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Reducts of structures and maximalclosed permutation groups
, 2013
"... Answering a question of Junker and Ziegler, we construct a countable first order structure which is not ωcategorical, but does not have any proper nontrivial reducts, in either of two senses (modeltheoretic, and grouptheoretic). We also construct a strongly minimal set which is not ωcategorical ..."
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Answering a question of Junker and Ziegler, we construct a countable first order structure which is not ωcategorical, but does not have any proper nontrivial reducts, in either of two senses (modeltheoretic, and grouptheoretic). We also construct a strongly minimal set which is not ωcategorical but has no proper nontrivial reducts in the modeltheoretic sense. 1
Imaginaries in pairs of algebraically closed fields
, 2006
"... We consider the theory (ACFp)P of pairs F < K of algebraically closed fields of a given characteristic p. We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form ∪ a∈B(F)Va(K)/Ga(F), where G, V, B are varieti ..."
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We consider the theory (ACFp)P of pairs F < K of algebraically closed fields of a given characteristic p. We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form ∪ a∈B(F)Va(K)/Ga(F), where G, V, B are varieties over the prime field G a group scheme over B and V is a scheme over B (both with irreducible fibres), G acts algebraically on V over B, and for generic b ∈ B(F) the action of Gb(F) on Vb(K) is generically free (namely regular on generic orbits). 1 Introduction and
From "Metabelian QVector Spaces" To New omegaStable Groups
, 1996
"... this paper is to describe (without proofs) an analogue of the theory of nontrivial torsionfree divisible abelian groups for metabelian groups. We obtain illustrations for "oldfashioned" model theoretic algebra and "new" examples in the theory of stable groups. We begin this pap ..."
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this paper is to describe (without proofs) an analogue of the theory of nontrivial torsionfree divisible abelian groups for metabelian groups. We obtain illustrations for "oldfashioned" 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 (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for 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.
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"... Mrank and meager groups by Ludomir N e w e l s k i (Wrocław) Abstract. Assume p ∗ is a meager type in a superstable theory T. We investigate definability properties of p ∗closure. We prove that if T has < 2 ℵ0 countable models then the multiplicity rank M of every type p is finite. We improve S ..."
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Mrank and meager groups by Ludomir N e w e l s k i (Wrocław) Abstract. Assume p ∗ is a meager type in a superstable theory T. We investigate definability properties of p ∗closure. We prove that if T has < 2 ℵ0 countable models then the multiplicity rank M of every type p is finite. We improve Saffe’s conjecture. 0. Introduction. Throughout the paper, T is a superstable theory in a countable language L, and we work within a monster model C = C eq of T. The general references are [Ba, Sh, Hru], see also [Ne2]. Suppose p is a regular stationary type. Associated with p is a closure operator clp defined by a ∈ clp(A) iff stp(a) is hereditarily orthogonal to p. Restricted to p(C),