I. Protasov, S. Slobodianiuk
Ultracompanions of subsets of a group

Comment.Math.Univ.Carolin. 55,2 (2014) 257-265.

Abstract:Let $G$ be a group, $\beta G$ be the Stone-\v Cech compactification of $G$ endowed with the structure of a right topological semigroup and $G^*=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^*$, we define the $p$-companion $\Delta _p(A)= A^*\cap Gp$ of $A$, and characterize the subsets with finite and discrete ultracompanions.

Keywords: Stone-\v Cech compactification; ultracompanion; sparse and discrete subsets of a group
AMS Subject Classification: 54D35 22A15 20F69