sectionally complemented lattice
Proposition 1.
Let L be a lattice with the least element 0. Then the following are equivalent
:
-
1.
Every pair of elements have a difference (http://planetmath.org/DifferenceOfLatticeElements).
-
2.
for any a∈L, the lattice interval [0,a] is a complemented lattice
.
Proof.
Suppose first that every pair of elements have a difference. Let b∈[0,a] and let c be a difference between a and b. So 0=b∧c and c∨b=b∨a=a, since b≤a. This shows that c is a complement of b in [0,a].
Next suppose that [0,a] is complemented for every a∈L. Let x,y∈L be any two elements in L. Let a=x∨y. Since [0,a] is complemented, y has a complement, say z∈[0,a]. This means that y∧z=0 and y∨z=a=x∨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∈L, the interval [a,1] is complemented, or, equivalently, the lattice dual L∂ is sectionally complemented.
References
- 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998)
Title | sectionally complemented lattice |
---|---|
Canonical name | SectionallyComplementedLattice |
Date of creation | 2013-03-22 17:58:46 |
Last modified on | 2013-03-22 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 |