Forking and invariant types in monadic NIP theories

Abstract

We obtain a strong decomposition theorem for invariant global types in a monadically NIP theory. From this, we prove that if an $n$-type $tp(\bar{a}/MC)$ does not fork over $M$ then $\bar{a}=(\bar{f},\bar{d})$ where $tp(\bar{d}/MC)$ is $M$-definable and $tp(\bar{f}/MC\bar{d})$ is finitely satisfied in $M$. Over arbitrary base sets $B$, we prove that an $n$-type $tp(\bar{a}/BC)$ does not fork over $B$ if and only if $tp(a_i/BC)$ does not fork over $B$ for each singleton $a_i\in\bar{a}$. We show that monadically NIP theories satisfy density of definable types among non-forking extensions.