To avoid any confusion, the notions of (strong) homogeneity as I understand them are as follows:

  • a model $M$ is said to be homogeneous if for any elementary partial function $f$ from the model into itself, such that the domain of $f$ is smaller than $M$ (in cardinality), $f$ can be extended by any one element of $M$ (while still being elementary);
  • if any such function can be extended to an automorphism, the model is said to be strongly homogeneous.

Now, it seems to me that a simple back-and-forth argument shows that the definitions are equivalent (just by extending domain and the range of the function until they cover the entire model). Is it correct? If not, are there any simple counterexamples?


Yes, this is exactly Proposition 4.2.13 (p.133) in Marker's Model Theory: An Introduction. As he puts it, "In homogeneous models, partial elementary maps are just restrictions of automorphisms."

The proof is indeed a back-and-forth argument.

  • $\begingroup$ Thanks! I've been given those two definitions separately (as special cases of kappa- (strong) homogeneities, which do differ), so I wasn't sure if I wasn't misunderstanding something. $\endgroup$ – groovin' Feb 18 '12 at 22:44

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