Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters

We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-Čech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-Čch compactification, our particular use of derived sets adapt and generalize methods recently used by Griffin arXiv:2311.09436 to characterize relative piecewise syndetic sets. As an application, we define an algebraically interesting subset of the Stone-Čech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal.