Atom (order theory)

In the mathematical field of order theory, given two elements "a" and "b" of a partially ordered set, one says that "b" covers "a", and writes "a" <: "b" or "b" :> "a", if "a" < "b" and there is no element "c" such that "a" < "c" < "b". In other words, "b" covers "a" if "b" is greater than "a" and minimal with this property, or equivalently if "a" is smaller than "b" and maximal with this property.

In a partially ordered set with least element 0, an atom is an element that covers 0, i.e. an element that is minimal among the non-zero elements. A partially ordered set with a least element is called atomic if every non-zero element "b" > 0 has an atom "a" below it, i.e. "b" ≥ "a" :> 0.

A partially ordered set is called relatively atomic (or "strongly 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.

A partially ordered set with least element 0 is called atomistic if every element is the least upper bound of a set of atoms. Every finite poset is relatively atomic, and every finite poset with 0 is atomic. But the linear order with three elements is not atomistic.

Atoms in partially ordered sets are abstract generalizations of singletons in set theory. 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.

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