A stationary type {\displaystyle p(x)} in a stable theory is said to be regular if it is orthogonal to all its forking extensions. If {\displaystyle p=\operatorname {tp} (a/C)}, this means that whenever {\displaystyle D\supseteq C} and {\displaystyle a\not \downarrow _{C}D} and {\displaystyle \operatorname {tp} (a/D)} is stationary, then {\displaystyle \operatorname {tp} (a/D)} and {\displaystyle \operatorname {tp} (a/C)} are orthogonal. Regularity is a parallelism invariant: if {\displaystyle p'} is a forking extension of {\displaystyle p}, then {\displaystyle p'} is regular if and only if {\displaystyle p} is regular.

Types of U-rank 1 are regular, because their non-forking types are algebraic, hence orthogonal to every type. More generally, any type whose U-rank is a power {\displaystyle \omega } is regular.

Proof. This comes from the fact that if {\displaystyle p} and {\displaystyle q} are types of rank {\displaystyle \omega ^{\alpha }} and rank {\displaystyle \beta <\omega ^{\alpha }}, then {\displaystyle p} and {\displaystyle q} are orthogonal. Indeed, if {\displaystyle U(a/C)=\omega ^{\alpha }} and {\displaystyle U(b/C)<\omega ^{\alpha }}, then by the Lascar inequalities, {\displaystyle U(a/bC)\oplus U(b/C)\geq U(ab/C)\geq U(a/C)=\omega ^{\alpha }}. Since {\displaystyle U(b/C)} is less than {\displaystyle \omega ^{\alpha }}, this forces {\displaystyle U(a/bC)\geq \omega ^{\alpha }}. Then {\displaystyle U(a/bC)\geq U(a/C)}, so {\displaystyle a\downarrow _{C}b}. QED

In DCF, the generic type of the home sort (which has rank {\displaystyle \omega }) and the generic type of the constant field (which has rank 1) are two examples of regular types.

If {\displaystyle p} is a regular type over a set {\displaystyle C}, there is a natural pregeometry structure on the set of realizations of {\displaystyle p}. A set {\displaystyle \{a_{1},\ldots ,a_{n}\}} is independent if and only if it is independent in the sense of stability theory, i.e., {\displaystyle a_{1}\downarrow _{C}a_{2}}, {\displaystyle a_{1}a_{2}\downarrow _{C}a_{3}}, and so on. If {\displaystyle S} is a set of realizations of {\displaystyle p}, a tuple {\displaystyle a} is in the closure of {\displaystyle S} if and only if {\displaystyle a\not \downarrow _{C}S}.

Regular types have weight 1.

Regular types are prevalent in superstable theories, in some sense… ::: ::: :::