A one-sorted infinite structure is said to be minimal if every definable subset is finite or cofinite. Here, "definable" means "definable with parameters." If this remains true in elementary extensions of , is called strongly minimal. A theory is strongly minimal if every model is minimal, or equivalently, every model is strongly minimal.

More generally, an infinite definable set in a structure is said to be minimal if for every definable set , either or is finite. If this remains true in elementary extensions, is said to be a strongly minimal set. In terms of Morley rank, is strongly minimal exactly when has Morley rank and degree 1.

## Example

The canonical examples of strongly minimal theories:

• The theory of an infinite set with no structure.
• The theory of infinite -vector spaces, for a field (or skew field).
• ACF

In each case, strong minimality follows from quantifier elimination results. For example, in ACF, any definable subset of the home sort must be a constructible subset of the affine line, hence a boolean combination of Zariski closed subsets of the affine line. But Zariski closed subsets of the affine line are either finite or everything.

The simplest example of a structure which is minimal, but not strongly minimal, is .

## Pregeometries from Minimal Sets

Lemma. If is a minimal structure, then the closure operator acl(-) on satisfies the exchange principle: if and are singletons and , and if , then .

Proof. By definition of , there must be some -formula such that holds and is finite, of size . Let be the -formula , the formula which says that holds and . Then holds, and has size at most for each .

If is finite, then , and we are done. Otherwise, is cofinite. Let . Let Then is -definable, and contains . If is finite, then , contradicting the hypotheses. So is infinite. Let be distinct elements of . Then has size for each . So has size at most , and in particular, is finite. This ensures that is non-empty. If , however, then , contradicting the fact that has size at most for every . QED

Since is always finitary closure operator, it follows that on minimal structures (and more generally, minimal sets), is the closure operator of some pre-geometry.

For ACF, this recovers the algebraic independence pregeometry. For the theory of -vector spaces, this recovers the pregeometry of linear independence in an infinite dimensional -vector space. For the theory of equality, this recovers the trivial pregeometry in which every set is closed.

The (false) Zilber Trichotomy Conjecture said (among other things) that these were essentially the only possible geometries for a strongly minimal set. Namely, if was a strongly minimal set, then the associated geometry had to be either

• Trivial, so every set is its own closure.
• The geometry associated to linear independence, i.e., the geometry associated to infinite dimensional projective space over a skew field .
• The geometry associated to algebraic independence.

Hrushovski found counterexamples to the Zilber Trichotomy Conjecture. However, there are a number of settings in which the Zilber trichotomy conjecture is known to hold. For example, it holds of the strongly minimal sets occurring in DCF and CCM. Moreover, it is known to hold in a totally categorical setting. In fact, if is -categorical, then the associated geometry must either be trivial, or must be the geometry associated to some finite field.

## Dimension theory in strongly minimal sets

Assume is strongly minimal. If is a set of parameters, then the operation yields a pregeometry on . A finite set is independent (over ) if for each . The rank of a finite tuple over can be defined to be the size of a maximal independent subset of . The rank of a -definable set is defined to be the maximum of for . It turns out that does not depend on the choice of defining parameters .

In the case of ACF, is the transcendence degree of over , and is the dimension of .

This notion of rank agrees with Lascar rank and Morley rank, and has many intuitive properties that one expects of dimension. Some of these are listed below:

• If , then .
• If is a definable surjection, then . If the fibers of this surjection are finite, then .
• if and only if is finite.
• .
• .
• Rank is definable in families: if is a formula, then the set of such that is definable for each .
• If is a definable surjection and every fiber has rank , then , unless is empty.
• .
• if and only if .
• If , then there is some -definable set containing with dimension .
• The rank function can also be extended to interpretable sets (to ) in a way which maintains all the above properties.

## Strong minimality and uncountable categoricity

Strongly minimal (complete, countable) theories are always uncountably categorical. This happens for essentially the same reason that ACF is uncountably categorical. If is a strongly minimal theory, then it can be shown that a model of is determined up to isomorphism by the cardinality of a basis of the associated pregeometry.

In more detail, here is a proof:

Proof. Let be a monster model of . If is any subset of , let be the type over asserting that is not in any finite -definable sets, i.e., that is not algebraic over . Then is consistent (because in an elementary extension of , all the new elements will not be algebraic over ). Also, is a complete type over . Indeed, if is a -definable set, then either is finite or cofinite. In the first case, is in , and in the second case, is in .

If , then , because if an element in an elementary extension of were not algebraic over , it would certainly not be algebraic over the smaller set . Consequently, the union of the ’s, as ranges over all sets, is , and for each . Let . The type is a global type, invariant under . From generalities on global invariant types, it follows that if and are two sequences such that for each , then and have the same type over the empty set. Equivalently, the map sending is a partial elementary map.

But the condition that for each just means that for each , or equivalently, that the set is independent.

So if and are two finite independent sets and is a bijection from to , then is a partial elementary map. More generally, if and are two independent sets of arbitrary size, and is any bijection from to , then is a partial elementary map. (It suffices to check that is a partial elementary map on finite subsets of .)

Now suppose and are two models of the same uncountable cardinality. We may assume that and sit inside . Let be a maximal independent subset of , and let be a maximal independent subset of . Then and . Since is countable, and similarly for . Since is uncountable, , and similarly, . Finally, since and have the same cardinality, . Let be a bijection from to . Then is a partial elementary map, so it can be extended to an automorphism of . Then So gives an isomorphism between and . QED

The converse is far from true. However, some variant of it is still true:

Fact. Suppose is a countable theory which is -categorical for some uncountable cardinal . Then any model of contains a strongly minimal set , with constructible over . The model is determined up to isomorphism by the size of a pregeometry basis in . Moreover, if and are two strongly minimal sets definable in , the underlying geometries of and are the same.

This fact is a key step in the proof of Morley's Theorem and the Baldwin-Lachlan theorem. ::: ::: :::