---
tag: EG1N82CV0R
kind: definition
name: Boolean algebras canonically defined
msc: ["06E05", "03G05", "08B05"]
defines: VY46T0A6VQ
uses: [0WG43AC0EV]
sources:
  - url: https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined
---

## Definition

Let $\mathbf{2} = (\set{0, 1}, \wedge, \vee, \neg, 0, 1)$ be the [[two-element Boolean algebra|0WG43AC0EV]].
A *term* is an expression built from variables with the binary symbols $\wedge, \vee$, the unary symbol $\neg$ and the constants $0, 1$.
An *equation* is a pair of terms, written $s = t$.
Let $A$ be a set with operations $\wedge, \vee, \neg$ and constants $0, 1$.
An equation *holds* in $A$ if both sides take the same value for every assignment of elements of $A$ to the variables.
In the canonical definition, a *Boolean algebra* is such a set $A$ in which every equation that holds in $\mathbf{2}$ also holds.

## Remarks

This class coincides with the class given by the usual axioms of a [[Boolean algebra|06GKBTAA5E]].
One direction is clear, since the usual axioms hold in $\mathbf{2}$; the other uses the fact that every equation true in $\mathbf{2}$ follows from those axioms.
In the language of universal algebra, the Boolean algebras form the variety generated by $\mathbf{2}$.

Example: the power set of a set $X$, with intersection, union and complement.
A subset $S$ of $X$ is the same as a function $X \to \set{0, 1}$, and the operations act point by point.
So an equation true in $\mathbf{2}$ holds at each point of $X$, and hence in the power set.

Non-example: the chain $0 < m < 1$ with $\wedge = \min$, $\vee = \max$, $\neg 0 = 1$, $\neg m = 0$ and $\neg 1 = 0$.
The equation $x \vee \neg x = 1$ holds in $\mathbf{2}$, but for $x = m$ it gives $m \vee 0 = m \ne 1$.