De Morgan algebra

Remarks

From clauses 2 and 3 one gets ${\sim}(a \wedge b) = {\sim} a \vee {\sim} b$, ${\sim} 0 = 1$ and ${\sim} 1 = 0$. So ${\sim}$ is an order-reversing bijection of $A$ onto itself.

Example: the interval $[0, 1]$ with $a \vee b = \max(a, b)$, $a \wedge b = \min(a, b)$ and ${\sim} a = 1 - a$. A chain is a distributive lattice, $1 - (1 - a) = a$, and $1 - \max(a, b) = \min(1 - a, 1 - b)$. It is not a Boolean algebra, since $\tfrac12 \wedge {\sim} \tfrac12 = \tfrac12 \ne 0$.

Non-example: the chain $0 < m < 1$ with ${\sim}$ the pseudocomplement, so ${\sim} 0 = 1$, ${\sim} m = 0$, ${\sim} 1 = 0$. Clause 2 fails, since ${\sim}{\sim} m = 1 \ne m$.