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*?