Antilexicographic Order/Examples

Examples of Antilexicographic Orders

Unit Square with Open Side

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

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

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

$\tuple {0, 0}$

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

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