Let $L$ be a bounded lattice. An element $a\in L$ is said to be central if $a$ is complemented and neutral. 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 $a\in \lbrace 0,1\rbrace$ , and therefore neutral.
• $\operatorname{Cen}(L)$ is a sublattice of $L$ .
• $\operatorname{Cen}(L)$ is a Boolean algebra.

Bibliography

1

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