Operads of hypergraphs (Q353937)

For all \(n \geq 1\), and any monoid \(G\), the authors construct two operad structures on the set of cubic \(n\)-multidimensional matrices. They both contain a suboperad \(HG_n\) obtained from a symmetry condition on cubic multidimensional matrices. If \(G=\{0,1\}\), with the monoid structure defined by \(1+1=1\), these symmetric multidimensional matrices are the incidence matrices of hypergraphs such that any edge is adjacent to at most \(n\) vertices, and the authors obtain in this way an operad structure on hypergraphs. It is also shown that these operads are \(Epi\)-operads, that is to say the right action of permutations can be extended to an action of surjections.