Hi there!
Let $X$ be a left $G$-set, and $\Delta=${$x_1,\ldots,x_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx_1,\ldots,Gx_n$} is the orbit space $X/G$.
Let us call $H_i$ the stabilizer of $x_i$, for all $i=1,2,\ldots, n$. Then the set $G/{H_i}$ of left $H_i$-cosets is the orbit $Gx_i$; according, the union $$T(H_1,\ldots, H_n):=\bigcup_{i=1}^nG/{H_i}$$ is another way to write down $X$ as the union of the $G$-orbits.
Somebody calls $T(H_1,\ldots, H_n)$ a coset geometry of rank $n$, or a Tits-Buekenhout geometry; interestingly enough, $T(H_1,\ldots, H_n)$ is a natural left $G$-set, and its identification with $X$ is a $G$-set morphism.
This line of thoughts looks so obvious that I cannot spot any mistake, so I'm asking:
Is it true that any $G$-set with $n$ orbits is a rank-$n$ coset geometry (in the sense I've just explained)?
If yes, who firstly realized that?