vector lattice
An ordered vector space whose underlying poset is a lattice^{} is called a vector lattice. A vector lattice is also called a Riesz space.
For example, given a topological space^{} $X$, its ring of continuous functions $C(X)$ is a vector lattice. In particular, any finite dimensional Euclidean space ${\mathbb{R}}^{n}$ is a vector lattice.
A vector sublattice is a subspace^{} of a vector lattice that is also a sublattice.
Below are some properties of the join ($\vee $) and meet ($\wedge $) operations^{} on a vector lattice $L$. Suppose $u,v,w\in L$, then

1.
$(u+w)\vee (v+w)=(u\vee v)+w$

2.
$u\wedge v=(u+v)(u\vee v)$

3.
If $\lambda \ge 0$, then $\lambda u\vee \lambda v=\lambda (u\vee v)$

4.
If $\lambda \le 0$, then $\lambda u\vee \lambda v=\lambda (u\wedge v)$

5.
If $u\ne v$, then the converse^{} holds for 3 and 4

6.
If $L$ is an ordered vector space, and if for any $u,v\in L$, either $u\vee v$ or $u\wedge v$ exists, then $L$ is a vector lattice. This is basically the result of property 2 above.

7.
$(u\wedge v)+w=(u+w)\wedge (v+w)$ (dual of statement 1)

8.
$u\wedge v=(u\vee v)$ (a direct consequence of statement 4, with $\lambda =1$)

9.
$(u)\wedge u\le 0\le (u)\vee u$
Proof.
$(u)\wedge u\le u$ and $(u)\wedge u\le u$ imply that $2((u)\wedge u)\le u+(u)=0$, so $(u)\wedge u\le 0$, which means $0\le ((u)\wedge u)=u\vee (u)$. ∎

10.
$(a\vee b)+(c\vee d)=(a+c)\vee (a+d)\vee (b+c)\vee (b+d)$, by repeated application of 1 above.
Remark. The first five properties are also satisfied by an ordered vector space, with the assumptions^{} that the suprema exist for the appropriate pairs of elements (see the entry on ordered vector space for detail).
Title  vector lattice 

Canonical name  VectorLattice 
Date of creation  20130322 17:03:13 
Last modified on  20130322 17:03:13 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06F20 
Classification  msc 46A40 
Synonym  Riesz Space 
Defines  vector sublattice 