---
tag: TTVCTW6TNC
kind: concept
name: Atom
msc: [06A06, 06B05]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(order_theory)
same_as:
  wikidata: Q261484
  wikipedia: Atom (order theory)
---

## Summary

In a [[partially ordered set|0QV2JS9RJ3]] with a [[least element|0AANKKNHS4]] $0$, an atom is an element just above $0$.
It is above $0$, and nothing lies strictly between $0$ and it.
Atoms are the minimal nonzero elements.
In the lattice of subsets of a set they are the one-element subsets, which is the origin of the name.
Atoms are used to define atomic and atomistic lattices and to describe Boolean algebras.

## Explanations

### L1 human

An atom is a smallest piece that is not nothing.
In the subsets of $\set{1, 2, 3}$, the empty set is nothing, and the atoms are $\set{1}$, $\set{2}$ and $\set{3}$.
In the positive integers ordered by divisibility, the bottom is $1$ and the atoms are the prime numbers, since the only divisors of a prime $p$ are $1$ and $p$.
Some orders have no atoms at all: in the interval $[0, 1]$, between $0$ and any $a > 0$ lies $a / 2$.