Translate-containment quasiorder

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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.