Dual of a Hypergraph

Question

I would like to know for a hypergraph $H=(V,\mathcal{E})$ if $H^*$ represents its dual then is $(H^*)^*= H$??

https://en.wikipedia.org/wiki/Hypergraph and Claude berge's books tells

$(H^*)^*= H$ but

https://books.google.co.in/books?id=Rwm7BQAAQBAJ&pg=PA241&lpg=PA241&dq=Dual+of+a+hypergraph&source=bl&ots=fGIs6ysZZi&sig=yp-Pcv0ON-tBc0XdiX8SDGD1nac&hl=en&sa=X&ved=0ahUKEwjH8O_vkOjLAhUPBI4KHdUqBP84ChDoAQg8MAU#v=onepage&q=Dual%20of%20a%20hypergraph&f=false

shows that $(H^*)^*\ne H$ . Which one is correct?? Is repeated edges allowed in dual of a hypergraph???

Example

I am using the definition of hypergraph in Berge.

Consider the hypergraph $H=(V,\mathcal{E})$, where $V=\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\}$ $\mathcal{E}= \{e_1,e_2,e_3,e_4\}=\big\{\{v_1,v_2,v_3\},\{v_2,v_3,v_4,v_7\},\{v_5,v_7,v_8\},\{v_6\}\big\}$.

$H^*=(V^*,\mathcal{E}^*)$ where $V^*=\{e_1,e_2,e_3,e_4\}$ and edge set $V_1,V_2,V_3,V_4,V_5,V_6,V_7,V_8$ where $V_1=\{ e_1\}$, $ V_2=\{ e_1, e_2 \},V_3=\{ e_1,e_2\},V_4=\{ e_2\},V_5=\{ e_4\},V_6=\{ e_3\},V_7=\{ e_2,e_4\},$ $V_8=\{ e_4\}.$

Answer 1

The issue is that hypergraph terminology is far less standardized than graph terminology, so the two links do not use the same definition.

In particular, graphs usually allow only one edge, and if you have multiple edges it is then a multigraph. Note that you have to change the underlying mathematical structure to handle multiple edges (e.g. for a graph $E \subseteq V \times V$ while for a multigraph $E: V\times V \to \mathbb{N}$, the edge relation is a function to integers).

You have the same distinction for hypergraphs, you can allow multiple edges (Wikipedia article) or not (your second link). In the first case the set of hyper-edges is a subset of the powerset of $V$, and similarly in the second case a function to $\mathbb{N}$.

On your own, its fine to say "by hypergraph I mean multiple edges allowed/not allowed", if you have to deal with both you can say hypergraph and multi-hypergraph.