Abstract
AbstractSome filter relative notions of size, $$\left( \mathscr {F},\mathscr {G}\right) $$
F
,
G
-syndeticity and piecewise $$\mathscr {F} $$
F
-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk (Semigroup Forum 79: 531–539, 2009). These notions are generalizations of the well studied notions of syndeticity and piecewise syndeticity. Since then, there has been an effort to develop the theory around the algebraic structure of the Stone–Čech compactification so that it encompasses these new generalizations. We prove one direction of a characterization of piecewise $$\mathscr {F}$$
F
-syndetic sets. This completes the characterization, as the other direction was proved by Christopherson and Johnson (Semigroup Forum 104: 28–44, 2021).
Publisher
Springer Science and Business Media LLC
Reference11 articles.
1. Baglini, L.L., Patra, S.K., Shaikh, M.M.: Dynamical notions along filters. New York J. Math. 29, 792–817 (2023)
2. Bergelson, V., Hindman, N., McCutcheon, R.: Notions of size and combinatorial properites of quotient sets in semigroups. Topol. Proc. 23, 23–60 (1998)
3. Christopherson, C., Johnson, J.H.: Algebraic characterizations of some relative notions of size. Semigroup Forum 104, 28–44 (2021)
4. Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Grundlehren der mathematischen Wissenschaften, vol. 211. Springer, Berlin (1974)
5. Ellis, R.: Distal transformation groups. Pac. J. Math. 8(3), 401–405 (1958)