Combinatorial Geometries and Infinitary Logics
In this chapter we introduce two of the key concepts that are used throughout
the text. In the first section we define the notion of a combinatorial geometry and
describe its connection with categoricity in the first order case. The second section
establishes the basic notations for infinitary logics.
1.1. Combinatorial Geometries
Shelahโ€™s notion of forking provides a notion of independence which greatly
generalizes the notion of Van der Waerden [VdW49] by allowing one to study
structures with a family of dimensions. (See Chapter II of [Bal88].) But Van der
Waerdenโ€™s notion is the natural context for studying a single dimension; in another
guise such a relation is called a combinatorial geometry. Most of this monograph is
devoted to structures which are determined by a single dimension; in the simplest
case, which we study first, the entire universe is the domain of a single combinatorial
One strategy for the analysis of โ„ต1-categorical first order theories in a countable
vocabulary proceeds in two steps: a) find a definable set which admits a nice
dimension theory b) show this set determines the model up to isomorphism. The
su๏ฌƒcient condition for a set ๐ท to have a nice dimension theory is that ๐ท be strongly
minimal or equivalently that algebraic closure on ๐ท forms a pregeometry in the
sense described below. It is natural to attempt to generalize this approach to study
categoricity in non-elementary contexts. In this chapter we review the first order
case to set the stage. For more detail on this chapter consult e.g. [Bue91]. In
the next few chapters, we study the infinitary analog to the first order concept
of strong minimality โ€“ quasiminimality. In Part 4 we show that the strategy of
reducing categoricity to a quasiminimal excellent subset can be carried out for
sentences of ๐ฟ๐œ”1,๐œ”.
Definition 1.1.1. A pregeometry is a set ๐บ together with a dependence relation
๐‘๐‘™ : ๐’ซ(๐บ) โ†’ ๐’ซ(๐บ)
satisfying the following axioms.
A1. ๐‘๐‘™(๐‘‹) =
{๐‘๐‘™(๐‘‹โ€ฒ): ๐‘‹โ€ฒ โŠ†๐‘“๐‘–๐‘› ๐‘‹}
A2. ๐‘‹ โŠ† ๐‘๐‘™(๐‘‹)
A3. ๐‘๐‘™(๐‘๐‘™(๐‘‹)) = ๐‘๐‘™(๐‘‹)
A4. (Exchange) If ๐‘Ž โˆˆ ๐‘๐‘™(๐‘‹๐‘) and ๐‘Ž โˆ• โˆˆ ๐‘๐‘™(๐‘‹), then ๐‘ โˆˆ ๐‘๐‘™(๐‘‹๐‘Ž).
If points are closed the structure is called a geometry.
Definition 1.1.2. A geometry is homogeneous if for any closed ๐‘‹ โŠ† ๐บ and
๐‘Ž,๐‘ โˆˆ ๐บ โˆ’ ๐‘‹ there is a permutation of ๐บ which preserves the closure relation (i.e.
an automorphism of the geometry), fixes ๐‘‹ pointwise, and takes ๐‘Ž to ๐‘.
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