compositon of relations and monoid

$\displaystyle X=\{1,2,3,4\}, $ $\displaystyle R,S \subset X \times X $ ,$\displaystyle R,S$ are relations, we define $\displaystyle R \bullet S \text{ to be the set:} (x,y) \in X \times X , \text{s.t there exsits one and only one}$ $\displaystyle z \in X : (x,z) \in S,(z,y) \in R $

Does this turn the set $\displaystyle X $ into a monoid?

I mean is $\displaystyle (R \bullet S) \bullet T =R \bullet (S \bullet T ) $?

I know the general compostion of relations is associative, but not sure about this one. any one can help?