Interpreting the action of the endomorphism monoid of the rationals

Abstract

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 $(\mathbb{Q},\le )$.