center of a lattice

Let $L$ be a bounded lattice. An element $a\\in L$ is said to be central if $a$ is complemented (http://planetmath.org/ComplementedLattice) and neutral (http://planetmath.org/SpecialElementsInALattice). The center of $L$ , denoted $\\operatorname{Cen}(L)$ , is the set of all central elements of $L$ .

Remarks.

•
$0$ and $1$ are central: they are complements of one another, both distributive and dually distributive, and satisfying the property
$a\\wedge b=a\\wedge c\\mbox{ and }a\\vee b=a\\vee c\\mbox{ imply }b=c\\mbox{ for all %}b,c\\in L$
where $a\\in\\{0,1\\}$ , and therefore neutral.
•
$\\operatorname{Cen}(L)$ is a sublattice of $L$ .
•
$\\operatorname{Cen}(L)$ is a Boolean algebra.

References

1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).