Atom (order theory)

In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 < a and there is no x such that 0 < x < a. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0. Let 0 has an atom a below it, that is, there is some a such that b ≥ a :> 0. Every finite partially ordered set with 0 is atomic, but the set of nonnegative real numbers (ordered in the usual way) is not atomic (and in fact has no atoms). A partially ordered set is relatively atomic (or strongly atomic) if for all a < b there is an element c such that a