Generalized Quantifiers: Scope Dominance and Branching

Let $P$ and $Q$ be proper upward monotone unary generalized quantifiers on a fixed domain $D$. We study the validity, over all structures $M$ with domain $D$, of \[ M \models Px\,Qy\,R(x,y) \rightarrow Qy\,Px\,R(x,y), \] called scope dominance. The iteration of two quantifiers lies between their bra