Pseudocomplement

Remarks

Equivalently, $a^*$ is the greatest element of the set $\set{x \in L : a \wedge x = 0}$, so it is unique when it exists. In a pseudocomplemented lattice, $a \le a^{**}$ and $a^{***} = a^*$ for all $a$.

Example: the open sets of a topological space $X$, ordered by inclusion. For an open set $U$, put $U^* = \operatorname{int}(X \setminus U)$. An open set $V$ satisfies $U \cap V = \emptyset$ exactly when $V \subseteq X \setminus U$, that is, when $V \subseteq \operatorname{int}(X \setminus U)$, since $V$ is open.

Non-example: in the diamond lattice $M_3$ with atoms $p, q, r$, the element $p$ has no pseudocomplement. The elements $x$ with $p \wedge x = 0$ are $0$, $q$ and $r$, and this set has no greatest element, since $q \vee r = 1$ and $p \wedge 1 = p \ne 0$.