D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, Journal of Symbolic Logic, 66 (2001), 127-143. Home/Search Document Details and Download
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Simplicity, And Stability In There - Byungh An Kim (1999) (1 citation) (Correct)
....general question is whether any simple T has elimination of hyperimaginaries (see Definition 1.3) This problem is related to the existence of canonical bases in M eq (see Definition 1. 1) For simple T , canonical bases exist as imaginaries if and only if T has elimination of hyperimaginaries [10]. Obviously in such T , elimination of hyperimaginaries implies the equivalence of the notions of Lascar strong type and strong type. B. Hart, A. Pillay and the present author showed the existence of canonical bases in the form of hyperimaginaries [3] When they worked together at the Fields ....
A. Pillay and D. Lascar, `Hyperimaginaries and automorphism groups', preprint (1998).
Lovely Pairs of Models - Itay Ben-Yaacov University Self-citation (Pillay) (Correct)
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D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, Journal of Symbolic Logic, 66 (2001), 127-143.
Imaginaries in Beautiful Pairs I - Anand Pillay Evgueni Self-citation (Pillay) (Correct)
....c P (C) c P (BC) The next fact reduces canonical bases in T P to canonical bases of some special kind of types. Fact 2.3 Let B be an elementary substructure of (M, P ) a M . Let d = Cb(stp L (a B#P (M) so d M) Then Cb(tp LP (a B) is L P interdefinable with Cb(tp LP (d B) By [3], for any (hyper)imaginary e in T P , there is a real tuple a and a model (M, P ) such that if c = Cb(tp (a M ) we have e dcl(c) and c acl(e) resp. c bdd(e) Since the tuple d above is in acl L (B#P (M ) we get the following characterization of imaginaries in T P up to ....
D. Lascar, A. Pillay, Hyperimaginaries and automorphism groups, Journal of Symbolic Logic, 66, (2001), 127-143.
Steven Buechler, Anand Pillay and Frank O. Wagner - Wa Gn Er Self-citation (Pillay) (Correct)
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Daniel Lascar & Anand Pillay, Hyperimaginaries and automorphism groups, preprint (1998).
Galois Groups of First Order Theories. - Casanovas, Lascar, Pillay, Ziegler (2000) Self-citation (Lascar Pillay) (Correct)
....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 introduced. Similar things were done in [7]. The main point was that the spaces S=EKP or even X=EKP are naturally equipped with compact Hausdorff topologies (the closed sets being precisely the typedefinable sets) There has been considerable attention paid to the issue of proving that EKP = E Sh in certain situations. For example in [1] ....
....with E X L and it is precisely the relation (on X) of being in the same orbit under Autf L ( M) ii) Similarly for EKP and AutfKP ( M) In particular, given a sort S and a complete type p(x) of that sort, E S KP jp = E p KP . Proof. i) is immediate. ii) is contained in Lemma 4. 18 of [7]. But we will give another proof. Work for simplicity in a sort S. It suffices to show that the equivalence relation on S of being in the same orbit under AutfKP ( M) is type definable over ; and thus has to be E S KP ) a and b are in the same orbit under AutfKP ( M) iff tp(a=e) tp(b=e) ....
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D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, to appear in Journal of Symbolic Logic. 18
From Stability To Simplicity - Kim, PILLAY (1998) Self-citation (Pillay) (Correct)
....PILLAY that canonical bases are the most complicated kind of hyperimaginaries one could have in simple theories. So for example, if all canonical bases exist as tuples of ordinary imaginaries, then it can be shown that every hyperimaginary is interdefinable with a tuple of ordinary imaginaries ([25]) In the stable case, we pointed out that Cb(p) is a kind of code for the set of stationary types which have a common nonforking extension with p. What is the corresponding family in the simple case Let Q be the collection of types q(x) # S(A) which are amalgamation bases (A varying) Let R 0 ....
D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, in preparation.
Galois Groups of First Order Theories. - Casanovas, Lascar, Pillay, Ziegler (2001) Self-citation (Lascar Pillay) (Correct)
....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 introduced. Similar things were done in [7]. The main point was that the spaces S EKP or even X EKP are naturally equipped with compact Hausdor# topologies (the closed sets being precisely the typedefinable sets) There has been considerable attention paid to the issue of proving that EKP = E Sh in certain situations. For example in [1] ....
....with E X L and it is precisely the relation (on X) of being in the same orbit under Autf L ( M) ii) Similarly for EKP and AutfKP ( M) In particular, given a sort S and a complete type p(x) of that sort, E S KP p = E p KP . Proof. i) is immediate. ii) is contained in Lemma 4. 18 of [7]. But we will give another proof. Work for simplicity in a sort S. It su#ces to show that the equivalence relation on S of being in the same orbit under AutfKP ( M) is type definable over # (and thus has to be E S KP ) a and b are in the same orbit under AutfKP ( M) i# tp(a e) tp(b e) ....
[Article contains additional citation context not shown here]
D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, to appear in Journal of Symbolic Logic. 19
Aspects of Geometric Model Theory - Pillay (2001) Self-citation (Pillay) (Correct)
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D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, to appear in Journal of Symbolic Logic.
Supersimple Theories - Steven Buechler Dept (1999) (2 citations) Self-citation (Pillay) (Correct)
....he or she so desired. 3 2 Preliminaries We start by recalling definitions and results regarding simple theories and hyperimaginaries. The basic material on simple theories is from [16] 7] 12] and also [1] for low theories. The material on hyperimaginaries and canonical bases is from [4] and [13]. But there are also a few new observations. T denotes a complete theory in a language L, and M a # saturated model of T for some suitable large #. Small means of cardinality strictly less than #. There is no harm in working in M eq in which there are additional sorts SE whenever E(x, ....
D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, preprint 1998.
A Note on Groups Definable in Difference Fields. - Kowalski, Pillay (2000) Self-citation (Pillay) (Correct)
....by the finite set z 1 of its M, b 1 , y 1 conjugates, another imaginary element, interalgebraic with a over M and in dcl(M, b 1 , y 1 ) In the # situation, the (in general infinite) set X of (M, b 1 , y 1 ) conjugates of z is not on the face of it another # tuple. However, as pointed out in [7], X can be identified with a # tuple: let z = z i ) i#I say. For each finite J # I, let z # J be the (finite) set of M, b 1 , y 1 conjugates of the J tuple (z j ) j#J , a single imaginary. Let z 1 be the # tuple (z # J ) J . Then z 1 is interdefinable with X (an automorphism fixes the ....
D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, to appear in Journal of Symbolic Logic.
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