Lexicographic order

Lexicographic order --- лексикографический порядок.

For [math]\displaystyle{ 1 \leq u \leq m }[/math], define a relation [math]\displaystyle{ \lt _{u} }[/math] in [math]\displaystyle{ R^{m} }[/math] by requiring that two vectors [math]\displaystyle{ \vec{i} = (i_{1}, \ldots, i_{m}) }[/math] and [math]\displaystyle{ \vec{j} = (j_{1}, \ldots, j_{m}) }[/math] satisfy

[math]\displaystyle{ \vec{i} \lt _{u} \vec{j}\mbox{ iff }i_{1} = j_{1}, \ldots, i_{u-1} = j_{u-1}, \mbox{ and }i_{u} \lt j_{u}. }[/math]

Let [math]\displaystyle{ \lt }[/math] denote the union of the relations [math]\displaystyle{ \lt _{u} }[/math] for [math]\displaystyle{ 1 \leq u \leq m }[/math]; it is called the lexicographic order in [math]\displaystyle{ R^{m} }[/math].