Semilattice of Weak Orders of a Set

摘要

Individual and collective preferences are often modelled using binary relations. Weak orders turn out to be especially useful for this purpose. The idea is to let X denote a set of alternatives and then rank X by preference. Thus xPy means y is preferable to x. The resulting binary relation P is often a weak order on X. Such relations also arise naturally in digital image processing. In its most general form, a (monochromatic) digital picture is simply a rectangular array of numbers that have spatial as well as numerical significance. Finally, weak orders arise in connection with the reconstruction of evolutionary trees. The underlying set here is a set of evolutionary units, and if the goal is to construct a binary tree on X that in some sense estimates the true evolutionary history of the currently existing members of X, then one can view this as constructing a nested sequence of weak orders on these members.