I got the results, which I'll describe below, for my own pleasure, without consulting anything, in 1971/2 in Ann Arbor, then I had a private seminar for a few kids at the U. of Texas at Austin in the fall of 1972, nothing official. The main topic was an elementary theory of diophantine approximations. Otherwise, I considered these things to be well known and even classic hence it was an end of the story to me. (I feel that Minkowski knew these things anyway). I also learned casually that there is some knowledge of this topic in computer science.

Let $\ SL(Z_{\ge 0})\ $ be the monoid of matrices with non-negative integer entries, which have determinant equal 1.

**THEOREM** $\ SL(Z_{\ge 0})\ $ is a free monoid which has a $2$-element free base
consisting of
$$ L\ :=\ \left[\begin{array}{cc} 1 & 0\\ 1 & 1\end{array}\right]\qquad
\mbox{and}\qquad R\ :=\ \left[\begin{array}{cc} 1 & 1\\ 0 & 1\end{array}\right]. $$

**PROOF** (*a sketch*) Consider the infinite table:

$$ \left[\begin{array}{cccccccccccccccc}
1/0 &&&&&&&&&&&&&&&& 0/1 \\ 1/0 &&&&&&&& 1/1 &&&&&&&& 0/1 \\
1/0 &&&& 2/1 &&&& 1/1 &&&& 1/2 &&&& 0/1 \\
1/0 && 3/1 && 2/1 && 3/2 && 1/1 && 2/3 && 1/2 && 1/3 && 0/1 \\
1/0 & 4/1 & 3/1 & 5/2 & 2/1 & 5/3 & 3/2 & 4/3 & 1/1
& 3/4 & 2/3 & 3/5 & 1/2 & 2/5 & 1/3 & 1/4 & 0/1 \\
\end{array}\right] $$
etc.

It's a pleasant(!) exercise to show that every matrix $\ M\in SL(Z_{\ge 0})\ $
appears in the above table in exactly one row, in an exactly one place -- just
remove the quotient signs ''$/$'', treat each fraction as a 2-dimensional (vertical) vector.

Observe that the fractions in each row go down from $\ \infty\ $ to $\ 0.\ $ I call a pair of such fractions (always reduced in this table) to be neighbors. Any reduced fraction which is between two neighbors has its denominator equal at least the sum of the denominators of the said neighbors.

**End of Proof** (well, of the sketch).

**REMARK** *I saw later a computer science text which features the above table, and a name was attached to this table.*

**REMARK** I remember that in one of the 2 volumes of *Development of the Minkowski Geometry of Numbers* by Harris Hancock, there was one line of the above table, perhaps only the half of the line between $0/1$ and $1/1$ (in this order, I think, which is reverse to the order above). I have a feeling that Minkowski knew what I have learned and you can see right now, that there is an increasing function of $[0;\infty]$ onto itself which maps non-negative rational numbers of the form $\ k/2^n\ $ onto the entire set of non-negative numbers $\ \mathbb Q_{\ge 0}\ := \mathbb Q\cap[0;\infty],\ $ and that this mapping maps $\ \mathbb Q_{\ge 0}\ $ onto the set of all non-negative numbers consisting of the rational numbers and of quadratic irrationals.

**REMARK** *This table is ***very** closely related to continued fractions (another exercise).

I don't know any references but for the two mentioned above (about Minkowski and a computer science textbook) -- sorry for this.