---
tag: 2XF2Y9B5E2
kind: definition
name: Atomless Boolean algebra
msc: ["06E05"]
defines: NFEEGWJFCK
uses: [0XADJKM699, TTVCTW6TNC]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(order_theory)
---

## Definition

Let $B$ be a [[Boolean algebra|0XADJKM699]] with order $\le$.
An [[atom|TTVCTW6TNC]] of $B$ is an element $a \ne 0$ such that no $b \in B$ satisfies $0 < b < a$.
The algebra $B$ is *atomless* if it has no atoms, that is, for every $a \ne 0$ in $B$ there is $b \in B$ with $0 < b < a$.

## Remarks

The one-element algebra, with $0 = 1$, is atomless in a trivial way.
Every other atomless Boolean algebra is infinite, since in a finite algebra with $0 \ne 1$ a minimal element of $B \setminus \set{0}$ is an atom.

Example: let $B$ be the algebra of finite unions of half-open intervals $[s, t)$ with $0 \le s < t \le 1$, together with $\emptyset$, as subsets of $[0, 1)$.
A nonzero $a \in B$ contains some $[s, t)$ with $s < t$; then $b = [s, (s + t)/2)$ lies in $B$ and satisfies $\emptyset \ne b \subsetneq a$.

Non-example: the power set of a nonempty set $X$ is not atomless, since each singleton $\set{x}$ is an atom.