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.