Algebraic characterizations of some relative notions of size

TL;DR

This paper provides algebraic characterizations of relative size notions in discrete semigroups, extending classical concepts like syndetic and thick sets, and introduces a duality framework for these notions.

Contribution

It introduces a unified algebraic framework for relative size notions in semigroups, generalizing classical concepts and connecting them through duality and composition.

Findings

Algebraic characterizations of relative size notions

Duality between different size notions

Framework for composing size notions to recover classical concepts

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, 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.