  We obtain algebraic characterizations of relative notions of size in a
discrete semigroup that generalize the usual combinatorial notions of syndetic,
thick, and piecewise syndetic sets. "Filtered" syndetic and piecewise syndetic
sets were defined and applied earlier by Shuungula, Zelenyuk, and Zelenyuk
[24]. Other instances of these relative notions of size have appeared
explicitly (and more often implicitly) in the literature related to the
algebraic structure of the Stone-\v{C}ech compactification. Building on this
prior work, we observe a natural duality and demonstrate how these notions of
size may be composed to characterize previous notions of size (like piecewise
syndetic sets) and serve as a convenient description for new notions of size.
