Isomorphism and Equality
A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.
A hypergraph is isomorphic to a hypergraph, written as if there exists a bijection
and a permutation of such that
The bijection is then called the isomorphism of the graphs. Note that
- if and only if .
When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. One says that is strongly isomorphic to if the permutation is the identity. One then writes . Note that all strongly isomorphic graphs are isomorphic, but not vice-versa.
When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality. One says that is equivalent to, and writes if the isomorphism has
and
Note that
- if and only if
If, in addition, the permutation is the identity, one says that equals, and writes . Note that, with this definition of equality, graphs are self-dual:
A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices. The set of automorphisms of a hypergraph H (= (X, E)) is a group under composition, called the automorphism group of the hypergraph and written Aut(H).
Read more about this topic: Hypergraph
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—Barbara Ehrenreich (20th century)