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On a question of Herzog and Rothmaler

Published online by Cambridge University Press:  12 March 2014

Anand Pillay  and
Martin Ziegler
Anand Pillay
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, USA, E-mail: pillay@math.uiuc.edu
Martin Ziegler
Affiliation:
Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany, E-mail: ziegler@uni-freiburg.de
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Extract

Herzog and Rothmaler gave the following purely topological characterization of stable theories. (See the exercises 11.3.4 – 11.3.7 in [2]).

A complete theory T is stable iff for any model M and any extension M ⊂ B the restriction map S(B) → S(M) has a continuous section.

In fact, if T is stable, taking the unique non-forking extension defines a continuous section of S(B) → S(A) for all subsets A of B, provided A is algebraically closed in Teq. Herzog and Rothmaler asked, if, for stable T, there is a continuous section for any subset A of B. Or, equivalently, if for any A, S(acleq(A)) → S(A) has a continuous section.

This is an interesting problem, also for unstable T. Is it true that for any T and any set of parameters A the restriction map S(acl(A)) → S(A) has a continuous section? We answer the question by the following two theorems.

Theorem 1. Let A be a subset of a model of T. Assume that the Boolean algebra of acl(A)-definable formulas is generated by

• some countable set of formulas,

• all A–definable formulas,

• all formulas which are atomic over acl(A).

Then S(acl(A)) → S(A) has a continuous section.

The conditions of the theorems are satisfied if, for example, L and A are countable, or, if there are only countably many non-isolated types over acl(A).

Theorem 2. There is a theory of Morley rank 2 and Morley degree 1 such that S(acl(∅)) → S(∅) has no continuous section.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 478 - 481
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[1]Monk, J. Donald and Bonnet, Robert (editors), Handbook of Boolean Algebras, vol. 3, North-Holland, 1989.Google Scholar
[2]Rothmaler, Philipp, Introduction to Model Theory, Gordon and Breach Science Publishers, 2000.Google Scholar