Let $V$ be a vector lattice over $\mathbb{R}$ , and $V^+$ be its positive cone. We define three functions from $V$ to $V^+$ as follows. For any $x\in V$ ,

• $x^+:=x\vee 0$ ,
• $x^-:=(-x)\vee 0$ ,
• $|x|:=(-x)\vee x$ .

It is easy to see that these functions are well-defined. Below are some properties of the three functions:

1. $x^+=(-x)^-$ and $x^-=(-x)^+$ .
2. $x=x^+-x^-$ , since $x^+-x^-=(x\vee 0)-(-x)\vee 0=(x\vee 0)+(x\wedge 0)=x+0=x$ .
3. $|x|=x^++x^-$ , since $x^++x^-=x+2x^-=x+(-2x)\vee 0=(x-2x)\vee (x+0)=|x|$ .
4. If $0\le x$ , then $x^+=x$ , $x^-=0$ and $|x|=x$ . Also, $x\le 0$ implies $x^+=0$ , $x^-=-x$ and $|x|=-x$ .
5. $|x|=0$ iff $x=0$ . The ``only if'' part is obvious. For the ``if'' part, if $|x|=0$ , then $(-x)\vee x=0$ , so $x\le 0$ and $-x\le 0$ . But then $0\le x$ , so $x=0$ .
6. $|rx|=|r||x|$ for any $r\in \mathbb{R}$ . If $0\le r$ , then $|rx|=(-rx)\vee (rx)=r\big((-x)\vee x\big)=r|x|=|r||x|$ . On the other hand, if $r\le 0$ , then $|rx|=(-rx)\vee (rx)=(-r)\big(x\vee (-x)\big)=-r|x|=|r||x|$ .
7. $|x|+|y|=|x+y|\vee |x-y|$ , since $$LHS=(-x)\vee x+(-y)\vee y=(-x-y)\vee (-x+y)\vee (x-y)\vee (x+y)=RHS.$$
8. (triangle inequality). $|x+y|\le |x|+|y|$ , since $|x+y|\le |x+y|\vee |x-y|=|x|+|y|$ .

Properties 5, 6, and 8 satisfy the axioms of an absolute value, and therefore $|x|$ is called the absolute value of $x$ . However, it is not the ``norm'' of a vector in the traditional sense, since it is not a real-valued function.