---
tag: 5SE8EMSDNY
kind: concept
name: Atomic lattice
msc: [06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(order_theory)
---

## Summary

A lattice with a least element $0$ is atomic if every nonzero element lies above at least one [[atom|TTVCTW6TNC]].
Every finite lattice is atomic, and so is the lattice of all subsets of a set.
The notion separates lattices that are built from minimal pieces from atomless ones such as the interval $[0, 1]$.
It is weaker than being [[atomistic|99DVVV7BH2]], where every element is a join of atoms.

## Explanations

### L1 human

In an atomic lattice, whatever nonzero element one picks, one can always find a smallest possible nonzero piece inside it.
In the subsets of a set, a nonempty set contains a one-element subset.
In the positive integers ordered by divisibility, every number $n > 1$ has a prime divisor.
In the interval $[0, 1]$ there is no smallest positive number, so it is not atomic.