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

Algebraic characterizations of some relative notions of size