Sh:300b

• Shelah, S. (2009). Universal Classes: Axiomatic Framework [Sh:h]. In Classification Theory for Abstract Elementary Classes II.
  Ch. V (B) of [Sh:i]
• Abstract:
  We define some frameworks to axiomatically define classes of models. The main one is AxFr_1, where \mathfrak{K}_\mathfrak{s} an abstract elementary class, omitting smoothness, but has free amalgamation. (That is, there is some M_3 – the free amalgamation of M_1,M_2 over M_0 – such that M_3 is the closure of M_1 \cup M_2.) Now universal classes failing a suitable order property can be expanded to such frames. We fit in older cases; then we define and study them a little. Lastly, we deal with \lambda-model-homogeneous M in \mathfrak{K}. In particular, we prove the equivalence of this property with saturation (assuming enough amalgamation).
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@inbook{Sh:300b,
 author = {Shelah, Saharon},
 title = {{Universal Classes: Axiomatic Framework  [Sh:h]}},
 booktitle = {{Classification Theory for Abstract Elementary Classes II}},
 year = {2009},
 note = {Ch. V (B) of [Sh:i]},
 refers_to_entry = {Ch. V (B) of [Sh:i]}
}