Relatively thin and sparse subsets of groups

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 $gA \cap A \in \mathcal{I}$ for every g ∈ G\{e}. A subset A of G is called $\mathcal{I}$-sparse if, for T every infinite subset S of G, there exists a finite subset F ⊂ S such that $\bigcap\nolimits_{g \in F} {gA \in \mathcal{F}}$. 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 ${\mathcal{P}_G}$.