Order, lattices, ordered algebraic structures Definition

Complete atomic Boolean algebra

Builds on Complete Boolean algebra Atom Atomic lattice

Definition

Let $B$ be a Boolean algebra. An atom of $B$ is an element $a \ne 0$ such that there is no $b$ with $0 < b < a$ . The algebra $B$ is a complete atomic Boolean algebra if:

$B$ is a complete Boolean algebra.
$B$ is atomic: for every $b \ne 0$ in $B$ there is an atom $a$ with $a \le b$ .

Remarks

If $B$ is complete and atomic, with set of atoms $A$ , then $b \mapsto \set{a \in A : a \le b}$ is an isomorphism from $B$ onto the power set of $A$ . So the complete atomic Boolean algebras are, up to isomorphism, the power set algebras.

Example: the power set of a set $X$ . It is complete, and every nonempty $Y \subseteq X$ contains a singleton $\set{x}$ , which is an atom.

Non-example: the algebra of regular open sets of $\R$ is complete but has no atom at all. Every nonempty regular open set contains an open interval, which contains two disjoint smaller open intervals, and these are regular open and strictly smaller. So clause 2 fails.

In Mathlib the notion is CompleteAtomicBooleanAlgebra.

Reading path

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Atom concept
Atomic lattice concept
Complete Boolean algebra concept
Complete atomic Boolean algebra definition

Tag	`MCW8V4RFYJ`
URI	`m2://MCW8V4RFYJ`
Kind	definition
Writing	W0 stub
MSC	06E10, 06E05
Defines	Complete atomic Boolean algebra
Uses	Complete Boolean algebra, Atom, Atomic lattice
Changed	2 November 2026

Complete atomic Boolean algebra

Definition

Remarks

Reading path

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