# compositon of relations and monoid

$X=\{1,2,3,4\},$ $R,S \subset X \times X$ , $R,S$ are relations, we define $R \bullet S \text{ to be the set:} (x,y) \in X \times X , \text{s.t there exsits one and only one}$ $z \in X : (x,z) \in S,(z,y) \in R$
Does this turn the set $X$ into a monoid?
I mean is $(R \bullet S) \bullet T =R \bullet (S \bullet T )$?