---
tag: Z344RY5XX1
kind: concept
name: Positive cone
msc: ["46A40", "06F20"]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_vector_space
---

## Summary

A positive cone in a real [[vector space|000AYPA9AJ]] is a subset closed under addition and under multiplication by nonnegative scalars.
It must also contain no line through the origin.
Positive cones are the same thing as compatible partial orders: the cone is the set of elements that are at least $0$.
So one can describe an order on a vector space just by saying which vectors are positive.
The same name is used for the set of nonnegative elements of an ordered group, ring or field.

## Explanations

### L1 human

In the plane, take the quarter plane of points with both coordinates nonnegative.
Adding two such points, or stretching one by a nonnegative factor, gives another such point.
The quarter plane contains no full line through the origin.
Saying that $u \le v$ when $v - u$ lies in the quarter plane gives the componentwise order.