---
tag: C8ZPFCX1SJ
kind: definition
name: Free Boolean algebra
msc: ["06E05", "03G05"]
defines: 1MF0TVHDZD
uses: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Free_Boolean_algebra
---

## Definition

Let $S$ be a set.
Let $F$ be a [[Boolean algebra|0XADJKM699]] and let $\iota \colon S \to F$ be a map.
The pair $(F, \iota)$ is a *free Boolean algebra* on $S$ if the following holds.
For every Boolean algebra $B$ and every map $g \colon S \to B$, there is exactly one homomorphism $\bar g \colon F \to B$ with $\bar g \circ \iota = g$.

## Remarks

It exists for every $S$ and is unique up to a unique isomorphism compatible with $\iota$.
For finite $S = \set{1, \ldots, n}$, one model is the algebra of all [[Boolean functions|0X05GS525E]] $\set{0, 1}^n \to \set{0, 1}$, with $\iota(i)$ the $i$-th coordinate function; it has $2^{2^n}$ elements and $2^n$ atoms.
For any $S$, one model is the algebra of clopen subsets of the product space $\set{0, 1}^S$.
The free Boolean algebra on a countably infinite set is atomless.

Example: for $S = \set{x}$, the algebra $\set{0, x, \neg x, 1}$ with $x$ and $\neg x$ incomparable is free.
A map $g$ extends by $\neg x \mapsto \neg g(x)$, $0 \mapsto 0$, $1 \mapsto 1$, and this extension is forced.

Non-example: the power set of $\set{1, 2}$ with $\iota(x) = \set{1}$ and $\iota(y) = \set{2}$ is generated by the image of $\iota$ but is not free on $\set{x, y}$.
For the map $g(x) = g(y) = 1$ into the two-element algebra, a homomorphism $\bar g$ would send $\set{1} \cap \set{2} = \emptyset$ to $1 \wedge 1 = 1$, but a homomorphism sends $\emptyset$ to $0$.