Atom (order theory)

Let <: denote the covering relation in a partially ordered set.

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).

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.

Thus, in a partially ordered set with greatest element 1, one says that

Fig. 2 : The lattice of divisors of 4, with the ordering " is divisor of ", 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 " is subset of " is an atomistic partially ordered set: each member set can be obtained as the union of all singleton sets below it.