# uniquely complemented lattice

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 non-trivial) properties of a uniquely complemented lattice:

## References

• 1 T.S. Blyth, , Springer, New York (2005).
• 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title uniquely complemented lattice UniquelyComplementedLattice 2013-03-22 17:58:15 2013-03-22 17:58:15 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 06B05 msc 06C15 uniquely complemented