Orthocomplemented lattice

Remarks

From clauses 2 and 3 follow the De Morgan laws $(a \vee b)^\perp = a^\perp \wedge b^\perp$ and $(a \wedge b)^\perp = a^\perp \vee b^\perp$.

Example: the lattice of closed linear subspaces of a Hilbert space $H$, with $M^\perp$ the orthogonal complement of $M$. Here $M \cap M^\perp = \set{0}$, the closed span of $M$ and $M^\perp$ is $H$, $M^{\perp\perp} = M$ for closed $M$, and $M \subseteq N$ implies $N^\perp \subseteq M^\perp$.

Non-example: the diamond lattice M3 with middle elements $a, b, c$ and the map $a \mapsto b$, $b \mapsto c$, $c \mapsto a$, $0 \mapsto 1$, $1 \mapsto 0$. Each image is a complement, but $(a^\perp)^\perp = c \ne a$, so clause 2 fails.