In a stable theory , a type over a set is said to be stationary if has a unique non-forking extension to every , or equivalently, to . In other words, is stationary if for every , if and and , then .

An equivalent condition is that is stationary if and only if there is some global -definable type such that .

Types over models are always stationary. More generally, if C is a set such that C = acleq(C) (or dcleq(C) = acleq(C)), then types over are stationary. That is, strong types are stationary.

In the setting of ACF, types correspond to irreducible varieties. Stationary types are exactly the types corresponding to geometrically irreducible varieties. Over algebraically closed sets, irreducible varieties are already geometrically irreducible.

Two stationary types are said to be parallel if they have the same nonforking global extension. Parallelism is an equivalence relation, the equivalence relation on stationary types generated by the relation " is a nonforking extension of ."

In the setting of ACF, two types are parallel if they have the same associated variety.

Canonical bases

If is a stationary type, the canonical basis of is the definable closure of the set of codes for the formulas occurring in the definition of the unique global non-forking extension of . That is, if denotes the monster, and denotes the unique non-forking extension of to , then for each formula , there is a -definable set such that if and only if . The canonical base is , where denotes the code for the definable set . The canonical base is always a small set, and depends only on , i.e., on the parallelism class of . Moreover, is uniquely determined by the following fact: an automorphism fixes pointwise if and only if fixes the parallelism class of . The canonical base is the smallest (definably closed) set over which is defined. It can also be characterized as the unique smallest definably closed set such that some type over is stationary and parallel to .

If is stationary, is always contained in , essentially because any automorphism which fixes pointwise must send to itself, and must therefore also fix the unique nonforking extension to . One always has .

Since strong types are always stationary, makes sense without any assumptions. This set is always contained in , and it is also in .

If is superstable, then the canonical base of any stationary type is in the definable closure of a finite set. Moreover, this property characterizes superstability. (Or does it?) ::: ::: :::