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.
References
 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title  sectionally complemented lattice 

Canonical name  SectionallyComplementedLattice 
Date of creation  20130322 17:58:46 
Last modified on  20130322 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 