Questions tagged [ordered-fields]
Ordered fields are fields which have an additional structure, a linear order compatible with the field structure. This tag is for questions regarding ordered fields and their properties, as well proofs related to un-orderability of certain fields.
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Does field isomorphism preserve order relation?
Let $(F,+_\mathrm{f},\cdot_\mathrm{f},<_\mathrm{f})$ and $(G,+_\mathrm{g},\cdot_\mathrm{g},<_\mathrm{g})$ be two ordered fields. Let $f:F\to G$ be a field isomorphism. (So, $f$ is bijective, and ...
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Zeno's Monoid: Has anyone got a reference for this?
Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F:
x*y = x+y - xy/k
Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element.
Moreover ...
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Permutations and partial orders
Consider the set of all permutations of $n$ numbers $\mathfrak{S}(n)$. Each permutation can be seen as a total ordering relation of $n$ elements $a_1,...,a_n$, such that
\begin{equation}
a_{\pi(1)}<...
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Notation $\text{RSpec}(K)$ for space of orderings of a field $K$
I'm reading notes from commutative algebra by Pete Clark.
In one of the examples of Galois connections he introduces notation $\text{RSpec}(K)$ for the set of all total orders on a field $K$, and ...
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Does every partially ordered commutative ring admit an order preserving homomorphism to a real closed field?
I hope the answer is yes. It would suffice to construct a homomorphism to a formally real field. I think it would then suffice to extend the partial order to a total ring order, but I'm not sure if ...
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Characterzation of the complex numbers
There is a characterization of the real number system：complete ordered field. And the complete ordered field is unique up to isomorphism. I'm trying to charaterise the complex numbers in a similar way....
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Non-lattice ordered fields that are not totally ordered
For the purposes of this question, totally ordered fields are ordered fields in which every element is comparable under the ordering: $x \geq y$ or $y \geq x$. Thus in this terminology, in an ordered ...
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Different intervals in an ordered vector space over $\mathbb{Q}$
Consider an ordered vector space $V$ over $\mathbb{Q}$ in the usual language of ordered vector spaces $(<,0,-,+,\lambda\cdot)_{\lambda \in \mathbb{Q}}$. For the duration of this question, the word ...
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Is it possible to replace hyperreal numbers with "good enough" alternatives?
The hyperreal numbers are undoubtedly interesting, generalizable, and have many nice properties, but are they really needed to solve the problems they solve? Would other, smaller fields work too?
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What is the order of a function in distribution theory?
Let $\mathscr{D}(\mathcal{O})$ be the space of basic functions, $\phi$, which is equipped
with convergence, and where $\phi$ vanish outside of the set $\mathcal{O}$.
Let $\mathscr{D}'(\mathcal{O})$ be ...
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Ordering of $\mathbb{R}[X]$ such that $0 < X < a$ for all positive numbers $a$
If $\mathbb{R}[X]$ denotes the field of real valued rational functions, then according to Bochnak, Coste and Roy's Real Algebraic Geometry, the unique ordering for which $X > 0$ and $X < a$ for ...
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Real-closed fields are the existentially closed ordered fields
While reading the proof in Shorter Hodges (Thm 7.4.4) that the theory of real-closed fields has quantifier-elimination, I came across the following claim (paraphased):
"Let $A$ and $B$ be real-...
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Compatibility of formally real extensions of an ordered field
My question
If $E$ is a formally real extension of an ordered field $F$, does $E$ always admit an ordering compatible with $F$?
Less ambitious: What if $E$ is a formally real simple algebraic ...
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Positive subset $\mathbb{R}_{>0}$ of $\mathbb{R}$ satisfying the order property
A field $\mathbb{F}$ is an ordered field if $\exists P\subseteq\mathbb{F}$ such that: 1. $\forall a,b\in P$, $a+b,ab\in P$. 2. Only one of the following is true $\forall a\in\mathbb{F}$, 1) $a\in P$ 2)...
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Adjoining greatest and smallest elements to uncountable ordered field
As titled. With $\mathbb{R}$, we can adjoin $+\infty$ and $-\infty$ as the greatest and smallest elements and define $\overline{\mathbb{R}}:=[-\infty, +\infty]$. But if we have an ordered field $\...
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