Nonstandard analysis of ordered sets

We introduce a nonstandard approach to the study of ordered setsX based on a classification of the elements of the ordered set *X into three types, upward, downward, and lateral, which may be thought of dynamically as arising from the possibilities of upward, downward, and lateral motion withinX. Initial applications include the characterization thatX has no infinite diverse subset iff *X has no lateral elements, a result subsequently exploited in work on the interval topology and order-compatibility, where we give a nonstandard proof of Naito's result that ifX has no infinite diverse subset, it has a unique order-compatible topology. We also describe how the completion of a nonempty linearly ordered setX may be obtained as a quotient of *X.