nLab
atomic Boolean algebra

Definition

Given an element $a$ of a Boolean algebra (or other poset) $A$, recall that $a$ is atomic in $A$ if $a$ is minimal among non-trivial (non-bottom) elements of $A$. That is, given any $b\in A$ such that $b\le a$, either $b=0$ or $b=a$.

A Boolean algebra $A$ is atomic if we have $b={\bigvee }_I{a}_i$ for every $b\in A$, where $\{a_i{\}}_I$ is some set of atoms in $A$.

Properties

• A complete atomic Boolean algebra is necessarily a power set; see CABA.