|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
sectionally complemented lattice
|
(Definition)
|
|
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.
- 1
- G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
|
"sectionally complemented lattice" is owned by CWoo.
|
|
(view preamble | get metadata)
Cross-references: interval, top, relatively complemented, distributive lattice, relatively complemented lattice, equivalent, complement, difference, complemented lattice, lattice interval, the following are equivalent, least element, lattice
This is version 3 of sectionally complemented lattice, born on 2008-04-07, modified 2008-04-08.
Object id is 10488, canonical name is SectionallyComplementedLattice.
Accessed 977 times total.
Classification:
| AMS MSC: | 06B05 (Order, lattices, ordered algebraic structures :: Lattices :: Structure theory) | | | 06C15 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
