Relatively thin and sparse subsets of groups

Abstract

Let $G$ be a group with identity $e$ and let $\mathcal{I}$ be a left-invariant ideal in the Boolean algebra $\mathcal{P}_G$ of all subsets of $G$. A subset $A$ of $G$ is called $\mathcal{I}$-thin if $gA \bigcap A \in \mathcal{I}$ for every $g \in G \ \{e\}$. A subset $A$ of $G$ is called $\mathcal{I}$-sparse if, for every infinite subset $S$ of $G$, there exists a finite subset $F \subset S$ such that $\bigcap_{g \in F}gA \in F$. An ideal $\mathcal{I}$ is said to be thin-complete (sparse-complete) if every $\mathcal{I}$-thin ($\mathcal{I}$-sparse) subset of $G$ belongs to $\mathcal{I}$. We define and describe the thin-completion and the sparse-completion of an ideal in $\mathcal{P}_G$.