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 $\mathrm{Cen}(L)$, 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

  $$a\wedge b=a\wedge c\text{ and }a\vee b=a\vee c\text{ imply }b=c\text{ for all }b,c\in L$$

  where $a\in \{0,1\}$, and therefore neutral.

• $\mathrm{Cen}(L)$ is a sublattice of $L$.

• $\mathrm{Cen}(L)$ is a Boolean algebra.

Title	center of a lattice
Canonical name	CenterOfALattice
Date of creation	2013-03-22 17:31:50
Last modified on	2013-03-22 17:31:50
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06B05
Defines	central element