Anatole Khelif

Equations in free groups are not finitely approximable

We give an equation in any free group F of rank > 2 that has a solution in any finite quotient of F, but has no solution in F.

Monochrome symmetric subsets of colored groups

In (Electron. J. Combin. 10 (2003); http://www.combinatorics.org/volume-10/Abstracts/v1oi1r28 html), the first author (Yuliya Gryshko) asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group G include a monochromatic symmetric subset of any cardinal <|G| for any finite coloring? Does every uncountable group G such that |B(G)| < |G| where B(G)={x ∈ G : x2 = 1}, admit a 2- coloring without monochromatic symmetric subsets of cardinality |G|? We answer the first question positively. Assuming the generalized continuum hypothesis (GCH), we give a positive answer to the second question in the abelian case. Finally, we build a counter-example for the third question and we give a necessary and sufficient condition for an infinite group G to admit 2-coloring without monochromatic symmetric subsets of cardinality |G|. This generalizes some results of Protasov on infinite abelian groups (Mat. Zametki 59 (1996) 468-471; Dopovidi NAN Ukrain 1 (1999) 54-57).

Existentially Closed Models via Constructible Sets: There are

We prove that there are 2 χ 0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]). A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated. Our main result uses constructible sets and can be put in an abstract general framework. Comparison with the standard methods which use forcing (cf. [4]) is sketched.

2^{\aleph_0}

We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.

Existentially Closed Pairwise Non Elementarily Equivalent Existentially Closed Ordered Groups

Let K be a (commutative totally) ordered field, let K[X 1,…, X n ] be the K-vector space of the polynomials with n variables. An operator T (i.e., an endomorphism of K[X 1,…,X n ] into itself) is said to be “positive” if the image of every positive polynomial is a positive polynomial, where a positive polynomial is a polynomial which takes only non-negative values. First we prove that in ℝ[X], the sum of the derivatives of a positive polynomial is a positive polynomial too. Then we give what we believe to be a good framework to prove that this result remains true for every ordered field and we propose generalizations.

Asymptotic Ideals (Ideals in the Ring of Colombeau Generalized Constants with Continuous Parametrization)

In [3], Keisler proves that if Q1 is a nonstandard model of the complete theory Th(Q) of (Q;+;×; 0; 1) and C1 is the algebraic closure of Q1, then (C1; Q1) is elementarily equivalent to (C;Q) where C is the algebraic closure of Q. But if Q r 1 is the real closure of Q1 and Qr the real closure of Q, then it is observed in [1] that (Q r 1; Q1) ≡(Qr ;Q). Victor Harnik asked the following question: if we have two nonstandard models Q1 and Q2 of Th(Q) are (Q r 1; Q1) and (Q r 2; Q2) elementarily equivalent? Let L be the language {0; 1;+;×} and let N be a nonstandard model of Peano Arithmetic, formulated in L. We refer to “formulas” of L meaning all the elements of N satisfying a formula F(x), naturally interpreting in the standard model N “x is the G< odel number of a formula”. Then “standard formula” means a “formula” which belongs to the standard part of N (i.e. N) and hence can be identi

From an Elementary Result on Real Closed Fields to Positive Operators on Ordered Fields

ed with a real formula. We call “standard formulas with arbitrary parameters” all the “formulas” of the form G(b) where G(x) is a “formula” with one free variable with a standard G< odel number and b is an integer (standard or nonstandard).

On nonelementarily equivalent pairs of fields

Resume Soit G un groupe libre non monogene. Soit Ĝ la completion profinie de G . Nous exhibons un enonce universel vrai dans G et faux dans Ĝ . Nous montrons que la theorie de Ĝ est instable et indecidable. Nous montrons egalement que certaines equations a parametres dans G sont resolubles dans G si elles sont resolubles dans Ĝ .

Le passage d'un groupe libre à sa complétion profinie : étude logique

The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings. First, we recall the classical setting. Let k be an algebraic number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sl n (θ) of ( n, n ) matrices of determinant 1 with coefficients in θ. The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index. Congruence subgroups of finite index of G are the kernels of the maps Sl n (θ) → Sl n (θ/ I ) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G . Every open set for this topology is open in the profinite topology. We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology. The Bass-Milnor-Serre theorem [1] consists of the two following statements: (A) If k admits a real embedding, then we have an exact sequence That is, Ĝ and Ḡ are isomorphic. (B) If k is totally imaginary, then one has an exact sequence where μ( k )is the group of the roots of unity of k .