Free monoids and forests of rational numbers

Abstract

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 = 1 0 1 1 and, which freely generate the monoid SL 2 (N 0 ) of 2 × 2 matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices Lu and Rv that freely generate submonoids of GL 2 (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.

The (u,v)-Calkin–Wilf forest

“…which is equivalent to the desired result. Theorem 2 (Nathanson's symmetry, [13]). Let z be a variable, and let u and v be positive integers.…”

“…Nathanson's symmetry was proved in [13] using induction on the row number. We conclude this section with two alternative proofs of Theorem 2.…”

“…In addition to enumerating the positive rationals, the Calkin-Wilf tree has many interesting properties and generalizations that have been explored by various researchers (for example, [3,4,5,6,8,9,10,12,13,14,15]). In particular, as in [12], we highlight the following four properties.…”

Left–right pairs and complex forests of infinite rooted binary trees

“…Motivated by the beautiful properties of the Calkin-Wilf tree (see [1], [2], [4], [5], [6], [7], [8], [9], [10], [13], [11], [14]), Nathanson ([12]) studied the more general construction F (L, R) described above. He posed the following problems:…”