uniquely complemented lattice
Recall that in a bounded distributive lattice^{}, complements^{}, relative complements, and differences of lattice elements, if exist, must be unique. This leads to the general consideration of general bounded lattices in which complements are unique.
Definition. A complemented lattice such that every element has a unique complement is said to be uniquely complemented. If $a$ is an element of a uniquely complemented lattice, ${a}^{\prime}$ denotes its (unique) complement. One can think of ${}^{\prime}$ as a unary operator on the lattice^{}.
One of the first consequences is
$${a}^{\prime \prime}=a.$$ 
To see this, we have that $a\vee {a}^{\prime}=1$, $a\wedge {a}^{\prime}=0$, as well as ${a}^{\prime \prime}\vee {a}^{\prime}=1$, ${a}^{\prime \prime}\wedge {a}^{\prime}=0$. So $a={a}^{\prime \prime}$, since they are both complements of ${a}^{\prime}$.
Below are some additional (and nontrivial) properties of a uniquely complemented lattice:

•
there exists a uniquely complemented lattice that is not distributive^{}

•
a uniquely complemented lattice $L$ is distributive if at least one of the following is satisfied:

(a)
${}^{\prime}$, as an operator on $L$, is order reversing;

(b)
${(a\vee b)}^{\prime}={a}^{\prime}\wedge {b}^{\prime}$;

(c)
${(a\wedge b)}^{\prime}={a}^{\prime}\vee {b}^{\prime}$;

(d)
(von Neumann) $L$ is a modular lattice^{};

(e)
(BirkhoffWard) $L$ is an atomic lattice.
In fact, the first three conditions are equivalent^{}, so that $L$ is distributive if it satisfies the de Morgan’s laws.

(a)

•
(Dilworth) every lattice can be embedded in a uniquely complemented lattice.
References
 1 T.S. Blyth, Lattices and Ordered Algebraic Structures^{}, Springer, New York (2005).
 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title  uniquely complemented lattice 

Canonical name  UniquelyComplementedLattice 
Date of creation  20130322 17:58:15 
Last modified on  20130322 17:58:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B05 
Classification  msc 06C15 
Defines  uniquely complemented 