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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