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Statement 13. [1], [8] ⟨Q, <⟩has the small index
property and Truss generic automorphisms.
By combining Statement 12 and Statement 13 we
conclude.
Theorem 14. ⟨Q, <⟩has the small index property
and Truss-generic automorphisms but ⟨Q, <⟩does
not have SI generic automorphisms.
Notice that we define SI generic for n-tuples of
automorphisms, n < ω. Similarly, we can also say
that an n-tuple (g1, g2, ..., gn)∈Gnis Truss generic
if the set {(gf
1, gf
2, . . . , gf
n) : f∈G}of its conju-
gates is a comeager in Gn. An unpublished result of
I.Hodkinson shows that ⟨Q, <⟩has no Truss generic
2-tuples of automorphisms, by Statement 12 it does
not have SI generic 2-tuples of automorphisms either.
3 A Few Examples
As we have mentioned earlier, there are four possi-
bilities for a countable model: to have both SI and
Truss generics; to have SI but not Truss generics; to
have Truss but not SI generics; and to have neither
SI nor Truss generics. We have already found mod-
els with both SI and Truss generics: the countable
random graph or a countable arithmetically saturated
model of True Arithmetic (Example 7). In the pre-
vious chapter we have shown that ⟨Q, <⟩has Truss-
generic automorphisms but ⟨Q, <⟩does not have SI
generic automorphisms. In this chapter we address
the other two remaining possibilities.
Let us consider a model Q× {0,1}where each
copy of Qis the countable model of the dense lin-
ear order without end-points. An automorphism of
Q×{0,1}preserves the partition {Q×{0},Q×{1}}
and is order preserving on each copy of Q. Then ev-
ery automorphism of Q×{0,1}either fixes each copy
of Qsetwise or sends Q× {0}to Q× {1}. We no-
tice that Q× {0,1}cannot have Truss generics. By a
similar argument as in the previous section we could
show Q× {0,1}has no SI generics. Therefore we
have an example of structure with neither Truss nor
SI generics.
Example 15. Q× {0,1}has neither Truss nor SI
generic automorphisms.
Notice that in Example 15 instead of Q×{0,1}we
might have considered Q×{0,1, . . . , n −1}, n < ω.
By a similar argument Q× {0,1, . . . , n −1}, n < ω
does not have Truss generics. On the other hand Q×ω
behaves differently, Q×ωhas Truss generics. See [1]
for the details.
Our next goal is to find a model with SI generics
but without Truss generics. We need the following
result.
Theorem 16. [2] Let Mbe a countable ω-stable ω-
categorical structure. Then Mhas SI generics.
Example 17. Let Mbe a countable ω-stable ω-
categorical structure. Then M× {0,1}has SI gener-
ics but it does not have Truss generics.
Proof. Let Mbe a countable ω-stable ω-categorical
structure. Since Mis countable ω-stable ω-
categorical then M×{0,1}is also countable ω-stable
ω-categorical. By Theorem 16 M× {0,1}has SI
generics. An automorphism of M× {0,1}preserves
the partition {M× {0}, M × {1}}. Thus every au-
tomorphism of M× {0,1}either fixes each copy of
Msetwise or sends M× {0}to M× {1}. We notice
that M× {0,1}cannot have Truss generics.
[2] and [4] show that the countable random graph
has an amalgamation base. [2] proves that if Mis
countable ω-categorical with an amalgamation base
then it has SI generics. Similar argument might be
applied to R× {0,1}. It can be shown that the base
consisting of finite sets is an amalgamation base for
R× {0,1}. Since R× {0,1}is ω-categorical, then,
using the mentioned result from [2], R× {0,1}has
SI generics. R× {0,1}does not have Truss generics
by the same argument as before. Hence we obtain the
following result.
Example 18. Let Rbe the countable random graph.
Then R× {0,1}has SI generics but it does not have
Truss generics.
4 Conclusion
Two types of generic automorphisms are defined
somewhat similar. Indeed if a model has both of SI
generics and Truss generics, they coincide. If Mis ei-
ther the countable random graph or a countable arith-
metically saturated model of True Arithmetic, then M
has both SI generic and Truss generic automorphisms.
We prove that the dense linear order has the small in-
dex property and Truss-generic automorphisms but it
does not have SI generic automorphisms. We also
construct examples: a countable structures which has
SI generics but it does not have Truss generics; a
countable structure which has no SI generics but it
has Truss generics; a countable structures which has
neither SI generics nor Truss generics.
References:
[1] Truss, J.K., Generic automorphisms of homo-
geneous structures, Proc. London Math. Soc.,
Vol.64, 1992, pp. 121–141.
[2] Hodges, W. and Hodkinson, I. and Lascar, D.
and Shelah, S., The small index property for ω-
stable ω-categorical structures and for the random
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DOI: 10.37394/23206.2024.23.8