Representations of the strong endomorphism monoid of finite $n$-uniform hypergraphs

In this paper, two exact representations of the strong endomorphism monoid of an arbitrary finite $n$-uniform hypergraph are described. It is proved that the strong endomorphism monoid of a finite $n$-uniform hypergraph is regular. We find all nonnegative integers $n$ such that $n$-uniform hypergraphs are determined by their strong endomorphisms.