What does compatible mean?
Wikipedia defines ordered fields in the following way:
An ordered field is a field together with a total ordering of its elements that is compatible with the field operations, i.e.:
A field $(F, + ,\cdot)$ together with a total order $\le$ on $F$ is an ordered field if the order satisfies the following properties:
if $a\le b$ then $a+c \le b+c$
if $0\le a$ and $0 \le b$ then $0 \le a\cdot b$
My question is what is the exactly definition of compatibility (if there is one)? In the multiplication case shouldn't we have if $a \le b$ then $a\cdot c \le b\cdot c$ following the sum one?
Thanks
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$\begingroup$The exact definition in this case is what you recalled.
No, you would not want that for the multiplication, as it is not what one wants or has in say the real numbers. $3 \le 5$ yet $(-1)3 > (-1)5$. It would even lead to absurdity right away: we have $-1 \le 1$. So $(-1)(-1) \le (-1)1$ and $1 \le -1$.
You could however rephrase the condition for multiplication that is given in the equivalent form that $a \le b$ implies $ca \le cb$ for $c \ge 0$.
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