In page 24 of the paper A survey on hypergraph product by Marc Hellmuth the lexicographic product of two hypergraph is defined as
Let $H_1 = (V_1, E_1)$ and $H_2 = (V_2, E_2)$ be two hypergraphs. The lexicographic product $ H = H_1 \circ H_2$ has vertex set $V (H) = V_1 \times V_2$ and edge set $E(H) = \{e ⊆ V (H) : p_1(e) ∈ E_1, |p_1(e)| = |e|\} ∪ \{\{x\} × e_2 | x ∈ V_1, e_2 ∈ E_2\}$.
In page 277 of The automorphism group of a product of hypergraphs by G&A HAHN Wreath product of two hypergraphs is defines as
Let $A$ and $B$ be two sets. For each $e\in P(A\times B)$ we denote the set of first coordinates of elements of $e$ by $e^1$ and the set of second coordinates of elements of $e$ by $e^2$.
Let $H_1=(V_1,\mathcal{E}_1)$ and $H_2=(V_2,\mathcal{E}_2)$ be two hypergraphs with disjoint vertex sets. The wreath product $H=(V,\mathcal{E})=H_1[H_2]$ of $H_1$ and $H_2$ is a hypergraph with vertex set $V=V_1\times V_2$ and a subset $e$ of $V\times V$ belongs to $\mathcal{E}$ if at least one of the following conditions holds.
(1) $e^1\in \mathcal{E}_1$ and $|e\cap (\{u\}\times V_2)|\le 1$ for each $u\in V_1$, or
(2) $|e^1|=1$ and $e^2\in \mathcal{E}_2.$
Is these two definition same??