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Saturday, March 27, 2021

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.

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