Complemented lattice

Summary

A complemented lattice is a bounded lattice in which every element $a$ has a complement: an element $b$ with $a \wedge b = 0$ and $a \vee b = 1$. The lattice of subspaces of a vector space and every Boolean algebra are complemented. Complements need not be unique; in a distributive lattice they are, and a complemented distributive lattice is the same as a Boolean algebra. Complemented modular lattices are the setting of projective geometry and of von Neumann's continuous geometry.