ideal

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. Definitions A subset of a partially ordered set is an ideal, if the following conditions hold: is non-empty, for every x in and y in P, implies that y is in  ( is a lower set), for every x, y in , there is some element z in , such that and  ( is a directed set).