  In this paper, we define the action of $M$, the monoid of embeddings of $({\mathbb Q}, \le)$, on $\mathbb Q$, in the monoid $(M, \circ)$. That is, we show that $\mathbb Q$ itself can be interpreted in $(M, \circ)$, and in addition, so can the action of $M$ on $\mathbb Q$. This is extended to the monoid $E$ of all endomorphisms of $({\mathbb Q}, \le)$. 
Interpreting the action of the endomorphism monoid of the rationals