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.