I'm trying to understand the following statement (found in this paper):
For every strongly connected component S ⊆ E of the digraph (A, E), if |S| = 1 then there is a loop incident to the node in S.
where $A$ is a set of nodes and $E$ is a set of edges of a digraph.
Does $|S|$ mean "number of edges in the SCC", or could it be something else ? The paper doesn't introduce this notation.
If so, then the only SCC with $|S|=1$ has to be a single node with a self referencing arc, like $A\to A$. Right?
I guess one or both the above aren't true, otherwise it seems trivial that any node with a self-referential arc must have a loop, which would be that self-referential arc itself. Or would this not classify as a "loop"?
$|S|$ is the number of nodes. So it is just saying that if the SCC has only one node, then there is a loop at that node.
This is clear from the next couple of lines which mention an SCC $\{k\}$ and then talk about nodes other than $k$. I assume the $S\subseteq E$ is a typo.