My colleague (I guess, investigating structure of specific semigroups) is looking for references about binary relations $R\subset X\times X$ such that $R\subset R\circ R$, that is for each $(v,u)\in R$ there exists $w\in X$ such that $(v,w)\in R$ and $(w,v)\in R$.
The issue has an equivalent graph theoretical formulation. A directed (not necessarily finite) graph $G$ has no parallel edges, but can have loops and for each edge $(v,u)$ of $G$ there exists a vertex $w$ of $G$ (which can coincide with $v$ or $u$) such that both $(v,w)$ and $(w,u)$ are edges of $G$.
Thanks.
An order $(S,\le)$ is order dense when
for all $a,b$ in $S$, exists $x$ with $a \le x$, $x \le b$.
Based upon that I would consider a relation $R$
with $R$ subset $R\circ R$ to be a dense relation.