Stone algebra

Remarks

Example: every bounded chain $C$ with $0 \ne 1$ is a Stone algebra. Its pseudocomplement is $0^* = 1$ and $a^* = 0$ for $a \ne 0$. If $a = 0$ then $a^* \vee a^{**} = 1 \vee 0 = 1$, and if $a \ne 0$ then $a^* \vee a^{**} = 0 \vee 1 = 1$.

Non-example: the lattice of open sets of $\R$, with $U^* = \operatorname{int}(\R \setminus U)$, satisfies clauses 1 and 2 but not 3. For $U = (0, \infty)$, one has $U^* = (-\infty, 0)$ and $U^{**} = (0, \infty)$, so $U^* \cup U^{**} = \R \setminus \set{0} \ne \R$.

Every Boolean algebra is a Stone algebra, with $a^* = \neg a$ and $a^{**} = a$.