Boolean algebras canonically defined

Remarks

This class coincides with the class given by the usual axioms of a Boolean algebra. 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 $m \vee 0 = m \ne 1$.