Free monoids and forests of rational numbers

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 $L_1=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ and $R_1=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, which freely generate the monoid $S{L}_2({\mathbf{N}}_0)$ 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 $G{L}_2({\mathbf{N}}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.

10 pages

References in corpus (2)

• The (u,v)-Calkin-Wilf Forest
• Orphans in Forests of Linear Fractional Transformations

Cited by in corpus (1)

• The (u,v)-Calkin-Wilf Forest