Question on $\aleph_0$-categorical nonhomogeneous structures

Question

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$. $\omega$-homogeneity means that any finite partial elementary mapping can be extended so that its domain includes any given element.

I am confused on this because it is well known that a $\aleph_0$-categorical structure is both atomic and countably saturated, and both atomic and countably saturated structures are $\omega$-homogeneous. This actually means that a $\aleph_0$-categorical structure is ultrahomogeneous. Where is wrong here?

Answer 1

You are confusing several notions of homogeneity. Saturated structures, and therefore also $\aleph_0$-categorical structures, are homogeneous, but not necessarily ultrahomogeneous. This means that every finite partial elementary mapping extends to an automorphism.

$\omega$-homogeneity is in fact an even weaker property: it says that any finite partial elementary mapping can be extended so that its domain includes any given element. However, this is equivalent to the property above for countable structures.

Ultrahomogeneity of $\omega$-saturated structures implies quantifier elimination, hence it is not implied by any standard model-theoretic properties that are invariant by expansion of the language with definable predicates.

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In more detail, let me try to deconfuse Macpherson’s terminology by reviewing the relevant properties (using more standard terminology that does not drop the ultra- prefixes) and their connections. In what follows, $M$ is a structure, and $\kappa$ is an infinite cardinal.

• $M$ is $\kappa$-homogeneous if for every partial elementary map $f\colon M\rightharpoonup M$ such that $|f|<\kappa$, and for every $a\in M$, there exists a partial elementary map $g\supseteq f$ such that $a\in\operatorname{dom}(g)$.

• $M$ is strongly $\kappa$-homogeneous if every partial elementary map $f\colon M\rightharpoonup M$ such that $|f|<\kappa$ extends to an automorphism of $M$.

• If $\kappa=|M|$, and $M$ is $\kappa$-homogeneous, it is in fact strongly $\kappa$-homogeneous. Such structures are simply called homogeneous.

• $M$ is $\kappa$-ultrahomogeneous if for every partial isomorphism $f\colon M\rightharpoonup M$ such that $|f|<\kappa$, and for every $a\in M$, there exists a partial isomorphism $g\supseteq f$ such that $a\in\operatorname{dom}(g)$.

• $M$ is strongly $\kappa$-ultrahomogeneous if every partial isomorphism $f$ such that $|f|<\kappa$ extends to an automorphism of $M$.

• $M$ is ultrahomogeneous if it is $\kappa$-ultrahomogeneous (or equivalently, strongly $\kappa$-ultrahomogeneous) for $\kappa=|M|$.

The basic properties are:

• If $M$ is $\kappa$-saturated, it is $\kappa$-homogeneous.

• If $M$ is atomic, it is $\omega$-homogeneous.

• The following are equivalent:

  1. $M$ is $\kappa$-ultrahomogeneous;
  2. $M$ is $\kappa$-homogeneous, and every partial isomorphism $M\rightharpoonup M$ is elementary.

• Likewise for strong $\kappa$-ultrahomogeneity.

• If $M$ is in a finite relational language, or if it is $\omega$-saturated, the following are equivalent:

  1. Every partial isomorphism $M\rightharpoonup M$ is elementary.
  2. $M$ has quantifier elimination.

• Consequently, if $M$ is in a finite relational language, or if it is $\omega$-saturated, the following are equivalent:

  1. $M$ is $\kappa$-ultrahomogeneous.
  2. $M$ is $\kappa$-homogeneous, and $M$ has quantifier elimination.