# Non-linear cyclic order of a field

A cyclic order of a group is non-linear if any cut of it is not compatible with the group operation.

A cut of a cyclically ordered set is a linear order $$<$$ such that

• $$a < b < c \implies [a, b, c]$$

for any elements $$a$$, $$b$$, $$c$$ of the set.

A cut of a cyclically ordered group is compatible with the group operation ($$+$$) iff:

• $$a < b \implies a + x < b + x$$ and $$x + a < x + b$$

for any elements $$a$$, $$b$$, $$x$$ of the group.

I am trying to find properties of a non-linear cyclic order on fields:

1. Is there an infinite field with a non-linear cyclic order on the additive group?
2. Is there a field with a non-linear cyclic order on the additive group such that $$[0, 1, -1]$$ and $$[0, x, 1]$$ for some element $$x$$?
• I assume all elements z = -z are non-negative and non-positive. – Alex C Mar 20 '17 at 21:36
• The standard compatibility with addition is required. Any compatibility with multiplication is optional. – Alex C Mar 21 '17 at 2:13
• If we are not requiring compatibility with multiplication, then there's little to no point in talking about rings or fields; the question should just ask about cyclically ordered abelian groups. – Hurkyl Mar 21 '17 at 14:48
• I don't believe cyclically orders are well-known; it would help if you give a definition of the term, and related terms. For example, what is a "nonnegative" element in this context? The term doesn't appear anywhere in Wikipedia's article. – Hurkyl Mar 21 '17 at 14:52
• 1. The compatibility with multiplication is not the same as for addition. It is adjusted to fit certain algebraic structures. I don't see any problems adjusting the compatibility for the strict cyclic order. 2. Nonnegative means not negative. The standard definition of a negative element a: $(0, -a, a)$. – Alex C Mar 21 '17 at 14:57

By picking any irrational real number $\alpha$, there is an injective map
$$\mathbb{Q} \to \mathbb{T} : x \mapsto e^{2 \pi i \alpha x}$$
and thus (?) $\mathbb{Q}$ inherits a cyclic ordering from the one on $\mathbb{T}$. Since, in the comments, you say you only care about compatibility with addition, I imagine this serves as an example to both of your questions.