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.

Atomic orderings

Let <: denote the cover relation in a partially ordered set.
A partially ordered set with a least element 0 is atomic 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 is not atomic.
A partially ordered set is relatively atomic if for all a < b there is an element c such that a <: c ≤ b or, equivalently, if every interval 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 if every element is the least upper bound of a set of atoms. The linear order with three elements is not atomistic.
Atoms in partially ordered sets are abstract generalizations of singletons in set theory. Atomicity provides an abstract generalization in the context of order theory of the ability to select an element from a non-empty set.

Coatoms

The terms coatom, coatomic, and coatomistic are defined dually. Thus, in a partially ordered set with greatest element 1, one says that