Gérard Leloup

Francois Lucas

c-regular cyclically ordered groups

Keywords: abelian cyclically ordered groups, regular, cyclic groups, groups of unimodular complex numbers, first order theory

published or not. The documents may come

1 Introduction.

Unless otherwise stated the groups are abelian groups. The definitions and basic properties of cyclically ordered groups can be found for example in [START_REF] Fuchs | Partially Ordered Algebraic Systems[END_REF], [START_REF] Giraudet | First order theory of cyclically ordered groups[END_REF] or [START_REF] Leloup | Autour des groupes cycliquement ordonnés[END_REF]. For the reader's convenience we recall them in Section 2. Being c-archimedean (i.e. isomorphic to a subgroup of the cyclically ordered group K of the complex numbers of module 1) for a cyclically ordered group is not a first order property. In this paper we define the c-regular cyclically ordered groups, and we prove that the class of all c-regular cyclically ordered groups is the lowest elementary class containing all the c-archimedean cyclically ordered groups. We also define the regular cyclically ordered groups, and we prove that the class of all regular cyclically ordered groups is the union of the class of all c-regular cyclically ordered groups and of the class of all linearly cyclically ordered groups.

We prove that an abelian cyclically ordered group G which is not c-archimedean is c-regular if and only if its linear part l(G) is a regular linearly ordered group and the cyclically ordered group K(G) = (G/l(G)) is divisible and its torsion group is isomorphic to the group of all roots of 1 in the field of all complex numbers. Another characterization is that its unwound is a regular linearly ordered group.

Every regular cyclically ordered group is either dense or discrete. We prove that any two dense c-regular cyclically ordered groups are elementarily equivalent if and only if they are elementarily equivalent in the language of groups, and that any dense c-regular group is elementarily equivalent to some c-archimedean group. In the case of discrete groups, we prove that every discrete c-regular group is elementarily equivalent to an ultraproduct of finite cyclic groups. Every discrete c-regular group G which is not c-archimedean contains an elementary substructure C such that K(C) ≃ U and l(C) ≃ Z. We define first order formulas D p n ,k (p a prime number, n ∈ N and k ∈ {0, . . . , p n -1}) such that any two discrete non-c-archimedean c-regular groups G and G ′ are elementarily equivalent if and only if G and G ′ satisfy the same formulas D p n ,k . If C and C ′ are the elementary substructures of G and G ′ , respectively, which we defined above, then G ≡ G ′ if and only if C ≃ C ′ . By means of the formulas D p n ,k which are satisfied by G, we define a family of subsets of N * , which we will call the family of subsets of N * characteristic of G, such that for every non principal ultrafilter U on N * , G is elementarily equivalent to the ultraproduct of the cyclically ordered groups Z/nZ modulo U if and only if U contains the family of subsets of N * characteristic of G. Furthermore, for every non-c-archimedean c-regular cyclically ordered group, there 1 2010 Mathematics Subject Classification. 03C64, 06F20, 06F99, 20F14.

1 exists an ultrafilter containing the family of subsets of N * characteristic of this group. Remark that the class of all discrete regular cyclically ordered groups is the lowest elementary class which contains all the cyclically ordered groups generated by one element. Furthermore, this class is divided into two disjoint elementary subclasses, the class of discrete c-regulars cyclically ordered groups, and the class of discrete regular linearly cyclically ordered groups, that is, the groups which are elementarily equivalent to the linearly cyclically ordered group Z.

In the last section, we deal with the same groups, wihtout a predicate for the cyclic order, to show which informations this predicate carries.

2 Cyclically ordered groups.

Recall that a cyclically ordered group is a group G (which is not necessarily abelian) together with a ternary relation R which satisfies:

(1) ∀(a, b, c) ∈ G 3 , R(a, b, c) ⇒ a = b = c = a (R is "strict"), (2) ∀(a, b, c) ∈ G 3 , R(a, b, c) ⇒ R(b, c, a) (R is cyclic), ( 3
) for all c ∈ G, R(c, •, •) defines a linear order on G\{c}, (4) ∀(a, b, c, u, v) ∈ G 5 , R(a, b, c) ⇒ R(uav, ubv, ucv) (R is compatible).

For example, let K be the group of all complex numbers of module 1, for e iθ1 , e iθ2 , e iθ3 in K with 0 ≤ θ i < 2π, (i ∈ {1, 2, 3}) we let R(e iθ1 , e iθ2 , e iθ3 ) if and only if θ 1 < θ 2 < θ 3 . The subgroup U of torsion elements of K, that is, the roots of 1 in the field C of all complex numbers, is a cyclically ordered group. On the other side, any linearly ordered group (G, <) can be equipped with a cyclic order: we set R(a, b, c) if and only if either a < b < c or b < c < a or c < a < b. In this case we say that (G, R) is a linearly cyclically ordered group (or sometimes linear cyclically ordered goup).

If g 0 is a positive, central and cofinal element of a linearly ordered group (G, <), then the quotient group G/ g 0 (where g 0 denotes the subgroup generated by g 0 ) can be cyclically ordered by setting R(a g 0 , b g 0 , c g 0 ) if and only if there exist a ′ ∈ a g 0 , b ′ ∈ b g 0 and c ′ ∈ c g 0 such that 1 G ≤ a ′ < g 0 , 1 G ≤ b ′ < g 0 , 1 G ≤ c ′ < g 0 and either a ′ < b ′ < c ′ or b ′ < c ′ < a ′ or c ′ < a ′ < b ′ . This cyclically ordered group is called the wound-round associated to G and g 0 . The theorem of Rieger ( [START_REF] Fuchs | Partially Ordered Algebraic Systems[END_REF], IV, 6, th. 21) states that every cyclically ordered group can be obtained in this way, that is for every cyclically ordered group (G, R), there exists a linearly ordered group uw(G) and an element z G which is positive, central and cofinal in uw(G), such that (G, R) is isomorphic to the cyclically ordered group uw(G)/ z G . uw(G) is the set Z × G together with the order (n, c) < (n ′ , c ′ ) if and only if either n < n ′ , or n = n ′ and R(1 G , c, c ′ ), or n = n ′ and c = 1 G . We set z G = (1, 1 G ), and the group law is given by:

(m, a) • (n, b) =            (m + n, b) if a = 1 G (m + n, a) if b = 1 G (m + n, ab) if R(1 G , a, ab) (m + n + 1, ab) if R(1 G , ab, a) (m + n + 1, 1 G ) if ab = 1 G = a
uw(G) is called the unwound of (G, R). For example, the unwound of K is isomorphic to (R, +).

If C is the greatest convex subgroup of the unwound uw(G) of a cyclically ordered group (G, R), then z -1 G < C < z G , hence C embeds trivially in G ≃ uw(G)/ z G , we denote by l(G) its image and we call it the linear part of G. The restriction of R to l(G) is a linearly cyclic order, where the order on l(G) is induced by the order on uw(G).

We will denote by K(G) the quotient group G/l(G). The class modulo l(G) of an element a ∈ G will be denoted by ā. K(G) is equipped with a structure of a cyclically ordered group, which is induced by the cyclic order on G, because if a, b, c in G belong to pairwise distinct classes, and d ∈ l(G), then R(a, b, c) ⇔ R(ad, b, c)&R(da, b, c). We know that K(G) embeds into the cyclically ordered group K.

We let R(x 1 , . . . , x n ), where x 1 , . . . , x n belong to G, stand for:

1≤i<j<k≤n R(x i , x j , x k ), which is equivalent to:

1≤i≤n-2 R(x i , x i+1 , x n )
In the following, (G, R) is an abelian cyclically ordered group. Since G is abelian, we will use the notation +, except in the case of subgroups of K, because it is customary to use the notation •.

One can prove the following property by induction.

2.1 Lemma. Let g 1 , . . . , g n be elements of G and θ 1 , . . . , θ n be the elements of the interval [0, 2π[ of R such that, for 1 ≤ j ≤ n, ḡj = e iθj . Let k be the integer part of (θ

1 + • • • + θ n )/(2π) and θ = (θ 1 + • • • + θ n ) -2kπ.
There exists g in G such that ḡ = e iθ and we have:

(0, g 1 ) + • • • + (0, g n ) = (k, g) within uw(G).
We will use this property in order in order to obtain a representation of n(a, x) ∈ uw(G) as a 2-uple from Z × G.

Recall that G (a priori non necessarily abelian) is said to be c-archimedean if l(G) = {0}. Świerczkowski ([Sw 59]) proved that G is c-archimedean if and only if for every x and y in G there exists an integer n > 0 such that R(0, nx, y) does not hold. Note that this is not a first order property. If G is c-archimedean, then it embeds into K, hence it is abelian. A c-convex subgroup of G is a subgroup H which satisfies for every a and b in G:

(b ∈ H ⇒ b = -b)&(b ∈ H & R(0, b, -b) & R(0, a, b)) ⇒ a ∈ H.
There exists a canonical one-to-one correspondence between the set of proper c-convex subgroups of G and the set of convex subgroups of l(G) which are different from {0} (note that l(G) is a c-convex subgroup). We see that G is c-archimedean if and only if it doesn't contain any proper c-convex subgroup. We will say that G is c-n-divisible if it is n-divisible and it contains a subgroup which is isomorphic to the subgroup of (U, •) generated by a primitive n-th root of 1. We see that this is a first order property. Now, G contains a subgroup which is isomorphic to the subgroup of (U, •) generated by a primitive n-th root of 1 if and only

if z G is n-divisible in the unwound uw(G). It follows that if uw(G) is n-divisible then G is c-n-divisible.
Conversely, let z ∈ uw(G) be such that z G = nz, then, since G is n-divisible, for every x ∈ uw(G) there exists y ∈ uw(G) such that x -ny belongs to the subgroup generated by z G , that is, there exists an integer k such that x -ny = kz G = nkz, so x = n(y + kz). Consequently: G is c-n-divisible if and only if uw(G) is n-divisible. We will say that G is c-divisible if it is divisible and it contains a subgroup which is isomorphic to (U, •).

c-regular groups.

3.1 Definitions. Let n ≥ 2 be an integer. We say that (G, R) is n-regular if for every x 1 , . . . , x n in G which satisfies R(0, x 1 , . . . , x n , -x n ), there exists x ∈ G such that (R(x 1 , nx, x n ) or nx = x 1 , or nx = x n ) and R(0, x, . . . , (n -1)x, x n ). We say that (G, R) is c-n-regular if for every x 1 , . . . , x n in G satisfying R(0, x 1 , . . . , x n ), there exists x ∈ G such that (nx = x 1 , or nx = x n , or R(x 1 , nx, x n )) and R(0, x, . . . , (n -1)x, x n ). We say that (G, R) is regular if it is n-regular for every n ≥ 2, and we say that (G, R) is c-regular if it is c-n-regular for every n ≥ 2.

Note that these are first order properties, and clearly if (G, R) is c-n-regular, then it is n-regular. Now from Lemma 3.5 it follows that a nontrivial linearly cyclically ordered group is not c-n-regular. One can prove that if (G, R) is a linearly cyclically ordered group, then it is a n-regular cyclically ordered group if and only if it is a n-regular linearly ordered group, that the subgroups of K are c-regular.

The proofs of the three following lemmas are left to the reader.

Lemma.

Assume that G is a nontrivial subgroup of K and let n ≥ 2. Then G is n-divisible and contains a primitive n-th root of 1 if and only if for every θ ∈ [0, π], if e iθ ∈ G, then e iθ/n ∈ G.

Note that the c-archimedean groups (i.e. the subgroups of K) are c-archimedean but they need not satisfy the conditions of Lemma 3.2.

Lemma.

Let n ≥ 2, assume that G is n-regular and is not c-archimedean. For every θ in [0, π], if e iθ ∈ K(G), then e iθ/n ∈ K(G).

3.4 Lemma. Let n ≥ 2 such that G is n-regular, then l(G) is n-regular, where the compatible total order on l(G) is derived from the definition of the positive cone of G. In particular, if G is regular, then l(G) is regular.

3.5 Lemma. Let n ≥ 2, assume that G is c-n-regular and is not c-archimedean, then K(G) contains a primitive n-th root of 1.

Proof. Assume that h 1 < • • • < h n < 0 are elements of l(G) such that 2h n < h 1 . In the case where l(G) is dense, clearly one can find such a sequence, and if l(G) is discrete with first positive element ǫ, let h n ∈ l(G) such that h n < -nǫ, and set

h 1 = h n -nǫ + ǫ, . . . , h n-1 = h n -ǫ. Since G is c-n-regular, there exists g ∈ G such that ng = h 1 , or ng = h n , or R(h 1 , ng, h n ), and R(0, g, . . . , (n -1)g, h n ). Note that ḡ is a n-th root of 1 in K(G), because ng = 0. Now l(G) is c-convex, so ng ∈ l(G), and h 1 ≤ ng ≤ h n < 0. Let k, 1 ≤ k ≤ n,
be such that kg ∈ l(G), and d be the gcd of k and n. Let a and b be two integers such that ak

+ bn = 1, then dg = akg + bng ∈ l(G), hence, if k is the lowest positive integer such that kg ∈ l(G), then k divides n. If k < n, since R(0, kg, h n ) holds, we have kg < h n < 0. Furthermore 2k ≤ n, hence ng ≤ 2kg < 2h n < h 1 : a contradiction. Consequently, k = n. Since for 1 ≤ k ≤ n -1 kg = 0, ḡ is a primitive n-th root. 3.6 Proposition. Let n ≥ 2, assume that l(G) is n-regular. If K(G) is n-divisible and contains a primitive n-th root of 1, then G is c-n-regular. Proof. K(G) is nontrivial because it contains a primitive n-th root of 1; since it is n-divisible, it is infinite, hence it is a dense subgroup of K.
Let g 1 and g 2 in G, assume: R(0, g 1 , g 2 ) and ḡ1 = ḡ2 . There exist θ 1 < θ 2 in [0, 2π] such that ḡ1 = e iθ1 and ḡ2 = e iθ2 . Since K(G) is dense, there exist g in G and θ in ]θ 1 /n, θ 2 /n[ such that ḡ = e iθ . Then ng = e inθ , with 0 ≤ θ 1 < nθ < θ 2 , and 0

< θ < • • • < (n -1)θ < θ 2 , hence R(g 1 , ng, g 2 ) and R(0, g, . . . , (n -1)g, g 2 ).
Now, let g 1 , . . . , g n in G be such that R(0, g 1 , . . . , g n ). If ḡ1 = ḡn , then by what has been proved above, there exists g ∈ G such that R(g 1 , ng, g n ) and R(0, g, . . . , (n -1)g, g n ). If ḡ1 = ḡn = 0 and g 1 ∈ l(G) + , g n ∈ -l(G) + , we set ḡ1 = e i0 and ḡn = e 2iπ , and we conclude in the same way. We assume now that ḡ1 = • • • = ḡn , we set ḡ1 = e iθ , with 0 ≤ θ ≤ 2π, and if θ = 0 we assume that either each of g 1 , . . . , g n belongs to l(G) + or each of g 1 , . . . , g n belongs to -l(G) + . In the first case, we set θ = 0, in the second case, we set θ = 2π. Note that in any case l(G) is nontrivial.

If θ = 0, i.e. both of g 1 , . . . , g n belong to l(G) + , then we have 0 < g 1 < • • • < g n . Since l(G) is n-regular, there exists g ∈ l(G) such that g 1 ≤ ng ≤ g n , and we have also 0 < g < • • • < (n -1)g < g n , hence (R(g 1 , ng, g n ) or ng = g 1 or ng = g n ) and R(0, g, . . . , (n -1)g, g n ).

Now assume that 0 < θ ≤ 2π. Since K(G) is n-divisible and it contains a primitive n-th root of 1, there exists an element g of G such that ḡ = e iθ/n . Since g 1 -ng, . . . , g n -ng belong to l(G), it follows from the relation R(0, g 1 , . . . , g n ) that g 1 -ng < • • • < g n -ng in l(G). After replacing g by ng -g 1 , if necessary, we can assume that 0 < g 1 -ng < • • • < g n -ng. Since l(G) is n-regular, there exists h ∈ l(G) such that g 1 -ng ≤ nh ≤ g n -ng. It follows: R(g 1 -ng, nh, g n -ng), or nh = g 1 -ng, or nh = g n -ng, then by compatibility R(g 1 , n(g + h), g n ) or n(g

+ h) = g 1 or n(g + h) = g n . Furthermore, since 0 < θ/n < • • • < (n -1)θ/n < θ, we have R(0, g + h, . . . , (n -1)(g + h), g n ).
This proves that G is c-n-regular.

3.7 Theorem. Assume that G is not c-archimedean and let n ≥ 2. The following assertions are equivalent.

1) G is n-regular and K(G) = {0}. 2) l(G) is n-regular, K(G) = {0} and for every θ ∈ [0, π] such that e iθ ∈ K(G) we have: e iθ/n ∈ K(G). 3) G is c-n-regular. 4) l(G) is n-regular, K(G) is n-divisible and contains a primitive n-th root of 1.
Proof. 1) ⇒ 2). If G is n-regular, then by Lemma 3.3, for every θ ∈ [0, π] such that e iθ ∈ K(G) we have e iθ/n ∈ K(G), furthermore K(G) is n-divisible, and by Lemma 3.4, l(G) is n-regular. 4) ⇒ 3) follows from Proposition 3.6. 3) ⇒ 1) follows from the definition, and from Lemma 3.2 we deduce 2) ⇔ 4). Proof. Straightforward.

3.9 Lemma. Let n ≥ 2, assume that there exist g ∈ G and x ∈ uw(G) such that ḡ = 1 and nx = (1, g), then e 2iπ/n ∈ K(G). It follows that if l(G) is nontrivial and uw(G) is n-regular, then e 2iπ/n ∈ K(G).

Proof. Since (1, g) > 0, by compatibility we have: 0 < x < (1, g). By the definition of the order on uw(G), there exists h ∈ G such that either x = (0, h) or x = (1, h). Now n(1, h) ≥ (n, nh) ≥ (n, 0) > (1, g), hence x = (0, h). Set h = e iθ , with 0 ≤ θ < 2π. According to Lemma 2.1, we have nθ = 2π, which proves that e 2iπ/n ∈ K(G). Now assume that l(G) is nontrivial and that uw(G) is n-regular. Let g 1 , . . . , g n in l(G), assume that 0

< g 1 < • • • < g n , hence (0, 0) < (1, g 1 ) < • • • < (1, g n ).
There exists y ∈ uw(G) such that (1, g 1 ) ≤ ny ≤ (1, g n ). Then ny admits a representation as ny = (1, g) with g ∈ G. By the definition of the order on uw(G), R(g 1 , g, g n ) holds, and by c-convexity of l(G), we have g ∈ l(G). Finally, from what we just proved, e 2iπ/n ∈ K(G).

3.10 Lemma. Let n ≥ 2, if uw(G) is n-regular, then G is n-regular.
Proof. Let g 1 , . . . , g n in G, assume that R(0, g 1 , . . . , g n ) holds. Since uw(G) is n-regular, and (0, 0) < (0, g 1 ) < • • • < (0, g n ), there exists x in uw(G) such that (0, g 1 ) ≤ nx ≤ (0, g n ). In the same way of in the proof of Lemma 3.9, nx admits a representation as nx = (0, g) with g ∈ G. It follows that x admits a representation as x = (0, h) with nh = g. We have (0, g 1 ) ≤ (0, nh) ≤ (0, g n ), hence R(g 1 , nh, g n ) or nh = g 1 , or nh = g n , which proves that G is n-regular.

3.11 Proposition. Let n ≥ 2, assume that l(G) is n-regular, and that K(G) is n-divisible and contains a primitive n-th root of 1. Then uw(G) is n-regular.

Proof. By assumption, K(G) is nontrivial, hence G is not equal to l(G). Hence if G is c-archimedean, then uw(G) is archimedean, hence it is regular. If G is not c-archimedean, then l(G) is nontrivial, hence G is infinite. K(G) is infinite because it is n-divisible. Let (0, 0) < (m 1 , g 1 ) < • • • < (m n , g n ) in uw(G).
Let q and r be the integers such that 0 ≤ r < n and m 1 = nq + r. If we can find x ∈ uw(G) such that (r, g 1 ) ≤ nx ≤ (m n -nq, g n ), then we have (m 1 , g 1 ) ≤ n(x + (q, 0)) ≤ (m n , g n ), which proves that we can assume that 0 ≤ m 1 < n. Since K(G) contains a primitive n-th root of 1, we have: e 2iπ/n ∈ K(G).

Let g ∈ G such that ḡ = e 2iπ/n , and set g ′ = m 1 g, then ḡ′ = e 2im1π/n , with 0 ≤ 2m 1 π/n < 2π, and n(2m 1 π/n) = 2m 1 π. By Lemma 2.1, there exists h ∈ l(G) such that n(0, g ′ ) = (m 1 , h). If ḡ1 = e iθ1 , since K(G) is divisible, one can find an element whose class is e iθ1/n in K(G). We add such an element to g ′ , then, if necessary, we add an element of l(G) in order to get g ′′ such that n(0, g ′′ ) = (m 1 , ng ′′ ) and R(0, ng ′′ , g 1 ) holds. Assume that m 1 = • • • = m n and ḡ1 = • • • = ḡn . By substracting n(0, g ′′ ) to each term, we get a sequence (0, 0) < (0, h 1 ) < • • • < (0, h n ), where the h k 's belong to l(G). By the definition of the order on uw(G), we have either 0

< h 1 < • • • < h n , or h 1 < • • • < h n < 0, or h n < 0 < h 1 .
In the first case, there exists h in l(G) such that h 1 ≤ nh ≤ h n . Hence R(h 1 , nh, h n ), or nh = h 1 , or nh = h n . In the second case we consider the sequence 0

< -h n < • • • < -h 1 , there exists h ∈ l(G) such that -h n ≤ -nh ≤ -h 1 , then h 1 ≤ nh ≤ h n ,
and we conclude in the same way as above. In the third case, we set for example h = h 1 , then h n < 0 < h 1 < nh, which implies R(h 1 , nh, h n ). In any case, we have proved that there exists

h ∈ l(G) such that h 1 ≤ nh ≤ h n . Then (0, h 1 ) ≤ (0, nh) = n(0, h) ≤ (0, h n ). Consequently (m 1 , g 1 ) ≤ n((0, h) + (0, g ′′ )) ≤ (m n , g n ). If the m k 's are not equal, note that we have m 1 ≤ • • • ≤ m n , and in this case m 1 < m n . Let 0 < h 2 < • • • < h n in l(G), then 0 < (m 1 , g 1 ) < (m 1 , g 1 + h 2 ) < • • • < (m 1 , g 1 + h n ). It follows that there exists h ∈ G such that (m 1 , g 1 ) ≤ n(0, h) ≤ (m 1 , g 1 + h n ), and since (m 1 , g 1 + h n ) < (m n , g n ), h is the required element.
If all of the m k 's are equal, but the g k 's are not, then R(0, g 1 , g n ) holds, take the element (0, h) that we just exhibit above.

3.12 Theorem. Let n ≥ 2. The following conditions are equivalent.

(

1) uw(G) is n-regular (2) G is c-n-regular (3) either G is c-archimedean, or l(G) is n-regular, K(G) is n-divisible and it contains a primitive n-th root of 1 (4) the quotient of G by every proper c-convex subgroup is c-n-divisible.
Proof. If G is c-archimedean, then it is either dense or finite, hence it is c-regular. Furthermore, uw(G) is archimedean, hence it is regular. Now, assume that G is not c-archimedean. The equivalence (2) ⇔ (3) has been proved in Theorem 3.7. If l(G) is n-regular, K(G) is n-divisible and contains a primitive n-th root of 1, then by Proposition 3.11, uw(G) is n-regular. If uw(G) is n-regular, since it contains a convex subgroup which is isomorphic to l(G), l(G) is n-regular. According to Lemma 3.10, G is n-regular, and by Lemma 3.3, K(G) is n divisible. Finally, by Lemma 3.9, e 2iπ/n ∈ K(G).

Let H be a proper c-convex subgroup (that is a convex subgroup of l(G) distinct from {0}). In the construction of the unwound, we note that uw(G/H) = uw(G)/H. By means of (1) ⇔ (2) we get:

G/H is c-n-divisible, for every c-convex subgroup H of G, if and only if uw(G)/H is n-divisible, for every c-convex subgroup H of G, if and only if uw(G) is n-regular, if and only if G is c-n-regular.
4 Elementarily equivalent c-regular cyclically ordered groups.

As being dense and being c-regular are first order properties, every cyclically ordered group which is elementarily equivalent to an infinite subgroup of K is dense and c-regular. Every ultraproduct of finite cyclically ordered groups is discrete and c-regular, since each factor satisfies both of these first order properties.

If G is discrete and infinite then l(G) is nontrivial, because every infinite and c-archimedean cyclically ordered group is dense. Furthermore, if G is c-regular, then by Lemma 3.3, K(G) is divisible, hence, it is infinite.

Preliminaries.

If A is an abelian group and p is a prime, we define the p-th prime invariant of Zakon of A, denoted by [p]A, to be the maximum number of p-incongruent elements in A. In the infinite case, we set [p]A = ∞, without distinguishing between infinities of different cardinalities. /[p n ]H. We pick a transcendent number e, and, for n and j in N * such that 1 ≤ j ≤ m ′ n , we pick pairwise distinct elements k nj in N * . Denote by M 0 the direct sum of the Z (pn) e kn j 's, where n ∈ N * and 1 ≤ j ≤ m ′ n , and Z (pn) denotes the localization of Z at the prime ideal (p n ). By [Za 61], for every n ∈ N * we have:

[p n ]M 0 = p m ′ n . Since Q ∩ M 0 = {0}, we have: Q + M 0 = Q ⊕ M 0 . We set: M = H ⊕ M 0 , then M contains H, H is pure in M , M is countable,
and for every prime p we have:

[p]M = ([p]H)([p]M 0 ) = p mp .

Proposition. Let A, B, T be linearly ordered abelian groups, where B is a convex subgroup of T . Then there exists an exact sequence of ordered groups 0 → B → T → A → 0 if and only if there exists a mapping θ from

A × A to B which satisfies: (*) ∀a ∈ A, θ(a, 0) = 0, (**) ∀(a 1 , a 2 ) ∈ A × A, θ(a 1 , a 2 ) = θ(a 2 , a 1 ), (***) ∀(a 1 , a 2 , a 3 ) ∈ A × A × A, θ(a 1 , a 2 ) + θ(a 1 + a 2 , a 3 ) = θ(a 1 , a 2 + a 3 ) + θ(a 2 , a 3 ) (in other words, θ is a 2-cocycle),
such that T is isomorphic to A × B lexicographically ordered and equipped with the operation + θ defined by:

∀(a 1 , b 1 , a 2 , b 2 ) ∈ A × B × A × B, (a 1 , b 1 ) + θ (a 2 , b 2 ) = (a 1 + a 2 , b 1 + b 2 + θ(a 1 , a 2 )).
We will say that this exact sequence is an extension of B by A, we set T = A -→ × θ B, and we will omit θ when θ ≡ 0.

Proof. See [START_REF] Jaffard | Extensions de groupes ordonnés[END_REF]. We recall the definition of the mapping θ. For every a ∈ A, we pick some r(a) in T whose image is a (a can be viewed as a class modulo B), by taking r(0) = 0, and we set, for every a 1 , a 2 in A, θ(a 1 , a 2 ) = r(a 1 + a 2 ) -r(a 1 ) -r(a 2 ). The condition (*) is satisfied because we set 0 = r(0), (**) follows from commutativity, and (***) follows from the associativity of addition. We get the isomorphism by defining the image of (a, b), where a ∈ A, b ∈ B, to be the element r(a) + b of T .

Proposition. 1) Let

A -→ × θ1 B 1 and A -→ × θ2 B 2 be two extensions of linearly ordered abelian groups, B = B 1 -→ × B 2 , θ = (θ 1 , θ 2 ) from A × A to B, then A -→ × θ B is an extension of linearly ordered abelian groups. Furthermore A -→ × θ1 B 1 and A -→ × θ2 B 2 embed canonically into cofinal subgroups of A -→ × θ B. 2) Let A 1 -→ × θ1 B and A 2 -→ × θ2 B be two extensions of linearly ordered abelian groups, A = A 1 -→ × A 2 , θ = θ 1 +θ 2 from A × A to B, then A -→ × θ B is an extension of linearly ordered abelian groups. Furthermore A 1 -→ × θ1 B and A 2 -→ × θ2 B embed canonically into subgroups of A -→ × θ B.
3) Let A and B ′ be two linearly ordered abelian groups, B be a subgroup of B ′ , and A -→ × θ B be an extension of linearly ordered abelian groups. Then there exists an extension of linearly ordered abelian groups

A -→ × θ ′ B ′ , where θ ′ extends θ, A -→ × θ B is a subgroup of A -→ × θ ′ B ′ , {0} -→ × θ ′ B ′ is a convex subgroup of A -→ × θ ′ B ′ ,
and the quotient group is isomorphic to A.

Proof. The proof is left to the reader.

Note that Proposition 4.2 remains true by assuming that A and T are cyclically ordered, and B is linearly ordered. Then analogues of 1) and 3) of Proposition 4.3 hold.

Proposition. Let A ′ and B be two linearly ordered abelian groups, A be a subgroup of A ′ , and A -→ × θ B be an extension of linearly ordered abelian groups. Then θ extends by induction to A

′ × A ′ by means of the following properties. Let a ′ ∈ A ′ \A. 1) If a ′ is rationally independent from A, we can extend θ to Za ′ + A (= Za ′ ⊕ A) in the following way: ∀(n 1 , a 1 , n 2 , a 2 ) ∈ Z × A × Z × A, θ(n 1 a ′ + a 1 , n 2 a ′ + a 2 ) = θ(a 1 , a 2 ).
2) If there exists a prime p such that pa ′ ∈ A, then for every b 0 ∈ B, θ extends to the subgroup generated by a ′ and A in the following way. The elements of the group generated by a ′ and A are the na ′ + a, where 0 ≤ n ≤ p -1 and a ∈ A. Denote by int(x) the integer part of a rational number x. The extension is defined by:

∀(n 1 , a 1 , n 2 , a 2 ) ∈ {0, . . . , p -1} × A × {0, . . . , p -1} × A, θ(n 1 a ′ + a 1 , n 2 a ′ + a 2 ) = θ(a 1 , a 2 ) + θ(int( n 1 + n 2 p )(pa ′ ), a 1 + a 2 ) + int( n 1 + n 2 p )b 0 .

So we get [p]B non isomorphic extensions of

(A × B, + θ ), each one depending on the class of b 0 modulo pB. A ′ × B is a subgroup of the divisible closure of A -→ × θ B. If b 0 = 0, then p(a ′ , 0) = (pa ′ , 0).
Proof. In case 1) as well as in case 2), one easily checks that the mapping θ that we define satisfies (*) and (**). In case 1), one can easily verify that (***) holds, it remains to prove (***) in case 2). In order to simplify the notations, we set:

ǫ 12 = int( n 1 + n 2 p ), ǫ 23 = int( n 2 + n 3 p ) and ǫ 123 = int( n 1 + n 2 + n 3 p ).
So let n 1 , n 2 , n 3 be in {0, . . . , p -1} and a 1 , a 2 , a 3 be in A. In order to compute θ(n

1 a ′ + a 1 + n 2 a ′ + a 2 , n 3 a ′ + a 3 ), we start from the representation of n 1 a ′ + a 1 + n 2 a ′ + a 2 as (n 1 + n 2 -ǫ 12 )(pa ′ ) + a 1 + a 2 + ǫ 12 (pa ′ ),
and we get:

θ(n 1 a ′ + a 1 + n 2 a ′ + a 2 , n 3 a ′ + a 3 ) = θ(a 1 + a 2 + ǫ 12 (pa ′ ), a 3 )+ θ(int((n 1 + n 2 -ǫ 12 p + n 3 )/p)(pa ′ ), a 1 + a 2 + a 3 + ǫ 12 (pa ′ )) +int((n 1 + n 2 -ǫ 12 p + n 3 )/p)b 0 = θ(a 1 + a 2 + ǫ 12 (pa ′ ), a 3 )+ θ((ǫ 123 -ǫ 12 )(pa ′ ), a 1 + a 2 + a 3 + ǫ 12 (pa ′ )) + (ǫ 123 -ǫ 12 )b 0 .
In the same way,

θ(n 1 a ′ + a , n 2 a ′ + a 2 + n 3 a ′ + a 3 ) = θ(a 1 , a 2 + a 3 + ǫ 23 (pa ′ ))+ θ((ǫ 123 -ǫ 23 )(pa ′ ), a 1 + a 2 + a 3 + ǫ 23 (pa ′ )) + (ǫ 123 -ǫ 23 )b 0 .
So, proving (***) reduces to proving that the following (E1) and (E2) are equal:

(E1) = θ(a 1 + a 2 + ǫ 12 (pa ′ ), a 3 ) + θ((ǫ 123 -ǫ 12 )(pa ′ ), a 1 + a 2 + a 3 + ǫ 12 (pa ′ ))+ (ǫ 123 -ǫ 12 )b 0 + θ(a 1 , a 2 ) + θ(ǫ 12 (pa ′ ), a 1 + a 2 ) + ǫ 12 b 0 and (E2) = θ(a 2 , a 3 ) + θ(ǫ 23 (pa ′ ), a 2 + a 3 ) + ǫ 23 b 0 + θ(a 1 , a 2 + a 3 + ǫ 23 (pa ′ ))+ θ((ǫ 123 -ǫ 23 )(pa ′ ), a 1 + a 2 + a 3 + ǫ 23 (pa ′ )) + (ǫ 123 -ǫ 23 )b 0 .
One can check that we have:

(E1) = θ(a 1 , a 2 ) + θ(a 1 + a 2 , ǫ 12 (pa ′ )) + θ(a 1 + a 2 + ǫ 12 (pa ′ ), a 3 ) +θ(a 1 + a 2 + ǫ 12 (pa ′ ) + a 3 , (ǫ 123 -ǫ 12 )(pa ′ )) + ǫ 123 b 0 . and (E2) = θ(a 2 , a 3 ) + θ(a 2 + a 3 , ǫ 23 )(pa ′ )) + θ(a 2 + a 3 + ǫ 23 (pa ′ ), a 1 ) +θ(a 2 + a 3 + ǫ 23 (pa ′ ) + a 1 , (ǫ 123 -ǫ 23 )(pa ′ )) + ǫ 123 b 0 .
The operation + θ being associative, we note that

[[[(a 1 , 0) + θ (a 2 , 0)] + θ (ǫ 12 (pa ′ ), 0)] + θ (a 3 , 0)] + θ ((ǫ 123 -ǫ 12 )(pa ′ ), 0) = (a 1 + a 2 + ǫ 12 (pa ′ ) + a 3 + (ǫ 123 -ǫ 12 )(pa ′ ), (E1)), and that [[[(a 2 , 0) + θ (a 2 , 0)] + θ (ǫ 23 (pa ′ ), 0)] + θ (a 1 , 0)] + θ ((ǫ 123 -ǫ 23 )(pa ′ ), 0) = (a 2 + a 3 + ǫ 23 (pa ′ ) + a 1 + (ǫ 123 -ǫ 23 )(pa ′ ), (E2)).
Now, calculations show that in order to prove that (E1) = (E2) it is sufficient to establish:

(ǫ 12 (pa ′ ), 0) + θ (ǫ 123 -ǫ 12 )(pa ′ ), 0) = (ǫ 23 (pa ′ ), 0) + θ (ǫ 123 -ǫ 23 )(pa ′ ), 0).
The 3-tuple (ǫ 123 , ǫ 12 , ǫ 23 ) can take the values (0, 0, 0), (2, 1, 1), (1, 0, 0), (1, 1, 0), (1, 0, 1) or (1, 1, 1), in any case, calculations prove that the equality holds, the details are left to the reader.

If n 1 +n 2 < p, then (n 1 a ′ , 0)+ θ (n 2 a ′ , 0) = ((n 1 +n 2 )a ′ , 0), and θ((p-1)a ′ , a ′ ) = θ(0, 0)+θ(pa ′ , 0)+b 0 = b 0 . Hence ((p -1)a ′ , 0) + θ (a ′ , 0) = (pa ′ , b 0 ) i.e. p(a ′ , 0) = (pa ′ , b 0 ). In particular, if b 0 = 0, then p(a ′ , 0) = (pa ′ , 0), and (pa ′ , 0) is divisible by p in the extension we obtained. In all cases, (a ′ , 0) belongs to the divisible hull of A -→ × θ B.

Now, the order relation on the quotient group is fixed, hence any morphism from A ′ × B onto A ′ × B, equipped with another operation, takes an element into an element which belongs to the same class modulo B. In particular, the image of (a

′ , 0) is (a ′ , b 1 ), for some b 1 ∈ B 1 . Then p(a ′ , b 1 ) = p(a ′ , 0) + p(0, b 1 ) = (pa ′ , b 0 ) + (0, pb 1 ) = (pa ′ , b 0 + pb 1 )
. Consider two operations extending the one of A × B, + θ and + θ ′ , characterized respectively by elements of B, say b 0 and b ′ 0 . So, if there exists a morphism between these two extensions, then b ′ 0 can be expressed as b 0 + pb 1 , for some b 1 ∈ B 1 , and conversely, which proves that there are (at least) the same number of non isomorphic extensions of the structure of group as the number of classes modulo pB in B.

If a ′ ∈ A ′ \A satisfies some relation na ′ ∈ A, we consider n as factored into prime powers, and we proceed in the same way as in 2). So, θ extends to every element of A ′ \A, and by Zorn's lemma we conclude that θ extends to A ′ . A ′ -→ × θ B is contained in the divisible hull of A -→ × θ B, because this holds at each step.

4.5 Definition. Let T be a discrete linearly ordered abelian group, with first positive element 1 T , and which contains a fixed element z T that we join to the language. Since 1 T is definable, we can assume that it lies in the language. For every prime p, n ∈ N * and k ∈ {0, . . . , p n -1}, we define the formula DD p n ,k : ∃x, p n x = z T + k1 T .

4.6 Lemma. Let T be a discrete linearly ordered abelian group, with first positive element 1 T , and containing a fixed cofinal positive element z T , that we join to the language. Assume that T / 1 T is divisible. 1) For every prime p and n ∈ N * , there exists exactly one integer k ∈ {0, . . . , p n -1} such that DD p j ,k holds in T . 2) Let p be a prime, n ∈ N * , and k ∈ {0, . . . , p n -1} such that DD p j ,k holds within T . We express k as a sum a 0 + a 1 p + • • • + a n-1 p n-1 , where, for 1 ≤ j ≤ n, a j ∈ {0, . . . , p -1}, then for every j ≤ n, and

k j = a 0 + • • • + a j-1 p j-1 , DD p j ,kj holds within T .
Proof. 1) Since T / 1 T is divisible, there exists x ∈ T such that the class of p n x modulo 1 T is equal to the class of z T . Let x 1 ∈ T , the class of x 1 is equal to the class of x if and only if x 1 -x ∈ Z • 1 T , and if this holds then p n x 1 -p n x ∈ p n Z • 1 T . So there exists a unique element of the class of x (that we still denote by x) such that p n x -z T ∈ {0, . . . , p n -1}, which proves 1).

2

) If p n x = z T + k1 T = z T + (a 0 + a 1 p + • • • + a n-1 p n-1 )1 T , then for j ≤ n we have p j (p n-j x -(a j + a j+1 p + • • • + a n-1 p n-j-1 )1 T ) = z T + (a 0 + • • • + a j-1 p j-1 )1 T , the proposition follows.
4.7 Proposition. Let T 1 and T 2 be discrete linearly ordered abelian groups such that T 1 / 1 T1 ≃ Q ≃ T 2 / 1 T2 , and containing an element z 1 and z 2 respectively, which is positive and cofinal (which is equivalent to saying that it is not contained in any proper convex subgroup). In the language of ordered groups together with a predicate for the distinguished element, the following conditions are equivalent. T 1 and T 2 are isomorphic T 1 and T 2 satisfy the same formulas DD p j ,k

T 1 ≡ T 2 .
Proof. Clearly, if T 1 and T 2 are isomorphic, then they are elementarily equivalent, and if they are elementarily equivalent then they satisfy the same formulas DD p j ,k , because these are first order formulas. Now, assume that they satisfy the same formulas DD p j ,k . We are going to define cocycles θ 1 and θ 2 such that T 1 ≃ Q -→ × θ1 Z and T 2 ≃ Q -→ × θ2 Z, then we will prove: θ 1 = θ 2 . We know that every element of Q has a unique representation as a finite sum r = n+ i j m ij p -j i , where n ∈ Z, the p i 's run over the increasing sequence of all primes, the m ij 's belong to {0, . . . , p i -1}, and j ∈ N * . Consequently, a generating subset of the additive group Q is {1} ∪ {1/p j ; p a prime and j ∈ N * }. We let z 1 ∈ T 1 be such that its class in T 1 / 1 T1 is 1, and for every n ∈ Z, we let nz 1 be the representative of the class n. If x belongs to the class 1/p and n ∈ Z, then there exists k ∈ Z such that p(x + n1 T1 ) = px + pn1 T1 = z 1 + k1 T1 + pn1 T1 , so we see that we can choose x in order to have 0 ≤ k ≤ p -1, we will denote this element by k p . For m ∈ {2, . . . , p-1}, we let mx be the representative of m/p. Assume that j ≥ 1 and that the representative x of the class 1/p j is fixed, we let the representative of the class 1/p j+1 be the element y which satisfies 0 ≤ py -x ≤ (p -1)1 T1 , we let k p j+1 be the integer such that k p j+1 1 T1 = py -x. For m ∈ {2, . . . , p -1}, we let my be the representative of the class m/p j+1 . In the sum of any two rational numbers expressed as r = n + i j m ij p -j i , the only interactions between the generators occur with 1/p n and 1/p n-1 (p a prime, n ∈ N * ). The mapping θ 1 is the isomorphism between T 1 and Q -→ × θ1 Z. For every integers m and n, we have θ 1 (m, n) = 0, and θ 1 is defined by induction in the same way as in 2) of Proposition 4.4. We see that θ 1 is uniquely determined by the k p j 's. We define θ 2 in the same way. Now, the k p j 's are uniquely determined by the formulas DD p j ,k , since DD p j ,k holds if and only if k = k p + k p 2 p + • • • + k p j p j-1 . It follows that θ 2 = θ 1 , and we have the isomorphism we were looking for. Note that if p divides z 1 , then the representative of the class of 1/p is the divisor of z 1 , in this case, k p = 0, the same holds if p j divides z 1 , hence the divisible hull of z 1 in T 1 is contained in the set of representatives.

4.2 Dense c-regular cyclically ordered abelian groups. Proof. Let x 1 , . . . , x n be elements of uw(G) such that x 1 , . . . , x n are pairwise p-incongruent (within G), then x 1 , . . . , x n are pairwise p-incongruent, hence [p]uw(G) ≥ [p]G. If [p]G is infinite, then the proposition is trivial. So we assume that [p]G is finite, and let x 1 , . . . , x n be a maximal family of pincongruent elements. In particular, for every i = j, x i -x j / ∈ z G . Let x ∈ uw(G), there exists some x i such that x is p-congruent to x i , i.e. there exists y ∈ uw(G) and k ∈ Z such that x -x i = py + kz G . If z G is divisible by p (in other words, if G contains a p-torsion element), this implies that x and x i are p-congruent (within uw(G)), hence x 1 , . . . , x n is a maximal p-incongruent family of elements of uw(G). It follows:

[p]uw(G) = [p]G. If z G is not p-divisible, then the elements x i + kz G , 1 ≤ i ≤ n, 0 ≤ k ≤ p -1,
are pairwise p-incongruent. Let x ∈ uw(G) and x i such that x = x i , then, by what we did above, x is p-congruent to one of the x i + kz G 's within uw(G), which proves that the family we defined is maximal, and that [p]uw(G) = p[p]G.

4.9 Proposition. Let G 1 and G 2 be two dense c-regular cyclically ordered groups such that G 1 is a subgroup of G 2 . Then G 1 is an elementary substructure of G 2 if and only if G 1 is pure in G 2 , and, for every prime p,

[p]G 1 = [p]G 2 . Proof. Trivially, if G 1 ≺ G 2 , then G 1 is pure in G 2 and, for every prime p, [p]G 1 = [p]G 2 ,
because these are first order properties. Conversely, we know that uw(G 1 ) embeds into uw(G 2 ). First we prove that uw(G 1 ) is pure in uw(G 2 ). Let (n, g) be an element of G 1 which is divisible by some prime p within uw(G 2 ). Since p(1, 0) = (p, 0), we can assume that 0 ≤ n < p, hence a divisor of (n, g) can be expressed as (0, h), for some h ∈ G 2 , and ph = g.

Since G 1 is pure in G 2 , it contains some element h ′ such that ph ′ = g. If h ′ = h, then h ′ h -1 is a p-torsion element within G 2 ,
and since G 1 is pure in G 2 , it contains a p-torsion element. We know that if a cyclically ordered group contains a p-torsion element, then its subgroup of p-torsion elements is cyclic of cardinal p (to see this, look at the subgroups of K). Consequently G 1 contains all the p-torsion elements of G 2 , and in particular, it contains h ′ h -1 , hence it contains h. Since any integer can be factored into prime powers, we deduce by induction that for every m ∈ N * , if m divides (n, g) within uw(G 2 ), then it divides (n, g) within uw(G 1 ) (note that the divisors are the same). We see also that for every prime p, G 2 contains a p-torsion element if and only if G 1 contains an p-torsion element, and by Lemma 4.8, we have [p]uw(G 1 ) = [p]uw(G 2 ). By [Za 61], we have: uw(G 1 ) ≺ uw(G 2 ). Since z G2 = z G1 ∈ uw(G 1 ), uw(G 1 ) still remains an elementary substructure of uw(G 2 ) in a language augmented with a predicate consisting of the constant z G , and by Theorem 4.1 of [START_REF] Giraudet | First order theory of cyclically ordered groups[END_REF], we have:

G 1 ≺ G 2 .
4.10 Proposition. Let G 1 and G 2 be two dense countable c-regular cyclically ordered groups having isomorphic torsion groups and such that, for every prime p, [p]G 1 = [p]G 2 . Then G 1 and G 2 are elementary substructures of the same cyclically ordered group.

Proof. Since G 1 and G 2 have the same prime invariants of Zakon and the same torsion groups, their linear parts have the same prime invariants of Zakon, hence, by Note 6.2 and Theorem 6.3 of [Za 61], they are elementary substructures of the same regular group L where for every u ∈ L there exists n ∈ N * such that nu can be expressed as g 1 + g 2 , for some g 1 ∈ G 1 and g 2 ∈ G 2 , the order being the lexicographic one. uw(G 1 )/l(G 1 ) and uw(G 2 )/l(G 2 ) embed into R, where the images of the classes of z G1 and z G2 are 1. Let R be the subgroup of R generated by the images of uw(G 1 )/l(G 1 ) and uw(G 2 )/l(G 2 ), R is divisible, because it is generated by two divisible subgroups. Let θ 1 and θ 2 be two mappings satisfying (*), (**) and (***) of Proposition 4.2, and such that uw(G 1 ) ≃ (uw(G 1 )/l(G 1 )) -→ × θ1 l(G 1 ) and uw(G 2 ) ≃ (uw(G 2 )/l(G 2 )) -→ × θ2 l(G 2 ). By Proposition 4.4, θ 1 and θ 2 extend to R. We set θ = (θ 1 , θ 2 ), in the same way as in Proposition 4.3. According to the same proposition, we get a structure of linearly ordered group R -→ × θ L where uw(G 1 ) and uw(G 2 ) are cofinal subgroups. By properties of regular groups, since L is regular and the quotient of R -→ × θ L by L is divisible, the group R -→ × θ L is regular, and it has the same prime invariants of Zakon as L and as uw(G 1 ) and uw(G 2 ). Since the theory of dense regular groups with fixed family of prime invariants of Zakon is model complete, it follows that uw(G 1 ) and uw(G 2 ) are elementary substructures of R -→ × θ L. Since these two subgroups contain (1, 0), they still remain elementary substructures in a language augmented with a predicate consiting of this element, and their wound-rounds G 1 and G 2 are elementary substructures of the wound-round associated to R -→ × θ L.

Proposition. For any choice of the family of prime invariants of Zakon and any subgroup of U,

there exists a countable c-archimedean dense cyclically ordered group G with family of prime invariants of Zakon and with torsion group so chosen.

Proof. Let T be the subroup of U, the unwound uw(T ) of T is a subgroup of Q, and for every prime p, n ∈ N, the element z T is p n -divisible if and only if ζ p n ∈ T . In the same way as at the beginning of Section 4.1, [p]uw(T ) = 1 if uw(T ) is p-divisible, and [p]uw(T ) = p otherwise. Let Z = {p ni i | i ∈ N * } be the given family of invariants of Zakon, where (p i ) is the increasing sequence of all primes and the n i 's belong to N ∪ {∞}. By Lemma 4.1, there exists a countable subgroup M of R such that for every i ∈ N * , [p i ]M = [p]uw(T )p ni i and uw(T ) is a pure subgroup of M . We set G = M/ z T . Since uw(T ) is pure in M , the groups G and T have the same torsion subgroup, which is T , and by Lemma 4.8, we have, for every i ∈ N * , [p i ]G = p ni i . If G is finite, hence it is contained in U, it suffices to consider the subgroup H = {e ir | r ∈ Q} of K, where Q is the group of rational numbers, and for every prime p we have [p]H = 1. Hence G ⊕ H is infinite, so it is dense, it has the same torsion subgroup as G, and, for every prime p,

[p](G ⊕ H) = [p]G • [p]H = [p]G.

Theorem. There exists an infinite c-archimedean cyclically ordered group which is elementarily equivalent to G if and only if G is c-regular and dense. Any two dense c-regular cyclically ordered groups are elementarily equivalent if and only if their torsion subgroups are isomorphic and they have the same family of prime invariants of Zakon.

These conditions depend on the first order theory of G by Lemma 4.8 and because the number of p-torsion elements of G depends on the first order theory of G.

Proof. If G is elementarily equivalent to some c-archimedean infinite cyclically ordered group, then it shares with this group the first order properties, in particular being dense and c-regular. Now, assume that G is c-regular and dense. By the theorem of Löwenheim-Skolem, there exists a countable elementary substructure G 1 of G. By Proposition 4.11, there exists a countable c-archimedean dense group G 2 having the same torsion group and the same family of prime invariants of Zakon as G 1 . By Proposition 4.10, there exists a cyclically ordered group

G ′ such that G 1 ≺ G ′ and G 2 ≺ G ′ . Since G 1 ≺ G, we have G 2 ≡ G.
Let G and G ′ be c-regular dense cyclically ordered groups. If they are elementarily equivalent, then their torsion subgroups are isomorphic and they have the same family of prime invariants of Zakon. Assume now that the torsion subgroups of G and G ′ are isomorphic and that they have the same family of prime invariants of Zakon. Let G 1 ≺ G and G ′ 1 ≺ G ′ be countable. By Proposition 4.10, there exists a cyclically ordered group

G ′′ such that G 1 ≺ G ′′ and G ′ 1 ≺ G ′′ . It follows G ≡ G ′ .
4.13 Remark. One can see, for example by Theorem 3.12, that every abelian c-divisible cyclically ordered group is c-regular, hence by Theorem 4.12 an abelian cyclically ordered group is c-divisible if and only if it is elementarily equivalent to U.

4.3 Discrete c-regular cyclically ordered abelian groups.

Proof. Trivially, if G 1 ≺ G 2 , then G 1 is pure in G 2 ,
and their positive cones have the same first positive element, since it is definable. Conversely, in the same way as in the proof of Proposition 4.9, we can show that uw(G 1 ) is pure in uw(G 2 ). Furthermore, uw(G 1 ) and uw(G 2 ) have the same lowest positive element, hence by [Za 61]: uw(G 1 ) ≺ uw(G 2 ). Since z G2 = z G1 ∈ uw(G 1 ), uw(G 1 ) still remains an elementary substructure of uw(G 2 ) in a language augmented with a predicate consisting of z G , and by Theorem 4.1 of [START_REF] Giraudet | First order theory of cyclically ordered groups[END_REF], we have:

G 1 ≺ G 2 .
4.15 Proposition. Assume that G is c-regular discrete and is not c-archimedean. Then G contains a discrete c-regular pure subgroup H such that K(H) = U and l(H) ≃ Z, and which is an elementary substructure of G.

Proof. Since G is discrete, it contains a smallest nontrivial c-convex subgroup 1 G , furthermore, 1 G is a pure subgroup of uw(G), it is the smallest nontrivial convex subgroup of uw(G). Since G is c-regular, uw(G) is regular and uw(G)/ 1 G is divisible. Denote by W the divisible hull of z G ⊕ 1 G within uw(G).

W is a pure subgroup of uw(G) and W/ 1 G is the divisible hull of ( z G ⊕ 1 G )/ 1 G within uw(G)/ 1 G (which is divisible), hence W/ 1 G ≃ Q. Furthermore, W contains a unique proper convex subgroup: 1 G , it follows that W is a regular subgroup of uw(G). Since W is pure in uw(G) and it contains the same first positive element 1 G , W is an elementary substructure of uw(G), and it still remains an elementary substructure in a language augmented with a predicate consisting of the element z G . Set H = W/ z G , by Theorem 4.1 of [START_REF] Giraudet | First order theory of cyclically ordered groups[END_REF], H is an elementary substructure of G. The linear part of H is 1 G , which is isomorphic to Z, and

K(H) = (W/ z G )/ 1 G ≃ Q/ 1 = U.
The proof of this proposition shows that the first order theory of the infinite c-regular discrete cyclically ordered groups is equal to the first order theory of the abelian cyclically ordered groups whose linear part is isomorphic to Z and with quotient isomorphic to U, in other words, to the first order theory of the abelian linearly ordered groups W which contain an unique convex subgroup, isomorphic to Z, and such that the quotient group is isomorphic to the ordered group Q and where the classe of 1 is fixed. Note that W is isomorphic to a subgroup of Q -→ × Q which contains Z -→ × {0} and such that {0} -→ ×Z is its only convex subgroup.

Definition.

If G is discrete and not c-archimedean, then the first positive element 1 G of G is definable, we can assume that it lies in the language. For a prime p, n ∈ N * and k ∈ {0, . . . , p n -1}, D p n ,k will be the formula: ∃x, R(0, x, 2x, . . . , (p n -1)x) ∧ p n x = k1 G .

Remark. Note that G contains a p k -torsion element whose class in

G/ 1 G is a p k -th root of 1 if and only if G satisfies the formula D p k ,0 , hence it contains an element of torsion n = p k1 1 • • • p kj j , whose class is a n-th root of 1 in G/ 1 G if

and only if it satisfies the formulas D p k 1

1 ,0 , . . . , D p k j j ,0 . In particular, G contains a subgroup isomorphic to U if and only if it satisfies the formulas D p k ,0 for every prime number p and k ∈ N * . 4.18 Lemma. Assume that G is c-regular discrete and infinite. For every p prime, n ∈ N * and k ∈ {0, . . . , p n -1}, G satisfies the formula D p n ,k if and only if uw(G) satisfies the formula DD p n ,k , where the fixed element of the language consists of z G .

Proof. The formula R(0, x, 2x, . . . , (p n -1)x) ∧ p n x = k1 G says that the jx's where 1 ≤ j ≤ p n -1 "do not turn full circle", but p n x "turns full circle", and is equal to k1 G . This is equivalent to saying that the image of x in uw(G) satisfies p n x = z G + k1 G .

4.19 Lemma. Assume that G is discrete, is not c-archimedean, and that G/ 1 G is c-divisible. 1) For every p prime and n ∈ N * , there exists exactly one integer k ∈ {0, . . . , p n -1} such that D p j ,k holds within G.

2) Let p be a prime, n ∈ N * , and k ∈ {0, . . . , p n -1} such that D p j ,k holds in G. We express k as

a 0 + a 1 p + • • • + a n-1 p n-1
, where, for 1 ≤ j ≤ n, a j ∈ {0, . . . , p -1}. Then for every j ≤ n, and

k j = a 0 + • • • + a j-1 p j-1 , D p j ,kj holds within G.
3) There exists an unique sequence (ϕ p ), where p runs over the increasing sequence of prime numbers and the ϕ p 's are mappings from N * into {0, . . . , p -1} such that for every prime p, every n ∈ N * and every k ∈ {0, . . . , p n -1}, D p n ,k holds within G if and only if

k = ϕ p (1) + ϕ p (2)p + • • • + ϕ p (n)p n-1 .
Proof. 1) and 2) follow from Lemma 4.18, Lemma 4.6, and from the fact that G is c-divisible if and only if uw(G) is divisible. 3) is a consequence of 1) and 2).

4.20 Definition. Assume that G is discrete, non-c-archimedean, and that G/ 1 G is divisible. For every prime p, let ϕ p be the mapping from N * into {0, . . . , p -1}, defined in 3) of Lemma 4.19, the sequence (ϕ p ), will be called the characteristic sequence of G.

4.21 Remark. Note that G contains a subgroup which is isomorphic to U if and only if for every prime p the mapping ϕ p is the 0 mapping. Proof. Trivially, if C 1 and C 2 are isomorphic, then they are elementarily equivalent, and if they are are elementarily equivalent then they satisfy the same formulas D p n ,k , since these are first order formulas. Now, assume that they satisfy the same formulas D p n ,k . By Lemma 4.18, uw(C 1 ) and uw(C 2 ) satisfy the same formulas

DD p n ,k . Since C 1 / 1 C1 ≃ U ≃ C 2 / 1 C2 , we have uw(C 1 )/ 1 C1 ≃ Q ≃ uw(C 2 )/ 1 C2 ,
and by Proposition 4.7, uw(C 1 ) and uw(C 2 are isomorphic in a language containing the element z C . Consequently, C 1 and C 2 are isomorphic. Finally, by Lemma 4.19 the last condition is equivalent to the second one.

Theorem. Any two non-c-archimedean c-regular discrete cyclically ordered groups are elementarily equivalent if and only if they satisfy the same formulas

D p n ,k .
Proof. Since these formulas are first order formulas, if two c-regular discrete cyclically ordered groups are elementarily equivalent then they satisfy the same formulas D p n ,k . Conversely, assume that G and G ′ are c-regular, discrete, and they satisfy the same formulas D p n ,k . Let H be the subgroup of G and H ′ be the subgroup of G ′ defined in the proof of Proposition 4.15. Since they are elementary substructures, they satisfy the same formulas D p n ,k , hence by Proposition 4.22, they are isomorphic. Hence G and G ′ have elementary substructures which are isomorphic, so they are elementarily equivalent.

Corollary. Any two c-regular discrete non-c-archimedean cyclically ordered groups are elementarily equivalent if and only if their characteristic sequences are equal.

Proof. This is a consequence of Theorem 4.23 and of the equivalence between the second and the fourth condition of Proposition 4.22, since if G is discrete, then G is c-regular if and only if G/ 1 G is divisible. 4.26 Proposition. For every sequence (ϕ p ) of functions from N * into {0, . . . , p -1}, where p runs over the increasing sequence of prime numbers, there exists a non-c-archimedean discrete c-regular cyclically ordered group H whose characteristic sequence is (ϕ p ).

Proof. Let (p n ) n∈N * be the increasing sequence of prime numbers. In the linearly ordered group Q -→ × Q we set z H = (1, 0) and 1 H = (0, 1). For n ∈ N * and i ∈ {1, . . . , n}, let

k i = ϕ pi (1) + ϕ pi (2)p + • • • + ϕ pi (n)p n-1 i
. According to the chinese remainder theorem, there exists k ∈ {0, 1, . . . , (p

1 • • • p n ) n -1} such that for i ∈ {1, . . . , n} k is congruent to k i modulo p n i . Denote by W n the subgroup of Q -→ × Q generated by 1 (p 1 • • • p n ) n , k (p 1 • • • p n ) n and 1 H . Hence (p 1 • • • p n-1 ) n 1 (p 1 • • • p n ) n , k (p 1 • • • p n ) n = 1 p n n , k p n n is congruent to 1 p n n , k n p n n
modulo 1 H , the same holds for the other i ∈ {1, . . . , n}. It follows that W n satisfies the formulas DD p n i ,ki , where i ∈ {1, . . . , n}. The sequence (W n ) n∈N * is an increasing sequence of subgroups of Q -→ × Q, denote by W the union of all the W n 's. One can prove that z H is divisible modulo 1 H within W , hence W/ z H ≃ Q, and that W is discrete with first element 1 H . Consequently, H = W/ z H is a discrete c-regular subgroup such that K(H) ≃ U, l(H) ≃ Z, the sequence (ϕ p ) is its characteristic sequence.

In Remark 4.21 we characterized the discrete c-regular cyclically ordered groups whose torsion subgroups are isomorphic to U, we can also characterize those which are divisible. The following result proves that this case can be seen as the opposite case.

4.27 Proposition. Let G be a discrete c-regular cyclically ordered group, p be a prime and ϕ p be the mapping from N * to {0, . . . , p -1} defined in 4.20. The following conditions are equivalent.

1) G is p-divisible. 2) 1 G is p-divisible. 3) ϕ p (1) = 0. 4) G doesn't contain any p-torsion element.
Proof. We have trivially: 1) ⇒ 2) and 4) ⇒ 3).

Assume that ϕ p (1) = 0. Hence, for every n ∈ N * , p doesn't divide k p n . By Bezout identity, there exist two integers u and v such that up n + vk p n = 1. We know that there exists x in G such that

p n x = k p n 1 G , hence p n (vx) = vk p n 1 G = (1 -up n )1 G , and p n (vx + u)1 G = 1 G , wich proves that 1 G is p-divisible. Let y ∈ G, since G is c-regular, the class of y modulo 1 G is p n -divisible, consequently there exists z ∈ G such that p n z -y ∈ 1 G , hence there exists an integer k such that p n z -k1 G = y. Since 1 G is p n -divisible, it follows that y is p n -divisible. We proved: 3) ⇒ 2) ⇒ 1).
Assume that G contains a p-torsion element y, then 2y, . . . , (p -1)y are p-torsion elements. By taking the lowest element of the set {y, 2y, . . . , (p -1)y} ordered by < 0 instead of y, we can assume: R(0, y, 2y, . . . , (p -1)y), and py = 0, so, by the definition of k p , we have: k p = 0, that is, ϕ p (1) = 0. We proved 3) ⇒ 4). Assume furthermore that 1 G is p-divisible, hence there exists z such that pz = 1 G . Let j ∈ {0, . . . , p -1} be such that R(jy, z, (j + 1)y) holds (z is not a multiple of y because it is not p-torsion), by setting z ′ = z -jy we have R(0, z ′ , y), wich implies R(0, z ′ , 2z ′ , . . . , pz ′ ), hence R(0, z ′ , 1 G ), a contradiction, because 1 G is minimal. It follows that 1 G is not p-divisible, hence G is not p-divisible. Consequently, 2) ⇒ 4), and the proposition is proved.

Proposition.

Let G be a non-c-archimedean cyclically ordered group, then G is c-regular discrete divisible if and only if there exists a discrete cyclic order R on the group Q such that (Q, R) is an elementary subextension of G.

Proof. First assume that G is c-regular divisible and discrete, then according to Proposition 4.15, there exists an elementary subextension H of G such that l(H) ≃ Z and

K(H) ≃ U. Q • 1 H is a subgroup of H, because H is divisible. Let x ∈ H, since H/ 1 H ≃ U,
there exist integers n and k such that nx = k1 H . Since H is divisible and torsion-free, (by Proposition 4.27), we have: x = k n 1 H , which proves:

H = Q • 1 H .
In order to prove the other implication, it suffices to show that if H = (Q, R) where R is a discrete cyclic order, then H is c-regular. Now, one can prove that l(H) = 1 H , hence it is regular, and that K(H) ≃ U, consequently, by Theorem 3.7, H is c-regular.

4.29 Definitions. For every prime p, let ϕ p be a mapping from N * into {0, . . . , p -1}. For every prime p and every n ∈ N * , denote by N p n ,ϕp the set p n N * -( n k=1 p k-1 ϕ p (k)). The set of the N p n ,ϕp 's will be called the family of subsets of N * characteristic of (ϕ p ). The family of subsets of N * characteristic of the characteristic sequence of G will be called the family of subsets of N * characteristic of G.

4.30 Proposition. 1) For every ultrafilter U on N * , there exists one and only one sequence (ϕ p ), where p runs over the increasing sequence of all primes and ϕ p is a mapping from N * into {0, . . . , p -1}, such that U contains the family of subsets of N * characteristic of (ϕ p ).

2) For every (ϕ p ), where p runs over the increasing sequence of prime numbers and ϕ p is a mapping from N * into {0, . . . , p -1}, there exists a non principal ultrafilter U on N * containing the family of subsets of N * characteristic of (ϕ p ).

Proof. 1) By a property of ultrafilters, if A ∈ U and A

= A 1 ∪ • • • ∪ A n is a finite partition of A,
then U contains exactly one of the subsets A k , 1 ≤ k ≤ n. Now, for p prime, n ∈ N * , and a 1 , . . . , a n in {0, . . . , p-1}, p n+1 N * -(

n k=1 a k p k-1 +jp n ), (0 ≤ j ≤ p-1) is a finite partition of p n N * -( n k=1 a k p k-1
), the result follows by induction.

2) Let p be fixed and 0

< n 1 < n 2 < • • • < n k , then N p n 1 ,ϕp ∩ • • • ∩ N p n k ,ϕp = N p n k ,ϕp , and if 1 < p 1 < p 2 < • • • < p k are prime, then by the chinese remainder theorem N p n 1 1 ,ϕp 1 ∩ • • • ∩ N p n 1 k ,ϕp k is infinite.
It follows that the intersection of every finite family of sets N p n ,ϕp has infinite cardinal. We join the elements of the filter of cofinite subsets to this family, then the property of non-empty intersection still remains satisfied, hence there exists an ultrafilter containing all the sets N p n ,ϕp and the filter of cofinite subsets. This ultrafilter is not principal, because it contains all the cofinite subsets.

4.31 Definition. Let U be an ultrafilter on N * , the family of subsets defined in 1) of Proposition 4.30 will be called the family of subsets of N * defined by U .

4.32 Theorem. 1) Let U be an non principal ultrafilter on N * , C be the ultraproduct of the cyclically ordered groups Z/nZ modulo U , p be a prime, n ∈ N * and k ∈ {0, . . , p n -1}. Then C satisfies the formula D p n ,k if and only if p n N * -k ∈ U . 2) Let U be a non principal ultrafilter on N * , C be the ultraproduct of the cyclically ordered groups Z/nZ modulo U . The family of subsets of N * defined by U is equal to the family of subsets of N * characteristic of C.

3) Let U 1 and U 2 be two non principal ultrafilters on N * , C 1 (resp. C 2 ) be the ultraproduct of the cyclically ordered groups Z/nZ modulo U 1 (resp. U 2 ). Then C 1 ≡ C 2 if and only if the family of subsets of N * defined by U 1 is the same as the family of subsets of N * defined by U 2 .

Proof. 1) C satisfies D p n ,k if and only if the set of all integers m such that Z/mZ satisfies D p n ,k belongs to U . Let m ∈ N * , there exists some j ∈ {0, . . . , p n -1} such that m ∈ p n N * -j, then there exists x ∈ N * such that m = xp n -j, and we have p n x = m + j, and 0 < x < 2x < • • • < (p n -1)x < m. In Z/mZ, this is equivalent to R(0, x, 2x, . . . , (p n -1)x) and p n x = j, hence Z/mZ satisfies D p n ,j , and for j ′ = j, 0 ≤ j ′ ≤ p n -1, Z/mZ does not satisfy D p n ,j ′ , since every discrete c-regular cyclically ordered group satisfies exactly one relation D p n ,j , for fixed p and n. It follows that Z/mZ satisfies D p n ,k if and only if m ∈ p n N * -k. Consequently, C satisfies the formula D p n ,k if and only if p n N * -k ∈ U .

2) Follows from 1) and from Lemma 4.19.

3) Follows from 2) and from Corollary 4.24.

4.33 Corollary. 1) If G is infinite, c-regular and discrete, and U is a non principal ultrafilter on N * , then G is elementarily equivalent to the ultraproduct of the cyclically ordered groups Z/nZ modulo U if and only if the family of subsets of N * characteristic of G is the same as the family of subsets of N * defined by U .

2) G is c-regular and discrete if and only if there exists an ultrafilter on N * such that G is elementarily equivalent to the ultraproduct of the cyclically ordered groups Z/nZ modulo U .

Proof. 1) Follows from Theorem 4.32 and from Corollary 4.24.

2) If there exists an ultrafilter on N * such that G is elementarily equivalent to the ultraproduct of the cyclically ordered groups Z/nZ modulo U , then it is c-regular and discrete, because these are first order properties. Conversely, assume that G is c-regular and discrete. If G is finite, then there exists n 0 ∈ N * such that G = Z/n 0 Z, then G is isomorphic to the ultraproduct of the cyclically ordered groups Z/nZ modulo the principal ultrafilter generated by {n 0 }. If G is infinite, the result follows from 1).

4.34 Proposition. For every subgroup S of U, there exists an ultraproduct of finite cyclic groups whose torsion subgroup is S.

Proof. This proposition is a consequence of Theorem 4.32. 3) Let p be a prime such that G is not p-divisible, then every class of G modulo l(G) contains an element which is not p-divisible and which is n-divisible for all integers n such that n and p are coprime. 4) Let p be a prime such that G is not p-divisible, then every class of uw(G) modulo l(G) contains an element which is not p-divisible and is n-divisible for all integers n such that n and p are coprime.

Proof. 1) Note that by Propositon 6.3 a) of [START_REF] Giraudet | First order theory of cyclically ordered groups[END_REF], K(G) ≃ K, since G is ω 1 -saturated and K(G) is infinite. Let x 0 ∈ G, assume first that x 0 / ∈ U, say x 0 = e iα for some irrational element α ∈ [0, 1[. For every n ∈ N * , let m n be the integer such that 2π mn n ≤ α < 2π mn+1 n . For any n ∈ N * , x ∈ G and θ ∈ [0, 1[ such that x = e iθ , we have: 2π n < θ < 2π n-1 ⇒ R(0, x, 2x, . . . , (n -1)x)&¬R(0, x, 2x, . . . , (n -1)x, nx)) ⇒ 2π n ≤ θ ≤ 2π n-1 , hence one can see that 2π mn n < θ < 2π mn+1 n implies [∀y, (R(0, y, 2y, . . . , (n -1)y)&¬R(0, y, 2y, . . . , (n -1)y, ny)) ⇒ R(0, m n y, x)]

[∃y, R(0, y, 2y, . . . , (n -1)y)&¬R(0, y, 2y, . . . , (n -1)y, ny)&R(0, x, (m n + 1)y)] which implies 2π mn n ≤ θ ≤ 2π mn+1 n . In any case, x = x0 if and only if x satisfies all of these first order formulas. The element x is divisible if and only if ∃y, ny = x holds for every n ∈ N * . We get a countable type. Take a finite subset of this type, and let m be lcm of all the n's that appear in some formula ∃y, ny = x. Then pick an element y of 1 m x 0 , and x = my, x satisfies this finite set of formulas. By ω 1 -saturation, the type is satisfied, and x 0 contains a divisible element. Now assume that x 0 ∈ U, say x 0 = e 2ikπ/n , where k and n are coprime, k < n. In K, x 0 is characterized by x 0 = 0, 2x 0 = 0, . . . , (n -1)x 0 = 0, nx 0 = 0, and for 2 ≤ l ≤ n -1, one of the relations R(0, x 0 , lx 0 ) or R(0, lx 0 , x 0 ). We are going to find sufficient conditions in order that these formulas be satisfied. For nx = 0, i.e. x ∈ l(G), we take all the formulas R(0, nx, knx), for k ≥ 2. For lx = 0, i.e. lx / ∈ l(G), we take the smallest integer k l such that R(0, k l lx 0 , lx 0 ) holds, and we take the formula R(0, k l lx, lx). x is divisible if and only if ∃y, qy = x holds for every q ∈ N * . This type is countable. Take a finite subset of this type, let m be the lcm of the q's such that ∃y, qy = x appear, and y ∈ 1 m x 0 . Then mny ∈ l(G). If mny does not belong to the positive cone of G, we take y ′ = y -mny, then mny ′ = mny -mnmny belongs to positive cone of G. Set x = my ′ , and the finite subset of formulas is satisfied. By ω 1 -saturation, the type is satisfied.

2) The elements of uw(G) can be expressed as (n, x), for some n ∈ Z and x ∈ G. In order to prove that (n, x) is divisible in uw(G), we can assume that n ≥ 0, and it is sufficient to prove that (n, x) is divisible by every integer m ≥ n. In this case, the divisor is some (0, y) with y ∈ G. Then m(0, y) = (n, x) is equivalent to: the sequence y, 2y, . . . , my "travels n times around" and furthermore my = x. (k + 1)y "being in the round which follows the round" of y is equivalent to: R(0, (k + 1)y, ky). Since m and the class of x are fixed, we know where are the jumps, hence we have a succession of R(0, (k + 1)y, ky) and R(0, ky, (k + 1)y) which is well-defined. The class of (n, x) modulo l(G) containing a divisible element can be rendered by: there exists z such that z -x ∈ l(G) and (n, z) is divisible. Finally z -x ∈ l(G) is equivalent to an infinite set of formulas R(0, z -x, q(z -x)), for q ≥ 2. This type is countable. In order to get an element which satisfies a finite subset of this type, we consider the lcm of all the m's which appear in this subset (we still denote it by we pick (0, t) in the class of 1 m (n, x) modulo l(G) (which exists since uw(G)/l(G) is divisible), then z = mt, satisfies the finite subset of formulas. The only formulas which appear belong to the language of cyclically ordered groups. By ω 1 -saturation, the type is satisfied, which gives us the required divisible element.

3) and 4) Let x 0 be an element of G which is not p-divisible, and we consider the countable type ∀y, py = x, and ∃y, ny = x, for every positive integer n prime to p. Consider a finite subset of this type, and let m be the lcm of all the n's prime to p which appear. Then x = mx 0 satisfies this finite subset. By ω 1 -saturation, G contains an element x which is not divisible by p and which is divisible by every integer which is prime to p. By 1), there exists y which is divisible and such that ȳ = x, hence x -y ∈ l(G) is not p-divisible and is divisible by every integer which is prime to p. Now, we add to x -y some divisible element of a fixed class of G or of uw(G).

4.39 Proposition. Assume that G is c-regular, dense and ω 1 -saturated. Then G contains a countable elementary substructure which is a c-archimedean group.

Proof. Denote by H the divisible hull of z G in uw(G). Since H is a pure subgroup of uw(G), we have for every prime p: [p]H ≤ [p]uw(G). By Lemma 4.1, there exists a countable subgroup M of R containing a pure subgroup isomorphic to H and such that for every prime p we have [p]M = [p]uw(G). We assume that uw(G)/l(G) = R by letting 1 being the class of z G ; M is a subgroup of uw(G)/l(G) that we embed into uw(G) in the following way. The image of 1 is z G . Every e kn j is a class modulo l(G), its image is some representative which is not divisible by p n and is divisible by every integer which is prime to p n . This element exists by Proposition 4.38. So, every e kn j satisfies the same divisibility relations within M as its image within uw(G). By the definition of M its image is a pure sbgroup of uw(G). Now, this image is indeed an ordered subgroup, because the images of the e kn j 's are the corresponding classes. M is a pure subgroup of uw(G) such that, for every prime p, [p]M = [p]uw(G), furthermore M is regular since it is archimedean. It follows that M ≺ uw(G). Consequently, M/ z G is a countable elementary substructure of G, which is a c-archimedean subgroup.

Proposition. Every discrete cyclically ordered group embeds into a c-regular discrete cyclically ordered group contained in its divisible hull.

Proof. By Proposition 4.36, uw(G) embeds into a minimal regular group contained in its divisible hull, we consider the wound-round associated to this divisible hull and z G . The minimality follows from Theorem 3.12.

5 With or without the cyclic order predicate?

It seems to be natural and usefull for studying R to consider its linear order, what about the study of K and its cyclic order? More generally, to determine the contribution of the cyclic order in the study of the c-regular cyclically ordered groups, we focus on the theory of those groups either in the language for groups without a predicate for the cyclic order or with the cyclic order predicate. In the strict language of groups, we know that any two abelian groups are elementarily equivalent if and only if they have the same invariants of Szmielew (cf. [Sz 55]). Here the torsion subgroups of our groups embed into U, hence the Szmielew invariants reduce to the Zakon invariants and the torsion subgroup. We can reformulate the second part of Theorem 4.12 as follows.

5.1 Proposition. Any two dense c-regular cyclically ordered abelian groups are elementarily equivalent if and only if they are elementarily equivalent groups.

The same holds for any two dense regular linearly ordered groups.

With respect to the discrete c-regular cyclically ordered groups, all groups having a fixed torsion subgroup have the same Szmielew invariants, it follows: 5.2 Proposition. Any two infinite discrete c-regular cyclically ordered abelian groups having a fixed torsion subgroup are elementarily equivalent in the language of groups. Two infinite discrete c-regular cyclically ordered abelian groups having a fixed torsion subgroup need not be elementarily equivalent in the language of cyclically ordered groups. Indeed, by Theorem 4.23, they are elementarily equivalent in the language of cyclically ordered groups if and only if they satisfy the same formulas D n,k . Now, by Lemma 4.19 and Proposition 4.26, they need not satisfy the same formulas D n,k .

In the case of linearly ordered groups, we have the following.

Proposition.

If G is a torsion-free abelian group, then there exists a structure of regular linearly ordered group on G with a smallest convex proper subgroup if and only if for every prime p the cardinal of every maximal family of pairwise p-incongruent elements is at most equal to the cardinal of R. Moreover, there exists a structure of discrete regular linearly ordered group on G if and only if, for every p prime, [p]G = 1 and there exists an element which is not divisible by any prime p.

Proof. Assume that for every prime p the cardinal of every maximal family of pairwise p-incongruent elements is at most equal to the cardinal of R and for each prime p let I p be such a family. Denote by H the divisible hull within G, of the subgroup generated by the union of all the subsets I p , where p run over the set of all prime numbers. The cardinal of H is at most equal to the cardinal of R, hence the cardinal of a basis of the Q-vector space H, where H is the divisible hull of H, is at most equal to the cardinal of R. We embed this basis into a rationally independent subset of R, so we get an embedding of the group H into R, hence we have an embedding of H into R. This embedding gives rise to a linear order on H, such that H is an archimedean group. Now, the quotient group G/H is torsion-free, hence there is a compatible linear order on G/H. Let H ′ be a set of representatives of the equivalence classes of G modulo H, we deduce a one-to-one mapping from G onto H ′ × H. The lexicographic order on H ′ × H induces a linear order on G, and in the same way as in [START_REF] Jaffard | Extensions de groupes ordonnés[END_REF], one can check that this order is compatible. Furthermore, G/H is divisible and H is archimedean, hence G is regular. Conversely, let C the smallest convex proper subgroup of G. Since G/C is divisible, for every prime p there is a maximal family I p of pairwise p-incongruent elements which is contained in C. Now, C is archimedean, hence card(C) ≤ card(R), so card(I p ) ≤ card(R). By [Za 61, Corollary 1.8] every maximal family of pairwise p-incongruent elements has the same cardinal, hence the cardinal of every maximal family of pairwise p-incongruent elements is at most equal to the cardinal of R.

If for every prime p we have [p]G = 1 and there exists an element g 0 which is not divisible by any prime, we let C be the subgroup generated by g 0 , it is a pure subgroup and G/C is divisible. We conclude in the same way as above. The converse is trivial. Now assume that G is a torsion-free abelian group such that there exists a prime p and a maximal family I p of elements which are pairwise p-incongruent and of cardinal greater than the cardinal of R and, for every prime q = p, G is q-divisible. Let H be the divisible hull within G of the subgroup generated by I p . Let C be a pure subgroup of H distinct from H. Hence one of the elements of the family I p does not belong to C, hence H/C is not divisible. Since the cardinal of H is greater than the cardinal of R, there cannot exist a structure of archimedean linearly ordered group on H. Hence for every linear order H contains a proper convex subgroup C, now H/C is not divisible, so, there does not exist any structure of regular linearly ordered group on H. For the same reason, there does not exist any structure of c-regular cyclically ordered group.

Turning to the cyclically ordered case we get: 5.4 Proposition. If G is a torsion-free abelian group, then there exists a structure of c-regular cyclically ordered group on G with a smallest convex proper subgroup if and only if for every prime p the cardinal of every maximal family of pairwise p-incongruent elements is at most equal to the cardinal of R. Moreover, there exists a structure of discrete c-regular cyclically ordered group on G if and only if, for every p prime, [p]G = 1 and there exists an element which is not divisible by any prime p.

Proof. Leg G be an abelian group such that for every prime p the cardinal of every maximal family of pairwise p-incongruent elements is at most equal to the cardinal of R and for each prime p let I p be such a family. Let H be the divisible hull within G of the subgroup generated by the union of all the I p 's (hence the cardinal of H is at most equal to the cardinal of R), let A be a Q-vectorial subspace of the divisible group G/H, of cardinal at most equal to the cardinal of R, and let B be a subspace such that G/H = A ⊕ B. Let a ∈ A\B, L := {x ∈ G | x + H ∈ B ⊕ Za}. Set A = Qa ⊕ A ′ , hence: G/L ≃ U ⊕ A ′ . So G/L embeds into K (the restriction of this embedding to U being identity), we equip L with a regular linear order, and applying [2, Lemma 5.4] we get a structure of c-regular cyclically ordered group on G. The remainder of the proof is similar to the proof of Proposition 5.3.

We know that an abelian group is orderable if and only if it is torsion-free, this is a consequence of its first order theory. According to [2, Theorem 5.8] being cyclically orderable is a consequence of the first order theory of a group. We are going to see that the class of infinite groups which can be equipped with a c-regular and discrete cyclic order and the class of infinite groups which can be equipped with a discrete regular linear order are not elementary.

If G is an infinite abelian group equipped with a structure of discrete c-regular cyclically ordered group, then its torsion subgroup embeds into U, and G contains an element e which is not divisible by any prime p and such that the quotient group G/Ze is divisible. The fact that G contains an element which is not p-divisible for every prime p is not a first order property in the language of groups (the following examples will prove this assertion), while this is a consequence of a first order formula of the language of cyclically ordered group (the positive cone has a smallest element).

We give examples of groups which are elementarily equivalent to discrete c-regular cyclically ordered groups in the language of groups, but which cannot be equipped with a structure of discrete c-regular cyclically ordered group. Let a n , n ∈ N * , be a rationally independent family of elements of K\U, and G be the direct sum of the Z (pn) a n 's (where (p n ) n∈N * is the increasing sequence of all primes, and Z (pn) is the localization of Z at the ideal (p n )). Every element of G is contained in a finite sum Z (pn 1 ) a n1 + • • • + Z (pn k ) a n k , hence it is p-divisible, for every prime p which is distinct from p n1 , . . . , p n k . Consequently, G does not contain any element which is not divisible by any prime, hence it cannot be equipped with a structure of discrete c-regular cyclically ordered group. However, as we recalled in Section 4.1, for every prime p, we have [p]G = p, hence G is elementarily equivalent in the language of groups to every discrete torsion-free c-regular cyclically ordered group. Note that G is also elementarily equivalent to Z, but it cannot be equipped with a structure of discrete regular linearly ordered group.

4. 1

 1 Lemma. Let H be a subgroup of Q, and, for every prime p, let m p ∈ N be such that [p]H ≤ p mp . Then there exists a countable subgroup M of (R, +) which contains H, such that H is pure in M and, for every prime p, [p]M = p mp .Proof. Denote by p 1 < p 2 < • • • < p n < • • • the increasing sequence of all primes, and, for every n ∈ N * , denote by m ′ n the integer which satisfies p

4. 8

 8 Lemma. Let p be a prime. If G contains a p-torsion element, then [p]uw(G) = [p]G, otherwise, [p]uw(G) = p[p]G.

4 .

 4 22 Proposition. Let C 1 and C 2 be two non-c-archimedean discrete cyclically ordered groups such that C 1 / 1 C1 ≃ U ≃ C 2 / 1 C2 . The following conditions are equivalent. C 1 and C 2 are isomorphic C 1 and C 2 satisfy the same formulas D p n ,k C 1 ≡ C 2 . The characteristic sequence of C 1 is equal to the characteristic sequence of C 2 .

4 .

 4 25 Corollary. If G is c-regular and discrete, then the first order theory of G is uniquely determined by the subgroup H defined in Proposition 4.15. Proof. By Theorem 4.23, the first order theory of G is uniquely determined by the formulas D p n ,k , now apply Proposition 4.22.

  4.14 Proposition. Let G 1 and G 2 be two discrete c-regular cyclically ordered groups such that G 1 is a subgroup of G 2 . Then G 1 is an elementary substructure of G 2 if and only if G 1 is pure in G 2 and the positive cones of G 1 and G 2 have the same first positive element.

  4.35 Theorem. 1) The class of all discrete c-regular cyclically ordered abelian groups is the smallest elementary class which contains all finite cyclic groups.2) The class of all discrete regular cyclically ordered abelian groups is the smallest elementary class which contains all cyclic groups. If G belongs to this class, then either G is a linearly cyclically ordered group which is elementarily equivalent to Z, or G is c-regular and elementarily equivalent to an ultraproduct of finite cyclic groups. Every linearly ordered abelian group T , having a smallest proper convex subgroup, embeds into a minimal regular groupT ′ which is contained into its divisible hull. If T is dense, then T ′ is dense, if T is discrete, then T ′ is discrete.In particular, every discrete linearly ordered abelian group embeds into a minimal discrete regular group which is contained in its divisible hull.Proof. Denote by C the smallest proper convex subgroup of T , then there exists a cocycle θ such that T ≃ (T /C) -→ × θ C. We embed (T /C) into its divisible hull and we extend T in the same way as in Proposition 4.4 (note that this embedding need not be unique) the details are left to the reader.4.37 Proposition. Every dense cyclically ordered group having a smallest proper c-convex subgroup embeds into a minimal c-regular cyclically ordered group which is contained in its divisible hull.Proof. By Proposition 4.36, uw(G) embeds into a minimal regular group contained in its divisible hull, we consider the wound-round associated to its divisible hull and z G . By Theorem 3.12, this extension is minimal. 4.38 Proposition. Assume that G is ω 1 -saturated and K(G) is infinite. 1) Every class of G modulo l(G) contains a divisible element. 2) Every class of uw(G) modulo l(G) contains a divisible element.

	Proof. Follows from what we just proved, from Theorem 3.8 and from the properties of regular linearly
	ordered abelian groups.
	4.36 Proposition.


Jean Geringer

Mathew Taylor

Digby D Macdonald

Prévision de l'épaisseur du film passif d'un acier inoxydable 316L soumis au fretting corrosion grâce au Point Defect Model, PDM. Predicting the steady state thickness of passive films with the Point Defect Model in fretting corrosion experiments

Some implants have approximately a lifetime of 15 years. The femoral stem, for example, should be made of 316L/316LN stainless steel. Fretting corrosion, friction under small displacements, should occur during human gait, due to repeated loadings and un-loadings, between stainless steel and bone for instance. Some experimental investigations of fretting corrosion have been practiced. As well known, metallic alloys and especially stainless steels are covered with a passive film that prevents from the corrosion and degradation. This passive layer of few nanometers, at ambient temperature, is the key of our civilization according to some authors. This work is dedicated to predict the passive layer thicknesses of stainless steel under fretting corrosion with a specific emphasis on the role of proteins. The model is based on the Point Defect Model (micro scale) and an update of the model on the friction process (micro-macro scale). Genetic algorithm was used for finding solution of the problem. The major results are, as expected from experimental results, albumin prevents from degradation at the lowest concentration of chlorides; an incubation time is necessary for degrading the passive film; under fretting corrosion and high concentration of chlorides the passive behavior is annihilated.

Résumé

Les implants orthopédiques de hanche ont une durée de vie d'environ 15 ans. Par exemple, la tige fémorale d'un tel implant peut être réalisée en acier inoxydable 316L ou 316LN. Le fretting corrosion, frottement sous petits déplacements, peut se produire pendant la marche humaine en raison des chargements répétés entre le métal de la prothèse et l'os. Plusieurs investigations expérimentales du fretting corrosion ont été entreprises. Cette couche passive de quelques nanomètres, à température ambiante, est le point clef sur lequel repose le développement de notre civilisation, selon certains auteurs. Ce travail vise à prédire les épaisseurs de cette couche passive de l'acier inoxydable soumis au fretting corrosion, avec une attention spécifique sur le rôle des protéines. Le modèle utilisé est basé sur le Point Defect Model, PDM (à une échelle microscopique) et une amélioration de ce modèle en prenant en compte le processus de frottement sous petits débattements. L'algorithme génétique a été utilisé pour optimiser la convergence du problème. Les résultats les plus importants sont, comme démontré avec les essais expérimentaux, que l'albumine, la protéine étudiée, empêche les dégradations de l'acier inoxydable aux plus faibles concentrations d'ions chlorure ; ensuite, aux plus fortes concentrations de chlorures, un temps d'incubation est nécessaire pour détruire le film passif.

Introduction

Le nombre de prosthèses totales de hanche augmentera dans les prochaines années. En effet, il a doublé entre 2000 et 2010, en France [START_REF] Macdonald | Evaluation des prothèses totales de hanche[END_REF]. Aux Etats-Unis, 250000 prothèses totales de hanche sont implantées annuellement, en 2012. Il est envisagé que ce nombre passe à 572000 en 2030. Il est important de noter le nombre de prosthèses totales de hanche devrait aussi augmenter d'un facteur 2 en 2030. De plus 1 américain sur 30 possède une prothèse totale de hanche. Enfin, les politiques de santé sur la mobilité concernent des patients de plus en plus jeunes. [START_REF] Kurtz | [END_REF]. Tous ces éléments vont dans le sens d'une augmentation drastique du nombre d'implants orthopédiques. Un des points clef lié à la prévision de la durée de vie des implants orthopédiques est du ressort de la compréhension de l'usure pour prédire la stabilité de l'implant et éviter la production de débris qui est responsable du rejet de l'implant. Quand une prothèse est implantée pour la première fois, la première étape liée à l'ostéointégration consiste en l'adhérence entre les tissus osseux et le matériau étranger. Après cette étape, en raison du stress shielding (masquage des contraintes), un descellement intervient et des micro-mouvements se produisent entre la tige fémorale et l'os. Ce dernier point implique des micromouvements dans un environnement aqueux corrosif, ce qui est appelé fretting corrosion [START_REF] Waterhouse | Fretting Corrosion[END_REF]. Comme indiqué précédemment, même si le métal a de meilleures performances mécaniques que celles de l'os, en fretting corrosion l'usure de l'acier inoxydable est plus grande que celle du polymère, pourtant beaucoup moins dur que l'acier inoxydable. Avant d'entreprendre des tests très proches des conditions réelles de sollicitation des prostheses totals de hanche, avec notamment la prise en compte du milieu physiologique complexe, des tests entre l'acier inoxydable 316L et le Polyméthacrylate de méthyle (PMMA) ont été entrepris ; ces matériaux ont des caractéristiques mécaniques très proches de celles des implants orthopédiques et de l'os [START_REF] Geringer | [END_REF]5]. Plusieurs résultats intéressants ont été fournis par des tests expérimentaux et les auteurs impliqués dans ce travail souhaitent apporter de nouvelles connaissances en utilisant une modélisation basée sur le Point Defect Model [6][7][8] L'épaisseur du film passif attendue doit être considérée comme un indicateur pour savoir si la couche passive existe ou non en présence d'albumine, film fin à la surface du film passif. Connaître avec la précision du nanomètre l'épaisseur du film passif ne serait pas raisonnable. Cependant cette approche est une bonne indication pour l'influence des conditions expérimentales sur la passivité de l'acier inoxydable étudié.

Résultats

L'approche expérimentale du fretting corrosion est décrite dans [START_REF] Geringer | [END_REF]5]. De ces investigations des résultats ont été obtenus sur l'influence de la concentration de chlorure avec une certaine teneur de protéines, i.e. l'albumine dans ce cas. Le contact étudié est l'acier inoxydable 316L, de forme parallélépipédique, contre un échantillon de PMMA, forme cylindrique. Le protocole de polissage a une grande importance, ainsi il est décrit précisément dans [9][10]. Les solutions sont composées de solutions de NaCl, d'une concentration allant de 10 -3 mol.L -1 , 10 -2 mol.L -1 , 10 -1 mol.L -1 à 1 mol.L -1 . 2 types de solutions salines ont été envisagées dans cette étude : sans albumine et avec, concentration de 20 g.L -1 . La modélisation tiendra compte de ces résultats expérimentaux. Les résultats expérimentaux pertinents sont les diagrammes de Nyquist provenant des expériences de spectroscopie d'impédance électrochimique. Durant les tests de fretting corrosion, toutes les 20 minutes, les diagrammes de Nyquist ont été enregistrés. La Figure 1 illustre les différents diagrammes obtenus lors d'une experience de fretting corrosion à potential imposé, -400mV/ECS, 20 g.L -1 d'albumine et 1 M de NaCl. Les diagrammes de Nyquist évoluent dans le sens d'une augmentation de la partie imaginaire, comme indiqué sur la Figure 1. La partie imaginaire, en valeur absolue, augmente donc avec le temps lors d'une expérience, ce qui peut être interprété par l'effet protecteur de l'albumine qui s'adsorbe au cours de l'expérience. A partir de ces résultats il est possible de comparer les résultats expérimentaux et ceux provenant de la modélisation grâce au PDM adapté au cas du fretting corrosion Dans cette section la procédure de modélisation va être décrite à partir du PDM. La Figure 2 résume les sept réactions du PDM qui seront utilisées dans le cadre de cette étude [11]. La dissolution du metal est prise en compte dans la partie anodique du circuit électrique équivalent. La relation (1) décrit l'épaisseur du film stationnaire dans des conditions de pseudo stationnarité. Seules les réactions 3 and 7 sont non conservatives et elles sont impliquées dans l'expression de l'épaisseur de film d'oxydes dans des conditions stationnaires. Le terme W est la contribution du fretting, usure mécanique. Ce terme est issu des mesures expérimentales et donné par la loi d'Archard [12]. [START_REF] Macdonald | Evaluation des prothèses totales de hanche[END_REF] Le coefficient de la loi d'Archard est un paramètre qui sera ajusté lors de la modélisation. Pour chaque spectre EIS, le potentiel est fixé et le courant (ou la densité de courant) est mesurée. De ce point le PDM peut être utilisé dans l'expression (2): [START_REF] Kurtz | [END_REF] The potential is imposed and the current density is measured, thus one may suggest that an equivalent electrical circuit could be suggested.

a V c pH C L L L k k W W t C t t Ω Ω + -   = - -= - -       ( ) ( ) ( ) 1
L a V b L c pH a V c pH a V c pH m bl V a V c pH a V c pH bl ol M n a V c pH k C k k I F k k C C k C χ χ χ δ χ δ χ δ χ     + + +     = - + - +         -            

Figure 1 :

 1 Figure 1: Diagrammes de Nyquist obtenus lors d'une experience de fretting corrosion, potentiel imposé -400 mV/ECS, 1 mol.L -1 de NaCl et 20 g.L -1 d'albumine. Avtf: avant fretting; Apf: après fretting, Pdtf(i): pendant l'expérience de fretting corrosion .

Figure 2 7

 2 Figure 2 7 réactions du Point Defect Model, approche déterministe [7].

Figure 3 bFigure 3

 33 Figure 3 épaisseur du film d'oxydes de l'acier inoxydable 316L en fonction du temps durant une expérience de fretting corrosion, 10 -2 mol.L -1 , 14,400 secondes a) modélisation sans albumine b) avec albumine. La Figure 4 rassemble les résultats obtenus des épaisseurs de film d'oxydes calculés grâce au PDM en tenant en compte le fretting. L'incertitude, i.e. l'intervalle de confiance à 95%, des résultats issus de la modélisation restent grands. Il est d'environ 2 nm. Comme indiqué précédemment, le temps d'incubation est d'environ 100 minutes. Cet élément doit être pris en compte pour expliquer la grande incertitude de ces résultats. C'est la raison pour laquelle la figure 5 représente les épaisseurs calculées après un temps de fretting corrosion de 120 minutes. Tout d'abord à 10 -3 mol.L -1 , l'albumine a un effet protecteur de la couche d'oxydes. Ce résultat est en accord avec la mesure du volume d'usure le plus faible en présence d'albumine faible a 0,1 mol.L -1 de force ionique, la tendance semble inverse : l'albumine ne protège pas le métal. Il est à noter que cette concentration est seuil. De nombreux phénomènes physiques doivent se produire à ce

Lss / cm Ionic strength / mol.L -1 Average values of oxides thickness during fretting corrosion experiments from 120 minutes of fretting corrosion

  Le second point concerne l'effet protecteur de l'albumine sur la dégradation de l'acier inoxydable 316L. Pour une concentration de 1 mol.L -1 de NaCl, l'albumine n'a pas d'effet sur l'épaisseur du film d'oxydes. Un effet protecteur a été mis en évidence par cette modélisation pour la plus faible concentration de NaCl, i.e. 10 -3 mol.L -1 . Cette modélisation a été entreprise pour la première fois, dans cette étude. La modélisation grâce aux algorithmes génétiques a permis d'obtenir des résultats dans des délais raisonnables de modélisation, i.e. moins d'une journée sur un ordinateur de bureau amélioré. Plusieurs investigations sont en cours pour affiner la modélisation par les algorithmes génétiques. En termes de matériaux, une nouvelle campagne est en cours pour étudier le comportement de l'alliage de Co-Cr-Mo en fretting corrosion.

					1.E-07
	seuil, ainsi le modèle choisi pourrait être mis en défaut. Le
	résultat le plus intéressant concerne la solution de Ringer.
	L'albumine protège le 316L durant une dégradation par
	fretting corrosion.
	3.E-07
	2.E-07
	Lss / cm
	1.E-07
	0.E+00
	0.001	0.010	0.100	0.156	1.000
	Figure 4: Valeurs moyennes de l'épaisseur de la couche
	d'oxydes calculées grâce au PDM-fretting. 0,156 M de
	force ionique pour la solution de Ringer (NaCl : 8,5 g.L -1 ,
	KCl : 0,25 g.L -1 , CaCl 2 , 2H 2 O: 0,22 g.L -1 , NaHCO 3 : 0,15 g.L -1 ); ces valeurs proviennent des 12 SIE enregistrés
	durant 1 expérience de fretting corrosion.
	5.E-07
	4.E-07
				OCP, 0g.L-1
				OCP, 20g.L-1
	3.E-07
	2.E-07
	1.E-07
	0.E+00
	0.001	0.010	0.100	0.156	1.000
	Figure 5: Les valeurs moyennes de la couche d'oxydes
	calculées par le modèle PDM-fretting. 0.156 mol.L -1 de
	force ionique correspond à celle de la solution de Ringer
	(NaCl : 8,5 g.L -1 , KCl : 0,25 g.L -1 , CaCl 2 , 2H 2 O: 0,22 g.L -1 , NaHCO 3 : 0,15 g.L -1 ); ces valeurs sont issues des 7
	dernières expériences d'EIS, après un temps d'incubation
		de 100 minutes.

Conclusions:

L'épaisseur quasi stationnaire du film passif a été calculée grâce au PDM-fretting. Deux points importants ont été mis en évidence dans cette étude. En premier lieu, un temps d'incubation a été mis mis en lumière pour tous les tests de fretting corrosion.

Ionic strength / mol.L -1 Average values of oxides thickness during fretting corrosion experiments

	OCP, 0g.L-1
	OCP, 20g.L-1