Given any relation $R$, that is, any set of ordered pairs, we can associate a unique digraph $D$ to our relation $R$ by setting $D=(\text{fld}(R),R)$ where $\text{fld}(R)=\text{dom}(R)\cup\text{rng}(R)=\bigcup\_{(x,y)\in R}\{x,y\}$. Therefore, in this way the study of binary relations can be put on an equal footing with the study of directed graphs. With that in mind, "independent vertex sets" can now be seen as generalizations of various other notions, for example an independent set in a partial order relation would just be an anti-chain, while an independent set in any equivalence relation would just be a section and if it was a maximal (ordered by set inclusion) independent set, then it would then be a complete system of representatives for said equivalence and so on. But this abstraction only works if one allows the vertices in loops to be elements of independent sets, so for any $X\subseteq \text{fld}(R)$ we'd have to write:
$$X\text{ is an independent set in }R\iff R\cap (X\times X)\subseteq \text{id}\_X=\{(x,x)\in X\times X:x\in X\}\\ \iff \forall u,v\in X\left[u\neq v\implies (u,v),(v,u)\not\in R\right]$$


Yet I have also seen another definition for independent vertex sets of a digraph or, equivalently, an associated binary relation $R$. Namely for any set $X\subseteq\text{fld}(R)$ I've seen some people write that:


$$X\text{ is an independent set in }R\iff R\cap (X\times X)=\emptyset\\\iff \forall u,v\in X\left[(u,v),(v,u)\not\in R\right]$$


Which would not allow any vertex $v\in \text{fld}(R)$ with $(v,v)\in R$ to be in an independent set of $R$ and thus the previous definitions of anti-chains, sections etc. could no longer hold since under the second definition partial order relations, equivalence relations, and many other reflexive relations can't have any non-empty independent sets. With that motivation for using the first definition in mind, are there any authors who likewise define independent vertex-sets in directed graphs where vertices contained in loops are permitted? Or do most authors use the second definition I listed?



In the context of directed graphs is it standard notation to allow an element of an independent vertex set to be contained in a loop?
Well, I personally don't think there is much to say here; regarding "are there any authors who define independent vertex-sets in directed graphs where vertices contained in loops are permitted": I know one. E.g., 



> 
> **[BJG2009]** Jørgen Bang-Jensen, Gregory Z. Gutin, *Digraphs: Theory, Algorithms and Applications* Springer Monographs in Mathematics, 2009, ISBN-13: 978-0857290410
> 
> 
> 


on page 21, second paragraph, clearly defines 'independent set' for *directed pseudographs* (this is [BJG2009]'s preferred synonym for what elsewhere is called 'quiver', a notion which allows arbitrary loops and multiple arcs, all with arbitrary multiplicities).


Therefore, *if one disregards the flaw described below*, then 



> 
> [BJG2009] uses your *first* definition.
> 
> 
> 


However, strictly speaking *[BJG2009]'s definition of 'independent set' is broken*, because the notation ' $H\langle Q\rangle$ ' used on page 21, where $H$ is allowed to be an arbitrary directed pseudograph, is defined on page 5, second paragraph below Proposition 1.2.1, only for $H$ being a *digraph* (a notion which forbids loops or parallel arcs). Hence 



> 
> $H\langle Q\rangle$, for $H$ a pseudograph, is *simply not defined* in [BJG2009]. 
> 
> 
> 


This small mistake in [BJG2009] is indicative of how unimportant an issue this is usually considered. There isn't any good answer to your question, I think, since the theory directed graphs is not well-developed enough yet. 


Quite tellingly, 



> 
> one of the few existing monographs on the topic ([BJG2009]) gets it wrong, 
> 
> 
> 


and then, as far as I can see, later on in the book one never ever notices that the definition is broken (there seems to be no place in the book where a notion of 'independent set in a pseudograph' is needed, hence the broken definition never makes itself felt).


