One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies:
- For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds.
- For all $x,y,z \in F$, if $x<y$ and $y<z$, then $x<z$.
- For all $x,y,z \in F$, if $x<y$, then $x+z < y+z$.
- For all $x,y,z \in F$, if $x<y$ and $0<z$, then $x \cdot z < y \cdot z$.
When $F$ is the real numbers $\mathbb R$, the usual "less than" is the only relation that satisfies all four axioms. I expect that all four axioms are necessary to know this: if we drop one of them, then other relations may "sneak in". There are easy constructions for two of the axioms:
- If we drop axiom 1, then we can declare that $x<y$ for all $x,y \in \mathbb R$: this satisfies the other three axioms.
- If we drop axiom 4, then we can declare that $x<y$ when $x$ is greater than $y$ in the usual sense: this satisfies the other three axioms.
Are there easy constructions showing that the other two axioms are necessary?
(Again, I want to stick with the real numbers. There are interesting examples over finite fields; for example, when $F = \mathbb F_3$, let $x<y$ exactly when $y=x+1$, and axioms 1,3,4 hold (but not 2). I am not interested in these right now.)