Positive Set in an Ordered Integral Domain

[alexfloo]

"Positive Set" in an Ordered Integral Domain

I'm currently reading Durbin's Modern Algebra, and I have a question about the positive set in ordered integral domains.

Durbin characterizes an ordere integral domain as one with a subset which is closed to addition and multiplication, and satisfies trichotomy (exactly one of a=0, a$\in$D, or -a$\in$D holds for each a).

One of the exercises asked me to prove that in the case of the integers, this "positive set" is unique (and in particular that it is the natural numbers). Is it the case that the positive set is always unique, or are there ordered integral domains whose order can be characterized in multiple ways?

 

[Erland]

Is it the case that the positive set is always unique, or are there ordered integral domains whose order can be characterized in multiple ways?

Consider the integral domain of all polynomials with integer coefficients. This can be ordered in infinitely many ways, one for each transcendental real number. If t is a transcendental real number, for each integer polynomial P(x) we stipulate that P(x) in D iff P(t)>0. This gives different orderings for different transcendental numbers.