Free monoids and forests of rational numbers

TL;DR

This paper explores how certain matrix-generated monoids create forests of infinite binary trees that uniquely partition positive rational numbers, revealing a new structural perspective on these numbers.

Contribution

It introduces a general framework for forests of rational numbers generated by free monoids of matrices, extending the classical Calkin-Wilf tree to broader cases with symmetry properties.

Findings

Forests partition positive rationals into infinite binary trees.

Matrices $L_u$ and $R_v$ generate these forests with symmetry.

Generalization of Calkin-Wilf tree to new matrix pairs.

Abstract

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$ and $R_1 = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $SL_2(\mathbf{N}_0)$ of $2 \times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $GL_2(\mathbf{N}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.