An ordering which treats numbers as if they were letters in a dictionary. As an example, suppose S is a set of monomials of degree ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'n' 'utf-8' math_unknown_error in ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'm' 'utf-8' math_unknown_error variables: that is,

then ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'x_1^{k_1} x_2^{k_2}ldots x_m^{k_m}' 'utf-8' math_unknown_error is less than or equal to (≤) ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'x_1^{l_1} x_2^{l_2}ldots x_m^{l_m}' 'utf-8' math_unknown_error if the first nonzero difference of corresponding exponents ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'k_j-l_j' 'utf-8' math_unknown_error satisfies ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'k_j-l_j leq 0' 'utf-8' math_unknown_error. For example, the following formula is a lexicographic linear ordering:

A lexicographic ordering for the field of complex numbers ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'C' 'utf-8' math_unknown_error is defined as follows: if ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'z_1=x_1 + y_1i, z_2 = x_2 + y_2i in C' 'utf-8' math_unknown_error, then ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'z_1 leq z_2' 'utf-8' math_unknown_error if and only if

In both of the above cases the order ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'leq' 'utf-8' math_unknown_error is reflexive, antisymmetric, transitive, and satisfies the trichotomy law, thus the order is a linear order and is called a lexicographic linear order.

More generally, if ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'O_1,O_2,ldots, Om' 'utf-8' math_unknown_error are ordered sets, the product set ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'p = extstyle prod_{j=1}^m O_j' 'utf-8' math_unknown_error is given a lexicographic ordering ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' '<' 'utf-8' math_unknown_error as follows: if ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'a=(a_1, a_2, ldots, a_m)' 'utf-8' math_unknown_error and ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'b = (b_1, b_2, ldots, b_m) in p' 'utf-8' math_unknown_error, then ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'a < b' 'utf-8' math_unknown_error if and only if the first non-zero difference of corresponding coordinates ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'a_j-b_j < 0' 'utf-8' math_unknown_error. When each ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'O_j' 'utf-8' math_unknown_error is linearly ordered, so is ./math/texvc '/var/www/htdocs/www/newtermwiki/cache/math' '/var/www/htdocs/www/newtermwiki/cache/math' 'p' 'utf-8' math_unknown_error.