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Amalgamations preserving ℵ1-categoricity

Published online by Cambridge University Press:  12 March 2014

Anand Pillay  and
Akito Tsuboi
Anand Pillay
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Altegeld Hall, Urbana, IL 61801, USA, E-mail: pillay@symcom.math.uiuc.edu
Akito Tsuboi
Affiliation:
Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan, E-mail: tsuboi@sakura.cc.tsukuba.ac.jp
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Extract

Let L0, L1 and L2 be countable languages with L ∩ L1 = L0. Let M0 be an L0-structure and Mi, an expansion of M0 to an Li,-structure (i = 1,2). We will call an L1 ∪ L2-structure M an amalgamation of M1 and M2 if M∣Li ≅ Mi, (i = 1,2). Let's consider the following problem.

(*) Suppose that both M1 and M2 belong to the class . Can we always find an amalgamation M in ?

Of course the existence of such an amalgamation depends on the class L. Some examples of and the answers are given below.

1. = Countably saturated strongly minimal structures with the DMP In [3], Hrushovski showed that any two strongly minimal theories formulated in totally different languages have a common extension which is still strongly minimal and with the DMP (DMP is the property that states that if a point is sufficiently close to ā, then φ(, ) has the same rank and the same degree as φ(, ā).) His proof essentially shows that if L0 = ∅ then any two countably saturated strongly minimal structures with the DMP have a strongly minimal amalgamation. Also he gave an example that shows the condition L0 = ∅ is necessary.

2. = ℵ1-categorical countable structures. Let M1 be the structure (ℚ, +) and let M2 be the {E, F}-structure defined by: (i) E is an equivalence relation which divides the universe into two infinite classes A and B, (ii) F is a bijection between A and B.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1070 - 1074
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

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