Def/Lax vector



=Primitive Lax Vectors= We say that a lax vector $\pm (a,b)$ is defines::primitive, if $GCD(a,b) = 1$. Observe that this notion is well-defined, i.e., it depends on only on the lax vector $\pm (a,b)$, and not on its representative $(a,b)$ or $(-a,-b)$ in $\ZZ^2$.

Let $\hat \QQ = \QQ \cup \{ \infty \}$ be the set whose elements are the rational numbers, together with the symbol $\infty$.

There is a natural bijection from the set of primitive lax vectors to the set $\hat \QQ$, given by the following:
 * For every rational number $q$, there exist unique integers $x,y$ such that $GCD(x,y) = 1$, $y > 0$, and $q = \frac{x}{y}$. To the rational number $q$, we associated the primitive lax vector $\pm (x,y)$.  To the element $\infty \in \hat \QQ$, we associate the primitive lax vector $\pm (1,0)$ (i.e., we view $1/0$ as $\infty$).
 * To each primitive lax vector $\pm (x,y)$, we associate the rational number $\frac{x}{y}$, unless $y = 0$, in which case we associate $\infty \in \hat \QQ$.