EDB — 071

[071] Given two ordered sets $(X,{\le }_X)$ and $(Y,{\le }_Y)$, setting $Z=X\times Y$, we define the lexicographic order ${\le }_Z$ on $Z$; let $z_1=⦇x_1,y_1⦈\in Z$ and $z_2=⦇x_2,y_2⦈\in Z$, then:

• in the case $x_1\ne x_2$ , then $z_1{\le }_Z{z}_2$ if and only if $x_1{\le }_X{x}_2$;

• in the case $x_1=x_2$ , then $z_1{\le }_Z{z}_2$ if and only if $y_1{\le }_Y{y}_2$.

This definition is then extended to products of more than two sets: given two vectors, if the first elements are different then we compare them, if they are equal we compare the second elements, if they are equal the thirds, etc.