Lexicographic Order/Examples

Examples of Lexicographic Orders

Unit Square with Open Side

Consider the lexicographic product of the real intervals $\hointr 0 1$ and $\closedint 0 1$ under the usual ordering:

$\struct {T, \preccurlyeq_l} := \struct {\hointr 0 1, \le} \otimes^l \struct {\closedint 0 1, \le}$

$\struct {T, \preccurlyeq_l}$ has one minimal element:

$\tuple {0, 0}$

which is also the smallest element: of $\struct {T, \preccurlyeq_l}$.

$\struct {T, \preccurlyeq_l}$ has no greatest element and no maximal elements.