Abstract
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.
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Acknowledgements
We gratefully acknowledge and thank Florian Richter for several important discussions along with valuable feedback and suggestion on earlier drafts of this article. We thank the referee for a careful reading and several suggestions that improved the exposition. We thank Jessica Christian for feedback on drafts of the introduction. We thank Baglini, Blass, and Di Nasso for helpful conversations on their work related to finite embeddability, and we also thank Anush Tserunyan and Andy Zucker for helpful discussions and interest. Finally, we also thank Vitaly Bergelson and Neil Hindman for helpful correspondence.
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Communicated by Jimmie D. Lawson.
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Christopherson, C., Johnson, J.H. Algebraic characterizations of some relative notions of size. Semigroup Forum 104, 28–44 (2022). https://doi.org/10.1007/s00233-021-10215-9
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DOI: https://doi.org/10.1007/s00233-021-10215-9