Lascar Rank in Expansions

RESUMO

We prove that the SU-rank of a supersimple theory is no greater than that of any of its supersimple expansions. A proof for finite rank theories already exists in the literature, but non-continuity presents a problem at limit ranks. We prove the non-decreasing of SU-rank in expansions type-by-type by reducing to finding, in the base theory, certain trees of forking extensions in which the formulas and dividing numbers which witness dividing have a sufficiently uniform pattern. In the case of countable ranks, we can give an especially nice description of this uniformity: for a countable ordinal $\alpha$, any type of SU-rank at least $\alpha$ in a supersimple theory has a chain of forking extensions in the reverse order type of $\alpha$.