4 1. COMBINATORIAL GEOMETRIES AND INFINITARY LOGICS Exercise 1.1.3. If ๐บ is a homogeneous geometry, ๐‘‹, ๐‘Œ are maximally inde- pendent subsets of ๐บ, there is an automorphism of ๐บ taking ๐‘‹ to ๐‘Œ . The most natural examples of homogeneous geometries are vector spaces and algebraically closed fields with their usual notions of closure. The crucial properties of these examples are summarised in the following definition. Definition 1.1.4. (1) The structure ๐‘€ is strongly minimal if every first order definable subset of any elementary extension ๐‘€โ€ฒ of ๐‘€ is finite or cofinite. (2) The theory ๐‘‡ is strongly minimal if it is the theory of a strongly minimal structure. (3) ๐‘Ž โˆˆ acl(๐‘‹) if there is a first order formula with parameters from ๐‘‹ and with finitely many solutions that is satisfied by ๐‘Ž. Definition 1.1.5. Let ๐‘‹,๐‘Œ be subsets of a structure ๐‘€. An elementary iso- morphism from ๐‘‹ to ๐‘Œ is 1-1 map from ๐‘‹ onto ๐‘Œ such that for every first order formula ๐œ™(v), ๐‘€ โˆฃ ๐œ™(x) if and only if ๐‘€ โˆฃ ๐œ™(๐‘“x). Note that if ๐‘€ is the structure (๐œ”,๐‘†) of the natural numbers and the successor function, then (๐‘€,๐‘†) is isomorphic to (๐œ” โˆ’ {0},๐‘†). But this isomorphism is not elementary. The next exercise illustrates a crucial point. The argument depends heavily on the exact notion of algebraic closure the property is not shared by all combinato- rial geometries. In the quasiminimal case, discussed in the next chapter, excellence can be seen as the missing ingredient to prove this extension property. The added generality of Shelahโ€™s notion of excellence is to expand the context beyond a com- binatorial geometry to a more general dependence relation. Exercise 1.1.6. Let ๐‘‹,๐‘Œ be subsets of a structure ๐‘€. If ๐‘“ takes ๐‘‹ to ๐‘Œ is an elementary isomorphism, ๐‘“ extends to an elementary isomorphism from acl(๐‘‹) to acl(๐‘Œ ). (Hint: each element of acl(๐‘‹) has a minimal description.) The content of Exercise 1.1.6 is given without proof on its first appearance [BL71] a full proof is given in [Mar02]. The next exercise recalls the use of combinatorial geometries to study basic examples of categoricity in the first order context. Exercise 1.1.7. Show a complete theory ๐‘‡ is strongly minimal if and only if it has infinite models and (1) algebraic closure induces a pregeometry on models of ๐‘‡ (2) any bijection between ๐‘Ž๐‘๐‘™-bases for models of ๐‘‡ extends to an isomorphism of the models. Exercise 1.1.8. A strongly minimal theory is categorical in any uncountable cardinality. 1.2. InfinitaryLogic Infinitary logics ๐ฟ๐œ…,๐œ† arise by allowing infinitary Boolean operations (bounded by ๐œ…) and by allowing quantification over sequences of variables length ๐œ†. Various results concerning completeness, compactness and other properties of these logics were established during the late 1960โ€™s and early 1970โ€™s. See for example [Bar68,
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