---
tag: J7KN14DYV3
kind: concept
name: Orthocomplemented lattice
msc: [06C15, 81P10]
specializes: [ZT47BK9Y6K]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_lattice
---

## Summary

An orthocomplemented lattice, or ortholattice, is a [[complemented lattice|ZT47BK9Y6K]] with a chosen complement $a^\perp$ for each element $a$.
The map $a \mapsto a^\perp$ must reverse the order and be its own inverse.
The main examples are [[Boolean algebras|0XADJKM699]], with the Boolean complement.
Another is the lattice of closed subspaces of a [[Hilbert space|HM2HHE3J51]], with the [[orthogonal complement|0XJXTBCB2T]].
The De Morgan laws hold in every ortholattice.
Ortholattices are the starting point of the lattice-theoretic approach to [[quantum logic|2E9DMB7A6Q]].

## Explanations

### L1 human

In a general complemented lattice an element can have many complements, with no preferred one.
An orthocomplementation picks one complement for each element in a coherent way, like taking the orthogonal complement of a subspace.
Taking it twice gives back the element, and a bigger element gets a smaller complement.
For a line $\ell$ through the origin in $\R^3$, the chosen complement is the plane perpendicular to $\ell$.