nLab atomic Boolean algebra

Definition

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

$A$ 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 $A$.

If $A$ is complete we can write it: if for every $b \in A$, we have $b = \bigvee \mathcal{A}(b)$ where $\mathcal{A}(b)$ is the set of all the atoms $a$ in $A$ such that $a \leq b$. Or: for every $b \in A$, we have $b \le \bigvee \mathcal{A}(b)$.

Properties

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