We prove that there exists no sentence F of the language of rings with an extra binary predicat I2 satisfying the following property: for every definable set X ⊆ ℂ2, X is connected if and only if ⊧ F, where I2 is interpreted by X. We conjecture that the same result holds for closed subset of ℂ2. We prove some results motivated by this conjecture.

The aim of this paper is to describe an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and (...) group theory. However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups. (shrink)

The aim of this paper is to describe an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and (...) group theory. However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups. (shrink)

Ne estas prima orda formulo, kiu definas la Zariskijajn slositojn inter la konstruitoj, malpli ke la konektojn inter la slositoj.

This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: 1. is the answer to the question P = ?NP the same in every real-closed field?2. if P ≠ NP for , does there exist a problem of which is NP but neither P nor NP-complete ?Some unclassical complexity classes arise naturally in the study of these questions. (...) They are still open, but we could obtain unconditional results of independent interest.Michaux introduced/const complexity classes in an effort to attack question . We show that , answering a question of his. Here A is the class of real problems which are algorithmic in bounded time. We also prove the stronger result: , where PAR stands for parallel polynomial time. In our terminology, we say that is A-saturated and PAR-saturated. We also prove, at the nonuniform level, the above results for every real-closed field. It is not known whether is P-saturated. In the case of the reals with addition and order we obtain P-saturation ). More generally, we show that an ordered -vector space is P-saturated at the nonuniform level ).We also study a stronger notion that we call P-stability. Blum, Cucker, Shub and Smale have shown that fields of characteristic 0 are P-stable. We show that the reals with addition and order are P-stable, but real-closed fields are not.Questions and and the/const complexity classes have some model theoretic flavor. This leads to the theory of complexity over “arbitrary” structures as introduced by Poizat. We give a summary of this theory with a special emphasis on the connections with model theory and we study/const complexity classes from this point of view. Note also that our proof of the PAR-saturation of shows that an o-minimal structure which admits quantifier elimination is A-saturated at the nonuniform level. (shrink)