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We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.

We investigate the relations between the geometric properties of tilings and the algebraic and model-theoretic properties of associated relational structures. Isomorphism and local isomorphism of tilings up to translation correspond to isomorphism and elementary equivalence of relational structures. In particular, two Penrose tilings, or two Robinson tilings, are elementarily equivalent. Classical results concerning the local isomorphism property and the “extraction preorder” for tilings are generalized to uniformly locally finite relational structures.Then, we define “equational structures”, which generalize both Cayley graphs of groups and relational structures associated to tilings, and for which we have an appropriate notion of free structure relative to a system of equations. For each finite system Σ of prototiles and local configurations, we give a finite set of local conditions characterizing the connected relational structures which are homomorphic images of Σ-tilings. It follows that tilings are free relative to finite systems of equations which express these conditions. We also prove that the theory of a tiling is superstable, model-complete, and can be axiomatized by ∀∃ sentences. One ∀∃ sentence suffices in the case of Penrose tilings or Robinson tilings.

We show that, for each finite group G, there exists an axiomatization of the class of abelian-by-G groups with a single sentence. In the proof, we use the definability of the subgroups M n in an abelian-by-finite group M, and the Auslander-Reiten sequences for modules over an Artin algebra.

We show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n + 1 alternations of quantifiers.

For any integers n >m ≥ 2, we say that a complete theory T is (m, n)-homogeneous if, for each model M of T, two n-tuples [abar],[bbar] in M have the same type if the corresponding m-tuples from [abar] and [bbar] have the same type. It was conjectured by H. Kikyo that, if M is an infinite group, with possibly additional structure, then the theory of M is not (m, n)-homogeneous. We prove a general result on structures with (m, n)-homogeneous theory which implies that, if M is a counterexample to this conjecture, then there exists an integer h such that each abelian subgroup of M has at most h elements. It follows that there exist an integer k such that M k = 1, and an integer l such that each finite subgroup of M has at most l elements.

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

We give an example of two groups G and H, generated by 2 and 3 elements, which satisfy G × Z ≅ H × Z.The groups G, H are semi-direct products of a torsion-free abelian group of rank 3 by an infinite cyclic group.

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular we prove that the elementary equivalence of two such groups G,H is equivalent to each of the following properties: 1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two elementarily equivalent polycyclic-by-finite groups A,B. It is not presently known if 1) implies elementary equivalence for any groups G,H., Comment: 15 pages. Minor changes in pages 1 to 3, following the remarks of a referee. The paper is presently published

We consider the class C \mathcal {C} which consists of the groups M M with M / M ′ M/M’ finitely generated which satisfy the maximal condition on direct factors. It is well known that any C \mathcal {C} -group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any C \mathcal {C} -group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for C \mathcal {C} -groups, a slightly different notion of decomposition, that we call J J -decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of J J -decompositions, several generalizations of results of R. Hirshon. For instance, we have Z × G ≅ Z × H \mathbb {Z} \times G \cong \mathbb {Z} \times H for any groups G G , H H which satisfy M × G ≅ M × H M \times G \cong M \times H for a C \mathcal {C} -group M M .

For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding curves as the curves without endpoint which are unions of increasing sequences of n-folding curves for n integer. We prove that there exists a standard way to extend any complete folding curve into a covering of the plane by disjoint such curves, which satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. This covering contains at most six curves., Comment: 39 pages containing 9 figures; to be read in pdf; minor changes to improve the readability, especially concerning the figures

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures: 1. the (n+2)-tuples (A 1…,A n+1,f),where n≥2 is an integerA 1…,A n+1 are disjoint finitely generated abelian groups and f A 1×…×A n →A n+1: is a n-linear map; 2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A n →B is a n-linear map; 3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A n →A is a n-linear map. For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.

In this paper, we show that any finitely generated abelian-byfinite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups G, H which satisfy these properties while G x Z and H x Z are not isomorphic. We also prove that a finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite. The definitions and results of model theory which are used here, and in particular the notions of elementary equivalence and elementary submodel, are given in [2]. Anyhow, for the sake of brevity, we denote by GU the ultrapower G'/U, for each group G and each ultrafilter U over a set I. Concerning group theory, the reader is referred to [9 and 10]. We obtain the following characterization: THEOREM. A finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite. This result is a consequence of Theorems 1 and 2 below. THEOREM 1. Any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. THEOREM 2. If G is a finitely generated nilpotent-by-finite, but not abelian-byfinite, group, then there exists an existential sentence, built up from the multiplication symbol, which is false in G and true in the profinite completion of G. Theorem 1 generalizes [7, Theorem 1.4], which was only valid for finitely generated finite-by-abelian groups. A finitely generated finite-by-abelian group is abelian-by-finite, since its center is a normal subgroup of finite index according to the result of P. Hall which is mentioned in [10, p. 12]. On the other hand, the finitely presented abelian-by-finite group (x, y; y= 1 yxy = X-1) is not finite-by-abelian as its center is trivial. By [11, Theorem 2.1 and 6, Theorem 5.5], the following properties are equivalent for two finitely generated finite-by-abelian groups G, H: (1) G and H are elementarily equivalent; Received by the editors July 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20A15; Secondary 03C60, 20F18. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $ 25 per page

We show that, if two finitely generated nilpotent groups are elementarily equivalent, and more generally if they satisfy the same sentences with one alternation of quantifiers, then, they are quite similar, though not necessarily isomorphic. For instance, each of them is isomorphic to a subgroup of finite index of the other and they have the same finite images, the same Mislin genus and the same Pickel genus.

In [O1], we gave algebraic characterizations of elementary equivalence for finitely generated finite-by-abelian groups, i.e. finitely generated FC-groups. We also provided several examples of finitely generated finite-by-abelian groups which are elementarily equivalent without being isomorphic.In this paper, we shall use our previous results to describe precisely the models of the theories of finitely generated finite-by-abelian groups and the elementary embeddings between these models.