Ordered field/Detailed/Definition

Ordered field

A field ${}K$ is called an ordered field, if there is a relation ${}>$ (larger than) between the elements of ${}K$, fulfilling the following properties (${}a\geq b$ means ${}a>b$ or ${}a=b$).

1. For two elements ${}a,b\in K$, we have either ${}a>b$ or ${}a=b$ or ${}b>a$.
2. From ${}a\geq b$ and ${}b\geq c$, one may deduce ${}a\geq c$ (for any ${}a,b,c\in K$).
3. ${}a\geq b$ implies ${}a+c\geq b+c$ (for any ${}a,b,c\in K$).
4. From ${}a\geq 0$ and ${}b\geq 0$, one may deduce ${}ab\geq 0$ (for any ${}a,b\in K$).