CHAPTER 1

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

Shelahโs 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

Waerdenโs 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

geometry.

One strategy for the analysis of โต1-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๏ฌcient condition for a set ๐ท to have a nice dimension theory is that ๐ท be strongly

minimal or equivalently that algebraic closure on ๐ท 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 ๐ฟ๐1,๐.

Definition 1.1.1. A pregeometry is a set ๐บ together with a dependence relation

๐๐ : ๐ซ(๐บ) โ ๐ซ(๐บ)

satisfying the following axioms.

A1. ๐๐(๐) =

โช

{๐๐(๐โฒ): ๐โฒ โ๐๐๐ ๐}

A2. ๐ โ ๐๐(๐)

A3. ๐๐(๐๐(๐)) = ๐๐(๐)

A4. (Exchange) If ๐ โ ๐๐(๐๐) and ๐ โ โ ๐๐(๐), then ๐ โ ๐๐(๐๐).

If points are closed the structure is called a geometry.

Definition 1.1.2. A geometry is homogeneous if for any closed ๐ โ ๐บ and

๐,๐ โ ๐บ โ ๐ there is a permutation of ๐บ which preserves the closure relation (i.e.

an automorphism of the geometry), fixes ๐ pointwise, and takes ๐ to ๐.

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http://dx.doi.org/10.1090/ulect/050/01