Translate-containment quasiorder

Definition

The translate-containment quasiorder is a quasiorder defined on the elements of a group as follows: for subsets $S$ and $T$ of a group $G$, $S\leq T$ if and only if there exist elements $x,y\in G$ such that $xSy\subseteq T$.

Roughly, the idea is that left and right translation by elements of the group should not change the size of a subset.

Note that translate-containment quasiorder is finer than left translate-containment quasiorder, right translate-containment quasiorder, and conjugate-containment quasiorder.

It defines a partial order on the set of subsets of a group upto translation equivalence.