  A hypergraph can be obtained from a simplicial complex by deleting some
non-maximal simplices. By [11], a hypergraph gives an associated simplicial
complex. By [4], the embedded homology of a hypergraph is the homology of the
infimum chain complex, or equivalently, the homology of the supremum chain
complex. In this paper, we generalize the discrete Morse theory for simplicial
complexes by R. Forman [5-7] and give a discrete Morse theory for hypergraphs.
We use the critical simplices of the associated simplicial complex to construct
a sub-chain complex of the infimum chain complex and a sub-chain complex of the
supremum chain complex, then prove that the embedded homology of a hypergraph
is isomorphic to the homology of the constructed chain complexes. Moreover, we
define discrete Morse functions on hypergraphs and compute the embedded
homology in terms of the critical hyperedges. As by-products, we derive some
Morse inequalities and collapse results for hypergraphs.
