There are two definitions of an ordered field: you can define it as a field with a total ordering and certain axioms relating the ordering to the field operations, or as a field with a "positive cone": a subset of the field satisfying certain axioms (see Orderered Field on Wikipedia). These definitions are equivalent: the total orders and the positive cones are in one-to-one correspondence.
I am wondering if the same equivalence holds for rings. I'm taking the definition of a totally ordered ring from Wikipedia, and I am taking the same definition for a positive cone as is given for fields: a positive cone on a ring $R$ is a subset $P$ of $R$ satisfying
- $P$ is closed under $+$ and $\cdot$.
- $x^2 \in P$ for all $x \in R$.
- $-1 \notin P$.
- either $x \in P$ or $-x \in P$ for all $x \in R$.
I have checked that the function $F$ taking a ring total order $\leq$ to the set $\{x \in R | x \geq 0\}$ and the function $G$ taking a positive cone $P$ to the relation $x \leq y \equiv y - x \in P$ are inverses.
I have also shown that if $\leq$ is a ring total order then $F(\leq)$ is a positive cone. However, I don't see how to show that if $P$ is a positive cone then $G(P)$ is a total order. In particular, I don't see how to show that $G(P)$ is anti-symmetric without using cancellation.
So I am wondering whether this equivalence holds or not. If not, is there a simple tweak to the positive cone style definition that makes it work? I'm interested because I think it is more natural to implement this style of definition in the algebra software I'm designing.