For a fixed set $X$ and a finite collection $E\_1,E\_2,\ldots,E\_k\subseteq X$, define the binary relation *adjacency* as follows: $E\_i,E\_j$ are adjacent
if their intersection is nonempty.
We term the transitive closure of this relation by
*transitive adjacency*
and define the *adjacent union* by
$$
\tilde\cup(E\_1,\ldots,E\_k):=
\begin{cases}
 \bigcup\_{i=1}^k E\_i, & \text{the $(E\_i)$ are transitively adjacent}
 \\
 \emptyset, & \text{else}
. 
\end{cases}
$$


Are there standard terms for *transitive adjacency* and *adjacent union*?



Is there a standard term for this graph/set theoretic concept?
You can think in terms of [intersection graphs](https://en.wikipedia.org/wiki/Intersection_graph). The transitive adjacency tells you when two vertices are adjacent. The adjacency union is then empty if and only if two of $E\_1,\ldots,E\_k$ are in distinct connected components.


