# 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).
Title center of a lattice CenterOfALattice 2013-03-22 17:31:50 2013-03-22 17:31:50 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 06B05 central element