1.1. Combinatorial Geometries 3

Quasiminimality In this chapter we introduce Zilber’s notion [Zil05] of an abstract quasiminimalexcellent class and prove Theorem 2.23: Lω1,ω-definable quasiminimal-excellent classes satisfying the countable closure condition are categorical in all powers. In the next chapter we expound Zilber’s simplest concrete algebraic example. In Chapter 25, we will place this example in the context of Shelah’s more general notion. An abstract quasiminimal class is a class of structures in a countable vocabulary that satisfy the following two conditions, which we expound leisurely. The class is quasiminimal excellent if it also satisfies the key notion of excellence which is described in Assumption 2.15. A partial monomorphism is a 1-1 map which preserves quantifier-free formulas. Assumption 2.1 (Condition I). Let K be a class of L-structures which admit a closure relation clM mapping X ⊆ M to clM (X) ⊆ M that satisfies the following properties. (1) Each clM defines a pregeometry (Definition 1.1.1) on M . (2) For each X ⊆ M , clM (X) ∈ K. (3) If f is a partial monomorphism from H ∈ K to H ′ ∈ K taking X ∪ {y} to X ′ ∪ {y′} then y ∈ clH(X) iff y ∈ clH′ (X ). Our axioms say nothing explicit about the relation between clN (X) and clM (X) where X ⊂ M ⊂ N . Note however that if M = clN (X), then monotonicity of closure implies clN (Y ) ⊆ M for any Y ⊆ M . And, if closure is definable in some logic L (e.g. a fragment of Lω1,ω) and X ⊆ M with M ≺L∗ N then clN (X) = clM (X). We will use this observation at the end of the chapter. Condition 2.1.3) has an a priori unlikely strength: quantifier-free formulas determine the closure; in practice, the language is specifically expanded to guarantee this condition. Remark 2.2. The following requirement is too strong: for any M,N ∈ K with M ⊆ N and X ⊆ M ,