all the basic products for graphs have been extended to hypergraphs[1].

is there a concept of a product of hypergraphs with the same vertex set? has this been studied?

normally the hypergraph product is between two hypergraphs $H_1 = (V_1, \mathscr{E}_1), H_2 = (V_2, \mathscr{E}_2)$. am asking about the case $V_1=V_2$. (have an idea for a defn of this but am looking for any other preexisting cases first. esp interested in factoring)

[1] Hypergraph products by Hellmuth

[2] mathoverflow, What are the Applications of Hypergraphs