sectionally complemented lattice
Suppose first that every pair of elements have a difference. Let and let be a difference between and . So and , since . This shows that is a complement of in .
Next suppose that is complemented for every . Let be any two elements in . Let . Since is complemented, has a complement, say . This means that and . Therefore, is a difference of and . ∎
Definition. A lattice with the least element 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 with the top element such that for every , the interval is complemented, or, equivalently, the lattice dual is sectionally complemented.
- 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
|Title||sectionally complemented lattice|
|Date of creation||2013-03-22 17:58:46|
|Last modified on||2013-03-22 17:58:46|
|Last modified by||CWoo (3771)|
|Defines||dually sectionally complemented lattice|