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 et al. (Semigroup Forum 79:531–539, 2009). 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–Čech 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.

我们在一个离散的半群中获得了大小的相对概念的代数特征，这些概念概括了通常的合集、厚和分段合集的组合概念。Shuungula 等人较早地定义并应用了“过滤”合成和分段合成集。（半群论坛 79：531–539，2009 年）。在与 Stone-Čech 紧化的代数结构相关的文献中，这些相对大小概念的其他实例已经明确（而且通常是隐含地）出现。在此先前工作的基础上，我们观察到自然二元性，并演示如何组合这些大小概念来表征先前的大小概念（如分段合成集），并作为对新大小概念的方便描述。