, including SC_{Zar}(K, d). The argument is based on an elementary representation theorem for d-subscaled lattices, combined with the local finiteness result of Section 4.

Given an arbitrary field K, a non-empty linear variety X ⊆ K^{m} is determined by the data of an arbitrary point P ∈ X and the vector subspace \overrightarrow{X} of K^{m}, via the relation X = P + \overrightarrow{X} (the orbit of P under the action of \overrightarrow{X} by translation). We call X a special linear variety (resp. a special linear set) if X is a linear variety such that \overrightarrow{X} is generated by a subset of the canonical basis of K^{m} (resp. if X is a finite union of special linear varieties). For example the empty set is a special linear set, as the union of an empty family of special linear varieties. The family L_{lin}(X) of all special linear subsets of X is a noetherian lattice (because it is the family of closed sets of a noetherian topology on X). So it has a natural structure of scaled lattice defined as in Example 2.9.

Remark 5.1 For every A ∈ L_{lin}(X), sc-dim A = dim_{L_{lin}(X)} A = the dimension of A as defined in linear algebra. If K is infinite then this dimension coincides with the Krull dimension of A. Moreover if A is ∨-irreducible in L_{lin}(X) then it is pure dimensionnal, hence it is sc-pure both in L_{lin}(X) and L_{Zar}(X). By Proposition 3.1 it follows that if K is infinite then L_{lin}(X) is an L_{SC}-substructure of L_{Zar}(X). Similarly if K is a henselian valued field of characteristic zero, a real closed field or an algebraically closed field then L_{lin}(X) is an L_{SC}-substructure of L_{def}(X).

In the following proposition K^{m} is identified to the subset K^{m} × {0}^{r} of K^{m+r}.

Proposition 5.2 Given any two special linear sets C ⊆ B ⊆ K^{m} and any N ≥ dim C there exists a positive integer r and a special linear set A ⊆ K^{m+r} of pure dimension n such that A ∩ B = C.

Proof: For any integer n let (e_{1}, …, e_{n}) be the canonical basis of K^{n}. If I is a subset of \{1, …, n\} we denote \overrightarrow{E}(I) the vector subspace of K^{n} generated by (e_{i})_{i∈I}.

Decompose C as a union of special linear varieties: C_{1}, …, C_{p} with each C_{i} = P_{i} + \overrightarrow{E}(J_{i}) and |J_{i}| = dim C_{i} ≤ n. Let r be larger than 0, N − m and every N − |J_{i}|. For each i take an arbitrary subset J′_{i} of cardinality N − |J_{i}| inside {m + 1, …, m + r} and let I_{i} = J_{i} ∪ J′_{i}. Choose an arbitrary point P_{0} ∈ K^{m+r} \ K^{m} and let I_{0} = \{1, …, N\}. Eventually let A_{i} = P_{i} + \overrightarrow{E}(I_{i}) for every i ≤ p, and A = A_{0} ∪ … ∪ A_{p}. By construction each A_{i} has dimension |I_{i}| = N, hence A is N-pure. At least A_{0} is non empty (it is the only one in case p = 0 that is if C is empty) hence A has pure dimension N. Clearly A∩K^{m} = C hence a fortiori A ∩ B = C.

Proposition 5.3 (Linear representation) Let K be an infinite field and let L be any finite subscaled lattice. Then there exists a linear set X over K and an L_{SC}-embedding φ: L → L_{lin}(X).

Remark 5.4 Since φ, in the above proposition, is an L_{SC}-embedding it preserves the sc-dimension hence dim X = sc-dim L. So if sc-dim L ≤ d we can indentify L and X with their respective L_{SC_{d}}-reduct, and φ is then also an L_{SC_{d}}-embedding.

Proof: Remember that L_{SC}^{∗} = L_{SC}\{1}. We prove by induction on the number n of non-zero ∨-irreducible elements of a subscaled lattice L that there exists an L_{SC}^{∗}-embedding φ of L into L_{lin}(K^{m}) for some m depending on L.

For n = 0, L is the one-element lattice {0} hence it is an L_{SC}^{∗}-substructure of L_{lin}(K).

Let n ≥ 1, assume the result proved for n − 1 and take a subscaled lattice L with non-zero ∨-irreducible elements a_{1}, …, a_{n}. Reordering if necessary we may assume that a_{n} is maximal among the a_{i}’s. Let a = a_{n}, b = ∨∨_{1≤i<n} a_{i}, c = a∧b and φ an L_{SC}^{∗}-embedding of L(b) into some L_{lin}(K^{m}) given by induction hypothesis. Since a is ∨-irreducible in L it is sc-pure. Moreover c ≪ a by TC_{1}, hence a has pure sc-dimension N for some N > sc-dim(c) by SC_{6}. Let B, C be the respective images of b, c by φ. Proposition 5.2 gives a positive integer r and a special linear set A ⊆ K^{m+r} of pure dimension N such that A ∩ B = C. Since we indentified K^{m} with K^{m}×{0}^{r} we consider L_{lin}(K^{m}) as an L_{SC}^{∗}-substructure of L_{lin}(K^{m+r}). So φ actually embeds L(b) into L_{lin}(K^{m+r}).

Every element x of L writes uniquely x_{a} ∨ x_{b} with x_{a} ∈ {0, a} and x_{b} ∈ L(b) (group appropriately the ∨-irreducible components of x, using the maximality of a_{n}) hence:

φ̄(x) = \{ \begin{matrix}φ(x_{b}) & if x_{a} = 0 \\ A ∪ φ(x_{b}) & if x_{a} = a \end{matrix}

is a well-defined L_{lat}^{∗}-embedding of L into L_{lin}(K^{m+r}). Moreover φ̄ is an L_{SC}^{∗}-embedding by Proposition 3.1. This finishes the induction.

We have constructed an L_{SC}^{∗}-embedding φ of L into L_{lin}(K^{m}) for some m. Then X = φ(1_{L}) is a special linear set, so φ induces an L_{SC}-embedding of L into L_{lin}(X).

For any infinite field K and positive integer d let SC_{lin}(K, d) be the class of d-scaled lattices L_{lin}(X) for every special linear variety X over K of dimension at most d.

Theorem 5.5 For any infinite field K and positive integer d, the universal theories of SC_{lin}(K, d) and SC_{Zar}(K, d) are exactly the theory of d-subscaled lattices.

If moreover K is a henselian valued field of characteristic zero, a real closed field or an algebraically closed field then the same holds for SC_{def}(K, d).

Proof: By Remark 5.1 it suffices to prove the theorem for SC_{lin}(K, d). Let T (K, d) be its universal theory.

The linear representation Proposition 5.3 shows that every finite d-subscaled lattice embeds into some L ∈ SC_{lin}(K, d) hence is a model of T (K, d). Since every finitely generated d-subscaled lattice is finite by Theorem 4.1, the modeltheoretic compactness argument then implies that any d-subscaled lattice is a model of T (K, d).

Conversely every L ∈ SC_{lin}(K, d) is a d-scaled lattice hence obviously its universal theory contains the theory of d-subscaled lattices.

6 Primitive extensions

This section is devoted to the study of minimal proper extensions of finite subscaled lattices. Let L_{0} be a finite subscaled lattice, L an L_{SC}-extension of L_{0} and x ∈ L. We introduce the following notation.

• For every a ∈ L_{0}, a^{−} = ∨∨{b ∈ L_{0} / b < a}.

• L_{0}⟨x⟩ denotes the L_{SC}-substructure of L generated by L_{0} ∪ {x}.

• g(x, L_{0}) = ∧∧{a ∈ L_{0} / x ≤ a}.

Clearly a ∈ I(L_{0}) if and only if a^{−} is the unique predecessor of a in L_{0} (otherwise a^{−} = a). We say that a tuple (x_{1}, x_{2}) of elements of L is primitive over L_{0} if there exists g ∈ I(L_{0}) such that:

1. x_{1}, x_{2} are sc-pure of the same sc-dimension.

2. Each g^{−} ∧ x_{i} ∈ L_{0}.

3. One of the following happens:

• x_{1} = x_{2} and g^{−} ∧ x_{1} ≪ x_{1} ≪ g.

• x_{1} ≠ x_{2}, x_{1} ∧ x_{2} ∈ L_{0} and g − x_{1} = x_{2}, g − x_{2} = x_{1}.

The above conditions imply that each x_{i} ∉ L_{0} and:

g = g(x_{1}, L_{0}) = g(x_{2}, L_{0})

We say that L is primitively generated over L_{0}, or simply that it is a primitive extension of L_{0}, if there exists (x_{1}, x_{2}) primitive over L_{0} such that L = L_{0}⟨x_{1}, x_{2}⟩ (then clearly L = L_{0}⟨x⟩_{1} = L_{0}⟨x⟩_{2}). By the following proposition such a tuple is necessarily unique.

Proposition 6.1 Let L_{0} be a finite subscaled lattice, L an extension generated over L_{0} by a primitive tuple (x_{1}, x_{2}), and let g = g(x_{1}, L_{0}).

Then L is exactly the upper semi-lattice generated over L_{0} by x_{1}, x_{2}. It is a finite subscaled lattice and one of the following happens:

1. x_{1} = x_{2}, sc-dim x_{1} < sc-dim g and I(L) = I(L_{0}) ∪ {x_{1}}.

2. x_{1} ≠ x_{2}, sc-dim x_{1} = sc-dim g and I(L) = (I(L_{0}) \ {g}) ∪ {x_{1}, x_{2}}.

Proof: Let L_{1} be the upper semi-lattice generated over L_{0} by x_{1}, x_{2}. In order to show that L_{1} = L it is sufficient to check that L_{1} is an L_{SC}-substructure of L.

Let p = sc-dim g and q = sc-dim x_{1} = sc-dim x_{2}. By TC_{1}, g^{−} ≪ g hence g^{−} ∧ x_{i} ≪ g since g^{−} ∧ x_{i} ∈ L_{0} by assumption. So sc-dim g^{−} < p. We will need the following facts, for every a ∈ L_{0}:

g ≰ a ⟹ a ∧ x_{i} ∈ L_{0} and a ∧ x_{i} ≪ x_{i}  (7)

Indeed g ∧ a = g^{−} ∧ a hence a ∧ x_{i} = (a ∧ g) ∧ x_{i} = a ∧ (g^{−} ∧ x_{i}). By assumption x_{i} ∧ g^{−} ∈ L_{0} hence a ∧ x_{i} ∈ L_{0} is proved. Moreover x_{i} ∧ (g^{−} ∧ a) ≤ x_{i} ∧ g^{−} hence it suffices to check that g^{−} ∧ x_{i} ≪ x_{i}. If x_{1} = x_{2} this is an assumption. Otherwise x_{1} = g−x_{2} and x_{2} = g−x_{1} are p-sc-pure by SC_{13}. Then x_{i} −g^{−} = x_{i} by SC_{5}, since sc-dim g^{−} < p, so g^{−} ∧ x_{i} ≪ x_{i}.

L_{1} is a sublattice of L then follows easily from (7), the distributivity law, and the fact that by assumption x_{1} ∧ x_{2} ∈ L_{0} ∪ {x_{1}}.

Since L_{0} is finite and L_{1} is generated by L_{0} ∪ {x_{1}, x_{2}} as an upper semilattice, it follows immediatly that L_{1} is finite and:

I(L_{1}) ⊆ I(L_{0}) ∪ {x_{1}, x_{2}}  (8)

So for any b′ < b ∈ cI(L_{1}) it is easily seen that sc-dim(b′) < sc-dim b in L. Corollary 3.3 then shows that L_{1} is an L_{SC}-substructure of L, hence L_{1} = L.

We turn now to the description of I(L). If x_{1} ≠ x_{2} then of course g = x_{1} ∨ x_{2} ∉ I(L). Conversely if x_{1} = x_{2} then (8) implies that g is ∨-irreducible in L, so:

g ∈ I(L) ⟺ x_{1} ≠ x_{2}  (9)

Assume that I(L_{0}) ⊈ I(L). Let b ∈ I(L_{0}) \ I(L) and let y_{1}, …, y_{r} (r ≥ 2) be its ∨-irreducible components in L_{1}. By (8), each y_{i} either belongs to L_{0} or to {x_{1}, x_{2}}, and at least one of them does not belong to L_{0}. We may assume without loss of generality that y_{1} = x_{1}. Then x_{1} ≤ b hence g ≤ b. If g < b then g ≪ b since b ∈ I(L_{0}) so b − g = b, but then we have a contradiction:

y_{1} ≤ b − g ≤ b − x_{1} = ∨∨_{i=2}^{r} y_{i}

So b = g. We have proved that:

I(L_{0}) \ {g} ⊆ I(L)  (10)

The conclusion follows by combining (8), (9), (10) and an obvious argument of cardinality.

It is not difficult to deduce from Proposition 6.1 that any primitively generated extension L of a finite subscaled lattice L_{0} is minimal, in the sense that there is no intermediate proper extension L_{0} ⊂ L_{1} ⊂ L. Only the converse, which we prove now, is actually needed in the remaining of this paper.

Proposition 6.2 Any finitely generated proper extension L of a finite d-subscaled lattice L_{0} is the union of a finite chain of primitively generated extensions of L_{0}.

Proof: Since L is finite by the local finiteness Theorem 4.1, it suffices to show that L contains a primitive extension of L_{0}. Take any element x minimal in I(L) \ L_{0}. Observe that if y is any element of L stricly smaller than x then all the ∨-irreducible components of y in L actually belong to L_{0}, so y ∈ L_{0}.

Let g = g(x, L_{0}). For every a ∈ L_{0}, if a < g then a ≰ x hence a ∧ x < x, so a ∧ x ∈ L_{0}. It follows that g^{−} ∧ x ∈ L_{0}, hence g^{−} < g. In particular g ∈ I(L_{0}).

Since x ∈ I(L) it is sc-pure, and moreover g^{−}∧x < x implies that g^{−}∧x ≪ x by TC_{1}. So if moreover x ≪ g then we have proved that (x, x) is primitive over L_{0}.

On the other hand if x ≪̸ g let x_{1} = x and x_{2} = g − x. Since g ∈ I(L_{0}) it is sc-pure, and so are x_{1} and x_{2}, with the same sc-dimension p = sc-dim g (indeed g − x ≠ 0 since x < g). Moreover x_{1} ∧ x_{2} ≪ x_{2} by TC_{3} so sc-dim x_{1} ∧ x_{2} < p hence g − x_{1} = x_{2} and g − x_{2} = x_{1} by TC_{1}, TC_{2} and the p-sc-purity of x_{1} and x_{2}. Eventually x_{1} ∧ x_{2} < x imply that x_{1} ∧ x_{2} ∈ L_{0}. We conclude that (x_{1}, x_{2}) is primitive over L_{0}.

7 Signatures

A triple (g, q, H) will be called a signature in a finite subscaled lattice L_{0} if and only if g ∈ I(L_{0}), q ≤ sc-dim g is a positive integer, and H is a subset of two (not necessarily distinct) elements h_{1}, h_{2} ∈ L_{0} such that:

• h_{1} ∨ h_{2} < g (hence h_{1} ∨ h_{2} ≪ g by TC_{1})

• sc-dim(h_{1} ∨ h_{2}) < q

• If q < sc-dim g then h_{1} = h_{2}.

By Proposition 6.1 and the definition, any extension L primitively generated over L_{0} by x_{1}, x_{2} determines a unique signature:

σ(L) = (g(x_{1}, L_{0}), sc-dim x_{1}, {x_{1} ∧ g^{−}, x_{2} ∧ g^{−}})

which we call the signature of L in L_{0}. It determines the extension L|L_{0} as follows.

Proposition 7.1 Two primitively generated extensions of a finite subscaled lattice L_{0} are isomorphic over L_{0} if and only if they have the same signature in L_{0}.

Proof: Let L (resp L′) be an extension of L_{0} generated over L_{0} by a primitive tuple (x_{1}, x_{2}) (resp. x′_{1}, x′_{2}). If they are isomorphic over L_{0} then obviously σ(L) = σ(L′). Conversely assume that:

σ(L) = σ(L′) = (g, q, {h_{1}, h_{2}})

Reordering if necessary we may assume that each h_{i} = x_{i} ∧ g^{−} = x′_{i} ∧ g^{−}. Let φ(x_{i}) = x′_{i} for each i ∈ {1, 2}, and φ(a) = a for every a ∈ I(L_{0}). By Proposition 6.1 this is a well defined bijection from I(L) to I(L′) which preserves the order, hence it extends uniquely to an isomorphism of upper semi-lattice φ: L′ → L. The L_{lat}-structure of a lattice being an extension by definition of its upper semi-lattice structure, this is an L_{lat}-isomorphism. Moreover by construction φ preserves the sc-dimensions of the ∨-irreducible elements of L and L′, hence by Proposition 3.1 it is an L_{SC}-isomorphism, whose restriction to L_{0} is the identity.

Proposition 7.2 Let σ = (g, q, {h_{1}, h_{2}}) be a signature in a finite subscaled lattice L_{0}. There exists a primitive extension L of L_{0} whose signature in L_{0} is precisely σ.

Proof: We only treat the case when q = sc-dim g. The case when q < sc-dim g (hence h_{1} = h_{2}) is similar, and left to the reader.

Let x_{1}, x_{2} be any two distinct elements in a set disjoint from I(L_{0}) and let:

I = (I(L_{0}) \ {g}) ∪ {x_{1}, x_{2}}

The order on I(L_{0}) \ {g} inherited from L_{0} can be extended to I by stating that x_{1} ≰ x_{2}, x_{2} ≰ x_{1}, and for every b ∈ I(L_{0}) \ {g} and every j ∈ {1, 2}:

• b < x_{j} ⟺ b ≤ h_{j}.

• x_{j} < b ⟺ g ≤ b

For every z ∈ I let z↑ = {y ∈ I / z ≤ y}. Let L be the upper-semilattice generated in the power set P(I) of I by all the z↑’s with z ranging over I. This is a sublattice of P(I) such that I(L) = {z↑ / z ∈ I}, that is:

I(L) = ({b↑ / b ∈ I(L_{0})} \ {g↑}) ∪ {x_{1}↑, x_{2}↑}

Let φ(g) = x_{1}↑ ∪ x_{2}↑, and φ(b) = b↑ for every b ∈ I(L_{0}) \ {g}. This application uniquely extends to an embedding of upper semi-lattice that we still denote φ from L_{0} to L, which is easily seen to be an L_{lat}-embedding. For every z ∈ I let:

D(z↑) = \{ \begin{matrix}sc-dim(z) & if z ∈ I(L_{0}) \ \{g\}, \\ sc-dim(g) & otherwise. \end{matrix}

This is a strictly increasing map from I(L) to ℕ hence it determines an L_{SC_{d}}-structure on L as in Example 2.9. Proposition 3.1 asserts that φ is an L_{SC_{d}}-embedding. By construction L is primitively generated over L_{0} by x_{1}↑, x_{2}↑, with signature σ in L_{0}.

The case when q < sc-dim g (hence h_{1} = h_{2}) is similar, and left to the reader.

8 Model-completion

We call super d-scaled lattice (resp. super scaled lattice) any d-subscaled lattice (resp. subscaled lattice) L which satisfy the following additionnal properties.

Scaling: L is a scaled lattice.

Catenarity: For every positive integers r ≤ q ≤ p and every elements c ≤ a, if a has pure dimension p and c has pure dimension r then there exists a q-pure element b such that c ≤ b ≤ a.

Splitting: For every elements b_{1}, b_{2}, a, if b_{1} ∨ b_{2} ≪ a then there exists non-zero elements a_{1} ≥ b_{1} and a_{2} ≥ b_{2} such that:

\{ \begin{matrix}a_{1} = a − a_{2} \\ a_{2} = a − a_{1} \\ a_{1} ∧ a_{2} = b_{1} ∧ b_{2} \end{matrix}

All these properties are clearly axiomatizable in L_{SC_{d}}, using only finitely many ∀∃-formulas. The name of the second one comes form the fact that in a subscaled lattice whose spectrum is a noetherian topological space, this property is equivalent to the usual notion of catenarity, namely that any two maximal chains in Spec(L) having the same first and last element have the same length.

Proposition 8.1 Every subscaled lattice L_{SC}-embeds in a superscaled lattice.

Proof: Let L_{0} be a finitely generated subscaled lattice and a, b_{1}, b_{2} ∈ L_{0} such that b_{1}∨b_{2} ≪ a. By Theorem 4.1, L_{0} is finite. Let g_{1}, …, g_{r} be the ∨-irreducible components of a in L_{0}, and for every i ≤ r let g_{i}^{−} be the unique predecessor of g_{i} in L_{0}, h_{i,1} = b_{1} ∧ g_{i}^{−} and h_{i,2} = b_{2} ∧ g_{i}^{−}.

σ_{1} = (g_{1}, sc-dim g_{1}, {h_{1,1}, h_{1,2}}) is a signature in L_{0}. Proposition 7.2 gives an extension L_{1} primitively generated over L_{0} by x_{1,1}, x_{1,2}, with signature σ_{1} in L_{0}. By construction x_{1,1} ∧ x_{1,2} belongs to L_{0} and is stricly smaller than g_{1} hence x_{1,1} ∧ x_{1,2} ≤ g_{1}^{−} For every i ≥ 2, g_{i} ∈ I(L_{0}) \ {g_{1}} ⊆ I(L_{1}), and moreover g_{i}^{−} is still the unique predecessor of g_{i} in L_{1}. Indeed let g_{i}^{†} be the union of every c ∈ I(L_{1}) strictly smaller than g_{i}. Neither x_{1,1} nor x_{1,2} are smaller than g_{i} because g_{1} ≰ g_{i}, so every such c must belong to I(L_{0}) by Proposition 6.1. It follows that g_{i}^{†} = g_{i}^{−}.

So we can repeat in L_{1} the same construction applied to g_{2}, h_{2,1}, h_{2,2}, and after r steps we obtain a chain of extensions (L_{i})_{i≤r} and non-zero elements x_{i,j} ∈ L = L_{r} such that for every i (1 ≤ i ≤ r):

g_{i} − x_{i,1} = x_{i,2} and g_{i} − x_{i,2} = x_{i,1}  (11)
x_{i,1} ∧ g_{i}^{−} = h_{i,1} and x_{i,2} ∧ g_{i}^{−} = h_{i,2}  (12)
x_{i,1} ∧ x_{i,2} ≤ g_{i}^{−}  (13)

Let x_{1} = ∨∨_{i≤r} x_{i,1} and x_{2} = ∨∨_{i≤r} x_{i,2}. For every i ≤ r and every k ≠ i, g_{i} ∧ x_{k,1} ≤ g_{i} ∧ g_{k} ≪ g_{i}, so by TC_{4} and (11):

g_{i} − x_{1} = g_{i} − ∨∨_{1≤k≤r} x_{k} = g_{i} − x_{i,1} = x_{i,2}

So by TC_{2}, a − x_{1} = ∨∨_{i≤r}(g_{i} − x_{1}) = x_{2}. Symmetrically a − x_{2} = x_{1}.

(12) and (13) imply that x_{i,1} ∧ x_{i,2} = x_{i,1} ∧ g_{i}^{−} ∧ x_{i,2} = h_{i,1} ∧ h_{i,2}. Similarly for every k ≠ i:

x_{i,1} ∧ x_{k,2} = x_{i,1} ∧ g_{i} ∧ g_{k} ∧ x_{k,2} = x_{i,1} ∧ g_{i}^{−} ∧ g_{k}^{−} ∧ x_{k,2} = h_{i,1} ∧ h_{k,2}

So eventually:

x_{1} ∧ x_{2} = ∨∨_{1≤i≤r} ∧∧_{1≤k≤r} x_{i,1} ∧ x_{k,2} = ∨∨_{1≤i≤r} ∧∧_{1≤k≤r} h_{i,1} ∧ h_{k,2} = b_{1} ∧ b_{2}

We have proved that for every finitely generated scaled lattice L_{0} and every a, b_{1}, b_{2} ∈ L_{0} such that b_{1} ∨ b_{2} ≪ a, there exists an L_{SC}-embeding φ from L_{0} to a subscaled lattice L in which there exists non-zero elements x_{1}, x_{2} such that, after identifying L_{0} to its image:

\{ \begin{matrix}x_{1} = a − a_{2} \\ x_{2} = a − a_{1} \\ x_{1} ∧ x_{2} = φ(b_{1}) ∧ φ(b_{2}) \end{matrix}

On the other hand, Theorem 4.1 and Proposition 5.3, show that every finitely generated subscaled lattice also L_{SC}-embeds into some L_{lin}(X), which is a catenary scaled lattice. The model-theoretic compactness argument then implies that every subscaled lattice L_{SC}-embeds in a superscaled lattice.

Proposition 8.2 Let L_{0} be a finite L_{SC}-substructure of a super scaled lattice L̂. Then for every signature σ in L_{0} there exists a primitive tuple (x_{1}, x_{2}) ∈ L̂ such that σ is the signature of L_{0}⟨x_{1}, x_{2}⟩ in L_{0}.

Proof: Let σ = (g, q, {h_{1}, h_{2}}) be a signature in L_{0}, let p = sc-dim g and r = sc-dim(h_{1} ∨ h_{2}). Observe that since g is ∨-irreducible in L_{0} it is sc-pure, and admits a unique predecessor g^{−} in L_{0}. Let y_{1}, y_{2} ∈ L̂ given by the splitting property applied to g, h_{1}, h_{2}. By construction y_{1} ∨ y_{2} = g, and since g has pure sc-dimension p so does each y_{i}. Moreover:

y_{1} ∧ h_{2} ≤ y_{1} ∧ y_{2} = h_{1} ∧ h_{2}

hence y_{1} ∧ (h_{1} ∨ h_{2}) = h_{1} ∨ (y_{1} ∧ h_{2}) = h_{1}. Since h_{1} ∨ h_{2} < g it follows that y_{1} ∧ g^{−} = h_{1} ∈ L_{0}, and symetrically y_{2} ∧ g^{−} = h_{2} ∈ L_{0}. So (y_{1}, y_{2}) is primitive over L_{0}, and the signature of L_{1} = L_{0}⟨y_{1}, y_{2}⟩ in L_{0} is (g, p, {h_{1}, h_{2}}).

If p = q then we are done. Assume now that q < p, hence h_{1} = h_{2}. For every i ≤ r, the catenarity gives an element x_{i} ∈ L̂ of pure sc-dimension p such that C^{i}(h_{1}) ≤ x_{i} ≤ y_{1}. Even in case r = −1 the catenarity gives x_{−1} ∈ L̂ of pure sc-dimension p such that x_{−1} ≤ y_{1}. Let x = ∨∨_{−1≤i≤r} x_{i}, by construction x has pure sc-dimension p and h_{1} ≤ x ≤ y_{1}. Then x ≤ g, and in view of their sc-dimension x ≪ g. Moreover:

h_{1} ≤ x ∧ g^{−} ≤ y_{1} ∧ g^{−} = h_{1}

hence h_{1} ∧ x = h_{1} ∈ L_{0} and in view of the sc-dimensions, h_{1} ≪ x. So (x, x) is a primitive tuple over L_{0}, of signature σ in L_{0}.

Theorem 8.3 The theory of super d-scaled lattices is the model-completion of the theory of d-subscaled lattices

Remark 8.4 Obviously the theorem remains true by replacing everywhere d-(sub)scaled lattices by (sub)scaled lattices of sc-dimension at most d (resp. exactly d).

Proof: Since the asioms of super d-scaled lattices are ∀∃, it follows from Proposition 8.1 that every existentially closed d-subscaled lattice is super d-scaled. By classical model-theoretic arguments, it is then sufficient to show that: given a d-superscaled lattice L̂, a finitely generated d-subscaled lattice L, and a common L_{SC_{d}}-substructure L_{0} of L and L̂, there exists an L_{SC_{d}}-embedding of L into L̂ whose restriction to L_{0} is the identity.

By the local finiteness Theorem 4.1, L_{0} is finite. By Proposition 6.2 an immediate induction allows us to assume w.l.o.g. that L is primitively generated over L_{0}. Let σ be the signature of L over L_{0}. By Proposition 8.2 there exists a primitively generated extension L_{1} of L_{0} in L̂ whose signature is σ. Eventually L is isomorphic to L_{1} over L_{0} by Proposition 7.1.

The completions of the theory of super d-scaled lattices are easy to classify. Let us say that a d-subscaled lattice is prime if it does not contain any proper d-subscaled lattice, or equivalently if it is generated by the empty set. Every prime d-subscaled lattice is finite. By Corollary 4.2 their exists finitely many prime d-subscaled lattices up to isomorphism.

Corollary 8.5 The theory of super d-scaled lattices containing (a copy of) a given prime d-subscaled lattice is ℵ_{0}-categorical, hence complete. It is also finitely axiomatisable, hence decidable. Since every completion of the theory of super d-scaled lattices is of that kind, the theory of super d-scaled lattices is decidable.

Proof: Let L, L′ be any two countable super d-scaled lattices containing isomorphic prime d-subscaled lattice L_{0} and L′_{0}. Any partial ismorphism between L and L′ (extending the given isomorphism between L_{0} and L′_{0}) can be extended by va-et-vient, using exactly the same argument as in the proof of Theorem 8.3. So the first statement is proved. The other ones are immediate consequences.

9 Atomic scaled lattices

A natural example of a super scaled lattice is not easy to find. Indeed if X is any topological space in which points are closed, the points of X are the atoms of L(X), and the splitting axiom imply that a super scaled lattice has no atom. In this section we explore a possible solution to this problem, which lead us to a new conjecture in p-adic semi-algebraic geometry.

Let L_{ASC} = L_{SC} ∪ {At_{k}}_{k∈ℕ^{∗}}with each At_{k} a new unary predicate symbol. We call sub-ASC-lattices the L_{ASC}-structures whose L_{SC}-reduct is a subscaled lattice and which satisfy the following lists of universal axioms:

ASC_{1} : (∀k ≠ l), ∀a, At_{k}(a) → ¬At_{l}(a)

ASC_{2} : (∀k), ∀a, a_{0}, …, a_{2^{k}}, At_{k}(a) ⟶

sc-dim a = 0⋀ [( ∨∨_{0≤i≤2^{k}} a_{i} = a) → ( ∨∨_{0≤i<j≤2^{k}} a_{i} = a_{j})]

ASC_{3} : (∀k, n), ∀a, a_{1}, …, a_{n},

[(a = ∨∨_{1≤i≤n} a_{i})⋀( ∧∧_{1≤j<i≤n} a_{i} ∧ a_{j} = 0)] ⟶

[At_{k}(a) ⟷ ⋁ ; ⋁_{l_{1}+···+l_{n}=k} ⋀ ; ⋀_{1≤i≤n}At_{l_{i}}(a_{i})]

For any L_{ASC}-structure L we denote by At_{k}(L) the set of elements a in L such that L ⊧ At_{k}(a), and we let At_{0}(L) = L \ ⋃_{k>0} At_{k}(L). If L is a sub-ASC-lattice then ASC_{1} asserts that (At_{k}(L))_{k∈ℕ} is a partition of L. For any a ∈ L we then define asc(a) as the unique k ∈ ℕ such that a ∈ At_{k}(L). The other axioms of sub-ASC-lattices have the following meaning:

• If asc(a) = k > 0 then ASC_{2} asserts that L(a) is a boolean algebra generated by n atoms a_{1}, …, a_{n} with n ≤ k, and ASC_{3} then implies that each asc(a_{i}) > 0 and for every b ∈ L(a):

asc(b) = ∑_{c∈C(b)} asc(c)

where C(b) is the set of atoms in L(a) (as well as in L) smaller than b.

• Conversely ASC_{3} implies that if L(a) is finite and asc(c) > 0 for each atom

c in L(a):

asc(a) = ∑_{c∈I(L(a))} asc(c)

In particular if a ≠ 0 then asc(a) > 0.

Remark 9.1 It follows immediatly that an L_{SC}-embedding of sub-ASC-lattices φ: L → L′ is an L_{ASC}-embedding if and only if for every k > 0 and every atom a ∈ L:

L ⊧ At_{k}(a) ⟺ L′ ⊧ At_{k}(φ(a))

Every natural example of sub-ASC-lattice L satisfy the following additionnal property, which imply ASC_{1} to ASC_{3}:

ASC_{0} : L is a scaled lattices, and for every k > 0, At_{k}(L) is the set of elements of L which are the join of exactly k atoms in L.

We call ASC-lattices^{4} the sub-ASC-lattices which satisfy ASC_{0}. Every subscaled lattice L admits a unique structure of ASC-lattice which is an extension by definition of its lattice structure. We denote by L^{At} this expansion of L.

Proposition 9.2 (Linear representation) Let K be an infinite field and let L_{0} be a finite sub-ASC-lattice. For any positive integer N there exists a positive integer m, not depending on N, and an L_{SC}^{∗}-embedding φ_{N}: L_{0} → L_{lin}^{At}(K^{m}) such that for every element a of L_{0} of sc-dimension zero:

asc(a) > 0 ⟹ asc(φ_{N}(a)) = asc(a)
asc(a) = 0 ⟹ asc(φ_{N}(a)) ≥ N

Proof: We prove by induction on tuples (r, s) ordered lexicographically that the proposition holds for every finite sub-ASC-lattice L_{0} having r non-zero ∨-irreducible elements, s of which have the same sc-dimension as L_{0}.

If r = 0 then s = 0 and the unique embedding of L_{0} = {0} into L_{lin}^{At}(K^{0}) has the required property. So let us assume that r ≥ 1 and that the result is proved for every (r′, s′) < (r, s). Let a_{1} …, a_{r} be the elements of I(L_{0}) ordered by increasing sc-dimension, so d = sc-dim L = sc-dim a_{r} ≥ 0.

Case d = 0. Then L is a boolean algebra, and a_{1}, …, a_{r} are its atoms. For every i ≤ r we choose a finite subset A_{i} of K \ ⋃_{j<i} A_{j} such that:

• If asc(a_{i}) > 0, A_{i} has asc(a_{i}) elements, so asc(A_{i}) = asc(a_{i}).

• If asc(a_{i}) = 0, A_{i} has N elements, so asc(A_{i}) = N.

Clearly m = 1 does not depend on N, and the map φ which maps each a_{i} to A_{i} extends uniquely to an L_{SC}^{∗}-embedding of L_{0} into L_{lin}^{At}(K) which has the required properties.

Case d > 0. The upper semi-lattice L_{0}^{−} generated by a_{1}, …, a_{r−1} is an L_{ASC}^{∗}-substructure of L_{0} to which the induction hypothesis applies. This gives a positive integer m not depending on N and an L_{SC}^{∗}-embedding φ: L_{0}^{−} → L_{lin}^{At}(K^{m}) having the required properties. One can extend φ to an embedding φ̄ of L_{0} into some L_{lin}^{At}(K^{m+p}) exactly like in the proof of Proposition 5.3. The integer m+p does not depend on N and φ̄ inherits from φ the required properties because all the elements of L with sc-dimension zero belong to L_{0}^{−}.

Let ASC_{Zar}(K, d) (resp. ASC_{lin}(K, d), resp. ASC_{def}(K, d)) the class of all ASC-lattices L^{At} with L ranging over SC_{Zar}(K, d), (resp. SC_{lin}(K, d), resp. SC_{def}(K, d)).

Corollary 9.3 For any infinite field K and positive integer d, the universal theories of ASC_{Zar}(K, d) and ASC_{lin}(K, d)) are exactly the theory of sub-ASC-lattices.

The same holds for ASC_{def}(K, d) if moreover K is a henselian valued field of characteristic zero, a real closed field or an algebraically closed field.

4Of course we will show that the sub-ASC-lattices are precisely the L_{ASC}-substructures of ASC-lattices.

Proof: Since ASC_{lin}(K, d) is contained in the other classes, all of which are contained in the class of ASC-lattices, it suffices to prove that every sub-ASC-lattice L_{ASC}-embeds into an ultraproduct of elements of ASC_{lin}(K, d).

For any positive integer N let φ_{N} : L_{0} → L_{lin}^{At}(K^{m}) be the L_{SC}^{∗}-embedding given by Proposition 9.2. We still denote by φ_{N} the induced L_{SC}-embedding of L_{0} into L_{lin}^{At}(X_{N}) where X_{N} = φ_{N}(1_{L_{0}}). Let U be a non principal ultrafilter in the boolean algebra of subsets of ℕ, and L = ∏_{N∈ℕ} L_{lin}^{At}(X_{N})/U. Then φ = ∏_{N∈ℕ} φ_{N}/U is an L_{SC}-embedding of L_{0} into the ultraproduct L. Let a be an atom of L_{0} and k = asc(a).

If k > 0 then for every N ≥ k, L_{lin}^{At}(X_{N}) ⊧ At_{k}(φ_{N}(a)) by construction. So L ⊧ At_{k}(φ(a)), that is asc(φ(a)) = k.

If k = 0, let l be any stricly positive integer. For every N ≥ l, L_{lin}^{At}(X_{N}) ⊧ At_{N}(φ_{N}(a)) by construction, hence lalin(X_{N}) |̸= At_{l}(φ_{N}(a)). So L |̸= At_{l}(φ(a)), and this being true for every l > 0 it follows that asc(φ(a)) = 0.

By Remark 9.1, φ is then an L_{ASC}-embedding.

Let us call super ASC-lattices those ASC-lattices which satisfy the following axioms:

Atomicity: Every element x is the least upper bound in L of the set of atoms of L smaller that x.

Catenarity: For every positive integers r ≤ q ≤ p and every c ≤ a, if a has pure dimension p and c has pure dimension r then there exists a q-pure element b such that c ≤ b ≤ a.

ASC-Splitting: For every b_{1}, b_{2}, a, if b_{1} ∨ b_{2} ≪ a and C^{0}(a) = 0 there exists non-zero elements a_{1} ≥ b_{1} and a_{2} ≥ b_{2} such that:

\{ \begin{matrix}a_{1} = a − a_{2} \\ a_{2} = a − a_{1} \\ a_{1} ∧ a_{2} = b_{1} ∧ b_{2} \end{matrix}

The class of super-ASC-lattices of sc-dimension at most d (resp. exactly d) is clearly axiomatisable by ∀∃-formulas in L_{ASC}. We are going to show that its theory is the model-completion of the theory of sub-ASC-lattices of dimension at most d (resp. exactly d).

Remark 9.4 An immediate consequence of the atomicity axiom is that for every elements x, y in a super ASC-lattice L such that y ≪ x, there are infinitely many atoms a ∈ L such that a ≤ x and a ∧ y = 0. Indeed y < x hence by the atomicity axiom there is an atom a_{1} ∈ L such that a_{1} ≤ x and a_{1} ≰ y. Then a_{1} ≤ x and a_{1} ∧ y < a_{1} hence a_{1} ∧ y = 0 (because a_{1} is an atom). Moreover sc-dim y ∨ a_{1} < sc-dim x because y ≪ x, hence y ∨ a_{1} < x so the same argument applies to x and y ∨ a_{1}. It gives another atom a_{2} ≤ x such that a_{2} ∧ y = 0, and so on.

Primitive tuples and primitive extensions are defined for sub-ASC-lattices exactly like for subscaled lattices. If L is an extension of a sub-ASC-lattice L_{0} and x ∈ L we denote now L_{0}⟨x⟩ the L_{ASC}-substructure of L generated by L_{0} ∨ {x} (that is the subscaled lattice generated by L_{0} ∪ {x} endowed with the L_{ASC}-structure induced by L).

We define ASC-signatures in a finite sub-ASC-lattice L_{0} as triples (g, p, H) with H a set of non-necessarily distinct ordered pairs (h_{1}, k_{1}), (h_{2}, k_{2}) such that (g, q, {h_{1}, h_{2}}) is a signature in the L_{SC}-reduct of L_{0}, k_{1}, k_{2} are positive integers and:

1. If q < sc-dim g then (h_{1}, k_{1}) = (h_{2}, k_{2}).

2. If q ≠ 0 then k_{1} = k_{2} = 0.

3. If k_{1} ≠ 0, k_{2} ≠ 0 and sc-dim g = 0 then asc(g) = k_{1} + k_{2}

4. If k_{1} = 0 or k_{2} = 0 then asc(g) = 0.

Example 9.5 Let L_{0} be a finite sub-ASC-lattice, and L an extension of L_{0} generated by a primitive tuple (x_{1}, x_{2}). Let (g, p, {h_{1}, h_{2}}) be the signature of L in L_{0} (in the sense of subscaled lattices) and k_{i} = asc(x_{i}). Then (g, p, {(h_{1}, k_{1}), (h_{2}, k_{2})} is easily seen to be an ASC-signature in L_{0}, uniquely determined by L. We call it the ASC-signature of L in L_{0}.

The same argument as in Proposition 7.1 shows (using Remark 9.1 in additition to Proposition 3.1) that two primitively generated extensions of a finite sub-ASC-lattice L_{0} are isomorphic over L_{0} if and only if they have the same signature in L_{0}.

Proposition 9.6 Let L_{0} be a finite L_{ASC}-substructure of an ℵ_{0}-saturated super ASC-lattice L̂. Then for every ASC-signature σ_{At} in L_{0} there exists a primitive tuple (x_{1}, x_{2}) ∈ L̂ such that σ_{At} is the signature of L_{0}⟨x_{1}, x_{2}⟩ in L_{0}.

Proof: Let σ_{At} = (g, q, {(h_{1}, k_{1}), (h_{2}, k_{2})}) be an ASC-signature in L_{0}, let σ = (g, q, {h_{1}, h_{2}}) and p = sc-dim g. In each of the following cases, the fact that the constructed tuple (x_{1}, x_{2}) is primitive over L_{0} and generates an extension whose signature in L_{0} is precisely σ_{At} is straighforward.

Case p ≥ 1. Then g is not an atom, so the ASC-splitting property applies to g, h_{1}, h_{2}. Moreover by Remark 9.4, there are infinitely many atoms a in L smaller than g and not in L_{0}. By ℵ_{0}-saturation it follows that there exists x_{0} ∈ L \ L_{0} such that sc-dim x_{0} = 0 and L |̸= At_{k}(a) for every k > 0, that is asc(x_{0}) = 0.

If q ≥ 1 then the construction in the proof of Proposition 8.2 applies here. It gives a primitive tuple (x_{1}, x_{2}) in L such that the L_{SC}-substructure L_{1} of L generated by (x_{1}, x_{2}) over L_{0} has signature σ in L_{0}. Moreover each k_{i} = 0 since p ≠ 0, and on the other hand each asc x_{i} = 0 since sc-dim x_{i} = p ≥ 1.

Otherwise q = 0 < p hence (h_{1}, k_{1}) = (h_{2}, k_{2}). If k_{1} ≠ 0 let x_{1} = x_{2} = a_{1} ∨ … ∨ a_{k_{1}} with the a_{i}’s being any distinct atoms in L smaller than g and not belonging to L_{0}. If k_{1} = 0 let x_{1} = x_{2} = x_{0}.

Case p = 0. Then q = 0 and h_{1} = h_{2} = 0.

If k_{1} and k_{2} are non-zero then asc(g) = k_{1} + k_{2} hence L(g) contains k_{1} + k_{2} atoms. Let x_{1} be the join of k_{1} of them, and x_{2} be the join of the others.

Eventually if k_{1} = 0 or k_{2} = 0 then asc(g) = 0 so L(g) contains infinitely many atoms, hence by saturation there exists an element x in L smaller than g such that both x and g − x are non-zero and asc(x) = asc(g − x) = 0. If k_{1} = k_{2} = 0 let (x_{1}, x_{2}) = (x, g − x). If k_{1} ≠ 0 let x_{1} = a_{1} ∨ … ∨ a_{k_{1}} with the a_{i}’s atoms in L smaller than g and x_{2} = g − x_{1}. If k_{2} ≠ 0 exchange k_{1} and k_{2}.

Proposition 9.7 Every sub-ASC-lattice embeds in a super-ASC-lattice.

Proof: Obviously every finitely generated substructure of a sub-ASC-lattices is finite by the local finiteness Theorem 4.1 because L_{SC} and L_{ASC} have the same function symbols.

Let L_{0} be a finitely generated sub-ASC-lattice, and a, b_{1}, b_{2} ∈ L_{0} such that b_{1} ∨ b_{2} ≪ a and C^{0}(a) = 0. So L_{0} is finite, let g_{1}, …, g_{r} be the ∨-irreducible components of a in L_{0}. As in the proof of Proposition 8.1 we construct a tower of L_{SC}-extensions (L_{i})_{i≤r} so that each L_{r} contains elements x_{1}, x_{2} satisfying

\{ \begin{matrix}x_{1} = a − a_{2} \\ x_{2} = a − a_{1} \\ x_{1} ∧ x_{2} = φ(b_{1}) ∧ φ(b_{2}) \end{matrix}

and each L_{i} is generated over L_{i−1} by a primitive tuple of elements x_{i,1} ≠ x_{i,2} such that g_{i} = g(x_{i,1}, L_{i−1}) = g(x_{i,2}, L_{i−1}). Proposition 6.1 then implies that:

I(L_{r}) = (I(L_{0}) \ {g_{1}, …, g_{r}}) ∪ {x_{i,1}, x_{i,2}}_{1≤i≤r}

By assumption all the g_{i}’s have non-zero sc-dimension, hence the x_{i,1}’s and x_{i,2}’s do the same. In particular L_{0} and L_{r} have the same elements of sc-dimension zero. Defining At_{k}(L_{r}) = At_{k}(L_{0}) for every k then endows L_{r} with an L_{ASC}-structure which makes it a sub-ASC-lattice and an L_{ASC}-extension of L_{0}.

On the other hand every L_{lin}^{At}(X) is an ASC-lattice satisfying the atomicity and catenarity properties, hence Corollary 9.3 implies that every finitely generated sub-ASC-lattice embeds in an ASC-lattice having these two properties.

The conclusion follows by the model-theoretic compactness argument.

Theorem 9.8 The theory of super ASC-lattices of sc-dimension at most d (resp. exactly d) is the model-completion of the theory of ASC-lattices of dimension at most d (resp. exactly d). It admits ℵ_{0} completions, each of which is decidable, and it is decidable.

Proof: The proof of the first statement is very similar to Theorem 8.3 and Corollary 8.5, with the only difference that in the embedding argument we have to assume the super ASC-lattice L̂ to be ℵ_{0}-saturated, in order to apply Proposition 9.6 in place of Proposition 8.2.

The last statement then follows from the remark that there are ℵ_{0} prime sub-ASC-lattices of dimension at most d (resp. exactly d). Indeed there are finitely many subscaled lattices of dimension at most d (resp. exactly d) and on each of them ℵ_{0} different structures of sub-ASC-lattices.

Let L be any super ASC-lattice and L′ the quotient of L by the equivalence relation x ∼ y if and only if x − y and y − x are the join of finitely many atoms. Then it is easily seen that L′ is a super scaled lattice. So the problem of finding a natural example of super scaled lattice boyls down to finding a natural example of super ASC-lattice. This and related ideas lead us to the following conjecture:

Conjecture 9.9 Let K be a p-adically closed field and A be an infinite definable subset of K^{n} which is open in its closure. Let (B_{k})_{k≤q} be a finite collection of closed definable subsets of Ā \ A. Then there exists a collection (A_{k})_{k≤q} of non-empty definable subsets of A clopen in A such that:

∀k ≤ q, \overline{A_{k}} = A_{k} ∪ B_{k}

It is an elementary exercise to deduce from this conjecture that L_{def}^{At}(X) models the ASC-splitting property for every definable set X over K. Moreover the L_{SC}-substructure of L_{def}^{At}(K^{n}) generated by the empty set is simply the two element lattice with the obvious L_{SC}-structure (because K^{d} is d-pure). This gives an explicit recursive axiomatisation of L_{def}^{At}(K^{n}).

Corollary 9.10 (Modulo Conjecture 9.9) Let K be a p-adically closed field, then L_{def}^{At}(K^{n}) is a super-ASC-lattice. In particular its complete theory is decidable and eliminates the quantifier in L_{ASC}.

References

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