D. Lascar, On the category of models of a complete theory, J. Symb. Logic 47 (1982) 249 -- 266.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....finite homomorphic image of H is jF j. So 2 (H) is a complement to ker in G, a contradiction. 2 In Section 7 of the notes, we indicate how results which are qualitative rather than quantitative can be obtained if we weaken the irreducibility conditions to G finiteness: a notion due to Lascar ([46]) and which in many ways seems to be the most attractive level of generality one could aim for. Definition 3.4.2 We say that a permutation structure W (or its automorphism group Aut(W ) is G finite if for all finite X W there is a smallest closed subgroup of finite index in Aut(W=X) 3.5 ....

D. Lascar, `On the category of models of a complete theory', J. Symb. Logic 47(1982), 249 -- 266.

....Theorem 2.9 (the Independence Theorem) for strong types, namely for p(x) a strong type rather than a complete type over a model. We have, however, been unable to prove this in general. Rather amazingly, a notion developed by Daniel Lascar in his study of the category of models of a complete theory [23], turned out to be exactly what was needed. Recall first some notation: by AutA (C) we mean the group consisting of those automorphisms of C which fix A pointwise. Definition 4.1. Let A be a subset of C. i) By LautA (C) the group of Lascar strong automorphisms of C over A) we mean the subgroup ....

....It is not di#cult to see that for a, b, n tuples say, the equivalence relation Lstp(a A) Lstp(b A) is the finest bounded equivalence relation on C n which is invariant under AutA (C) we just say A invariant) In fact bounded corresponds here to of cardinality at most 2 T A . In [23], Lascar defined when a theory T is G compact. The first author has pointed out an equivalent (but more palatable) definition in [19] Definition 4.3. The theory T is G compact, if for any A # C, we have: FROM STABILITY TO SIMPLICITY 29 (i) LautA (C) is a closed subgroup of AutA (C) and (ii) ....

D. Lascar, On the category of models of a complete theory, The Journal of Symbolic Logic, vol. 82 (1982), pp. 249--266.

..... Further Q is a group under multiplication: if x; y 2 Q, let c 2 Pn be generic to (x; y) then yc 2 Pn , and by a dimension argument, yc is generic to x; so xyc 2 Pn . 2 The compact Lascar group A remarkable connection between compact groups and first order theories was discovered by Lascar [Lascar]. He associated to each first order theory a certain quotient of the automorphism group of a saturated model. For a large class of theories, the G compact theories, he showed that this quotient has the structure of a compact topological group. We will repeat here Lascar s ideas in a slightly ....

....Examples of Poizat involving actions 9 of real algebraic groups show that any compact real algebraic group can be a Lascar group of some theory. Recent work of Kim and Pillay [K1] KP] has shown that simple theories are G compact, and that the Lascar group described here agrees with the one in [Lascar]. The question of the existence or construction of non G compact theories appears not to have been investigated. 2.1 The Kim Pillay space Let U be an 0 saturated structure. A relation on U is said to be 0 definable if it is determined by a formula without parameters. If a binary relation on U ....

D. Lascar, `On the category of models of a complete theory', J. of Symbolic Logic 47 (1982) 249-266.

....which is a countable union of closed sets. As such Gal L (T ) is a kind of descriptive set theoretic invariant of T . In many cases (such as when T is stable) all these equivalence relations and Galois groups coincide. Gal L (T ) the Lascar group, was introduced by the second author in [6]. He also introduced the notion of a G compact theory and remarked that all known theories were G compact. Essentially G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [5] where Lascar strong types (E L classes) took the ....

....Essentially G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [5] where Lascar strong types (E L classes) took the place of strong types. Kim [4] subsequently showed that simple theories are G compact. The second author, in [6], defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdorff) topological 2 group. In [2] Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was ....

D. Lascar,The category of models of a complete theory, Journal of Symbolic Logic 47(1982), 249-266.

....which is a countable union of closed sets. As such Gal L (T ) is a kind of descriptive set theoretic invariant of T . In many cases (such as when T is stable) all these equivalence relations and Galois groups coincide. Gal L (T ) the Lascar group, was introduced by the second author in [6]. He also introduced the notion of a G compact theory and remarked that all known theories were G compact. Essentially G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [5] where Lascar strong types (E L classes) took the ....

....G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [5] where Lascar strong types (E L classes) took the place of strong types. Kim [4] subsequently showed that simple theories are G compact. 2 The second author, in [6], defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdor#) topological group. In [2] Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was ....

D. Lascar,The category of models of a complete theory, Journal of Symbolic Logic 47(1982), 249-266.

....sort S. Roughly speaking, taking the projective limit of these groups as X varies, yields groups Gal L (T ) GalKP (T ) and Gal T Sh which are invariants of the bi interpretability type of T . Precise statements and definitions are given below. Gal L (T ) the Lascar group, was introduced in [5]. He also introduced the notion of a G compact theory and remarked that all known theories were G compact. Essentially G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [4] where Lascar strong types (E L classes) took the ....

....G compact. Essentially G compactness of T means that Gal L (T ) GalKP (T ) Additional interest was generated by the work on simple theories [4] where Lascar strong types (E L classes) took the place of strong types. Kim [3] subsequently showed that simple theories are G compact. Lascar in [5] defined a topology on Gal L (T ) in the case where T is G compact, making Gal L (T ) into a compact (Hausdorff) topological group. In [2] Hrushovski gave another account of the topology, working directly with GalKP (whether T is G compact or not) In fact in that paper the EKP notation was ....

D. Lascar,The category of models of a complete theory, JSL 1982.

....by an NSF grant 1 This work was partially done during a visit of the two authors in the Centre de Recerca Matem ematica, Institut d Estudis Catalans. The authors wish to express their gratitude for its support and hospitability 1 groups. In section 4 we study a certain group introduced in [5], which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in section 3. In sections 3 and 4 there is some overlap with parts of Hrushovski s paper [2] In section 5, we show that if T satisfies some mild assumptions, then for a suitably saturated ....

....) for each i. Then by Lemma 3,3 applied to Y n and E n we obtain a compact Hausdorff topology n on X n , which clearly refines the product topology on X n . But the latter is also compact and Hausdorff, so the topologies agree. It is worth at this stage giving a link with the notions from [5] (although we will do this systematically in the next section) Suppose M 0 is an elementary extension of M , let Y 0 be the solution set of Phi in M 0 and E 0 the solution set of Psi in M 0 . Let X 0 = Y 0 =E 0 . Note that there is a canonical bijection between X and X 0 (each ....

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D. Lascar, The category of models of a complete theory, Journal of Symbolic Logic 47 (1982), 249-266.

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D. Lascar, The category of models of a complete theory, Journal of Symbolic Logic 47 (1982), 249-266.

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D. Lascar, On the category of models of a complete theory, J. Symb. Logic 47 (1982) 249 -- 266.

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D. Lascar, Category of models of a complete theory, Journal of Symbolic Logic, Journal of Symbolic Logic 82(1982), 249-266.

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D.Lascar, `On the category of models of a complete theory', J. Symb. Logic 47(1982), 249 -- 266.

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