# sectionally complemented lattice

###### Proposition 1.

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

1. 1.

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

2. 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 $b\leq a$. 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.

## References

• 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title sectionally complemented lattice SectionallyComplementedLattice 2013-03-22 17:58:46 2013-03-22 17:58:46 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 06C15 msc 06B05 DifferenceOfLatticeElements sectionally complemented dually sectionally complemented lattice