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.
Fig. 2: The lattice of divisors of 4, with the ordering "isdivisorof", is atomic, with 2 being the only atom and coatom. It is not atomistic, since 4 cannot be obtained as least common multiple of atoms.
Fig. 1: The power set of the set {x, y, z} with the ordering "issubsetof" is an atomistic partially ordered set: each member set can be obtained as the union of all singleton sets below it.
A partially ordered set with a least element 0isatomic if every element b > 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 (orstrongly atomic) if for all a < b there is an element c such that a <: c ≤ b or, equivalently, if every interval [a, b] is atomic. Every relatively atomic partially ordered set with a least element is atomic. Every finite poset is relatively atomic.
A partially ordered set with least element 0 is called atomistic (not to be confused with atomic) if every element is the least upper bound of a set of atoms. The linear order with three elements is not atomistic (see Fig. 2).
Atoms in partially ordered sets are abstract generalizations of singletonsinset theory (see Fig. 1). Atomicity (the property of being atomic) provides an abstract generalization in the context of order theory of the ability to select an element from a non-empty set.
