# Is Hyper-graph Isomorphism preserve the size of edges or Rank of Hyper-graph?

Informally, hypergraph is a generalization of a graph in which an edge can join any number of vertices.

A hypergraph $$G=(V, E)$$ is a two tuple, where $$V$$ is the set of vertices and $$E$$ is a set contain subsets of the vertex set of $$V$$. An example of hyper-graph is given below and for example edge $$e_3$$ is a subset contain $$v_3,v_5,v_6$$ and similarly for other edges. Isomorphism : Two hyper graphs $$G(V,E)$$ and $$H(V,E')$$ are isomorphic if there is a permutation $$g$$ on $$V$$ such that, $$\forall$$ $$e \in E$$, $$e\in E \iff g(e) \in E'$$

Question : Is the hypergraph isomorphism preserve the size of edge (edge is a subset of vertex set here) i.e. an edge $$e$$ that contain say $$l$$ vertices will be mapped to edge $$g(e)$$ whose size is also $$l$$ or it is not required.

Reference: https://en.wikipedia.org/wiki/Hypergraph

Whether it's expressly stated or not, it must be the case that hypergraph automorphisms send an edge containing $\ell$ points to another edge containing $\ell$ points.
This follows from the fact that $g: V \to V$ is a permutation (i.e., bijection). The very definition of two sets having the same size is that there is a bijective map from one to the other. Here, $g$ restricted to any subset of $V$ (say, $E$) is a bijective map from $E$ to $g(E)$, hence $E$ and $g(E)$ contain the same number of vertices.