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 $$G$$, $$S \le T$$ 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.