Atom (order theory)
Using the covering relation <:, an atom is simply an element a such that 0 <: a (a covers zero).
That is, an atom is an element that is minimal among the non-zero elements.
A partially ordered set with a least element is atomic if every non-zero element b > 0 has an atom a below it, i.e. b ≥ a :> 0.
A partially ordered set is 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 (cf. pic.2).
Atoms in partially ordered sets are abstract generalizations of singletons in set theory (cf. pic.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.
The terms coatom, coatomic, and coatomistic are defined dually; thus in a partially ordered set with greatest element 1:
- A coatom is an element covered by 1
- The set is called coatomic if every b < 1 has a coatom c above it
- The set is called coatomistic if every element is the greatest lower bound of a set of coatoms.