Image equivalence classes

The image endofunctor of the category of sets $Im : Sets \to Sets$ maps any function $f: A \to B$ to $Im(f) : \wp(A) \to \wp(B)$. The image of $Im(f)$ is a power set equal to $\wp(I)$ where I is the image of $f$. The image is therefore just another power set. On the other hand, the equivalence classes of $Im(f)$ are a type of set system which you may not have seen before.

Proposition. the equivalence classes of $Im(f) : \wp(A) \to \wp(B)$ are upper bounded convex sets whose minima are the maximal cliques of a cocluster graph.

The family of maximal representatives of $Im(f)$ forms a partition topology, and so these equivalence classes are equivalent to those of the closure of a partition topology. If we treat a partition as a cluster graph, then clearly the minimal generators of any image are maximal independent sets because equivalence-equal elements are redundant.

By complementation, they are maximal cliques of a cocluster graph. Which means that the minimal sets are a maximal clique family. This is interesting because not all sperner families are maximal clique families. The upper boundedness of image equivalence classes is a special case of the fact that $Im(f)$ is the lower adjoint of a monotone Galois connection. The partition topology of maximal representatives is the image of the inverse image which is an upper adjoint.

Example 1.

Example 2.

Example 3.