The functions square and abs are equivalence relation isomorphic to one another because they both partitioned the set of all real numbers into positive and negative segments. If there is a one to one correspondence between all equivalence classes induced by an equivalence relation then that equivalence relation also induces a partition of members.
The category of equivalence relations is partially ordered by a coarseness relation that determines rather or not an equivalence relation is coarser, finer, or equal to another equivalence relation.
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