Lexicographic order

A nonzero vector is lexicographically positive if its first non-zero coordinate is positive. The vector $LaTeX: x$ is lexicographically greater than the vector $LaTeX: y$ if $LaTeX: x-y$ is lexicographically positive, and this defines the lexicographic order in $LaTeX: \mathbb{R}^n.$ This is a total ordering in that every two vectors are either equal, or one is lexicographically greater than the other.
This was first used in mathematical programming to resolve cycling in the simplex method. It also provides a way to obtain solutions for multiple objectives with the property that $LaTeX: x$ is a Pareto maximum if $LaTeX: f(x)$ is lexicographically greater than or equal to $LaTeX: f(y)$ for all feasible $LaTeX: y$.