sectionally complemented lattice

Proposition 1.

Let $L$ be a lattice with the least element $\mathrm{0}$. Then the following are equivalent:

1. Every pair of elements have a difference (http://planetmath.org/DifferenceOfLatticeElements).

2. for any $a\in L$, the lattice interval $[0,a]$ is a complemented lattice.

Proof.

Suppose first that every pair of elements have a difference. Let $b\in [0,a]$ and let $c$ be a difference between $a$ and $b$. So $0=b\wedge c$ and $c\vee b=b\vee a=a$, since $b\le a$. This shows that $c$ is a complement of $b$ in $[0,a]$.

Next suppose that $[0,a]$ is complemented for every $a\in L$. Let $x,y\in L$ be any two elements in $L$. Let $a=x\vee y$. Since $[0,a]$ is complemented, $y$ has a complement, say $z\in [0,a]$. This means that $y\wedge z=0$ and $y\vee z=a=x\vee y$. Therefore, $z$ is a difference of $x$ and $y$. ∎

Definition. A lattice $L$ with the least element $0$ satisfying either of the two equivalent conditions above is called a sectionally complemented lattice.

Every relatively complemented lattice is sectionally complemented. Every sectionally complemented distributive lattice is relatively complemented.

Dually, one defines a dually sectionally complemented lattice to be a lattice $L$ with the top element $1$ such that for every $a\in L$, the interval $[a,1]$ is complemented, or, equivalently, the lattice dual $L^{\partial }$ is sectionally complemented.

Title	sectionally complemented lattice
Canonical name	SectionallyComplementedLattice
Date of creation	2013-03-22 17:58:46
Last modified on	2013-03-22 17:58:46
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06C15
Classification	msc 06B05
Related topic	DifferenceOfLatticeElements
Defines	sectionally complemented
Defines	dually sectionally complemented lattice