Interpreting the action of the endomorphism monoid of the rationals

TL;DR

This paper explores how the monoid of embeddings and endomorphisms of the rational numbers can be interpreted within their own algebraic structures, revealing deep connections between order-preserving maps and their algebraic actions.

Contribution

It demonstrates that the rational numbers and their actions can be interpreted within the monoids of embeddings and endomorphisms, extending previous understanding of their algebraic structure.

Findings

Q can be interpreted in (M, ◦)

Action of M on Q can be interpreted in (M, ◦)

Extended interpretation to the monoid E of all endomorphisms

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