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**Additional info for Categorical foundations: special topics in order, topology, algebra, and Sheaf theory**

**Example text**

2 CHAPTER 6. CHARACTERIZATIONS Balanced1 Lyndon words We next introduce an important class of words first defined by Marston Morse and Gustav A. Hedlund in 1940. A word w ∈ {x, y}∗ is balanced1 if for each pair u, v of factors of w of equal length, one has |u|x − |v|x ≤ 1, or equivalently |u|y − |v|y ≤ 1. Serge Dulucq and Dominique Gouyou-Beauchamps [DGB1990] have shown that the set of balanced1 words is exactly the set of factors of Christoffel words, or equivalently of Sturmian words (cf. [Lot2002, Chapter 2]).

By refining the method of the above proof one can obtain the following result relating the Christoffel morphisms with the automorphisms of the free group F2 . Recall that an element w ∈ F2 is positive if w ∈ {x, y}∗ ⊆ F2 . An endomorphism f of F2 is a positive morphism if both f (x) and f (y) are positive. 5 (Wen, Wen [WW1994]). The Christoffel morphisms of {x, y}∗ are exactly the positive morphisms of the free group x, y . Chapter 6 Characterizations By now we have presented several characterizations of Christoffel words— discretization of line segments, Cayley graphs of cyclic groups, palindromization and the positive primitive elements of F2 .

It remains to show that |w1 | and |w2 | are relatively prime. 4, |w1 |x |w2 |x |w1 |y |w2 |y ∈ SL2 (Z). This implies that det |w1 | |w2 | |w1 |y |w2 |y = det |w1 |x |w2 |x |w1 |y |w2 |y = 1. 3). 2. 1 (Pick’s Theorem). Let P be a simple polygon (that is, the boundary of P has no self-intersections) with vertices in Z × Z. Then the area of P is i + 21 b − 1, where i is the number of integer points in the interior of P and b is the number of integer points of the boundary of P . 2. Suppose i, j, k, l are positive integers.