Ordered field

if ${\displaystyle a\leq b}$ then ${\displaystyle a+c\leq b+c,}$ and

if ${\displaystyle 0\leq a}$ and ${\displaystyle 0\leq b}$ then ${\displaystyle 0\leq a\cdot b.}$

the field ${\displaystyle \mathbb {Q} }$ of with its standard ordering (which is also its only ordering);

rational numbers

the field ${\displaystyle \mathbb {R} }$ of with its standard ordering (which is also its only ordering);

real numbers

any subfield of an ordered field, such as the real or the computable numbers, becomes an ordered field by restricting the ordering to the subfield;

algebraic numbers

the field ${\displaystyle \mathbb {Q} (x)}$ of ${\displaystyle p(x)/q(x)}$, where ${\displaystyle p(x)}$ and ${\displaystyle q(x)}$ are polynomials with rational coefficients and ${\displaystyle q(x)\neq 0}$, can be made into an ordered field by fixing a real transcendental number ${\displaystyle \alpha }$ and defining ${\displaystyle p(x)/q(x)>0}$ if and only if ${\displaystyle p(\alpha )/q(\alpha )>0}$. This is equivalent to embedding ${\displaystyle \mathbb {Q} (x)}$ into ${\displaystyle \mathbb {R} }$ via ${\displaystyle x\mapsto \alpha }$ and restricting the ordering of ${\displaystyle \mathbb {R} }$ to an ordering of the image of ${\displaystyle \mathbb {Q} (x)}$. In this fashion, we get many different orderings of ${\displaystyle \mathbb {Q} (x)}$.

rational functions

the field ${\displaystyle \mathbb {R} (x)}$ of ${\displaystyle p(x)/q(x)}$, where ${\displaystyle p(x)}$ and ${\displaystyle q(x)}$ are polynomials with real coefficients and ${\displaystyle q(x)\neq 0}$, can be made into an ordered field by defining ${\displaystyle p(x)/q(x)>0}$ to mean that ${\displaystyle p_{n}/q_{m}>0}$, where ${\displaystyle p_{n}\neq 0}$ and ${\displaystyle q_{m}\neq 0}$ are the leading coefficients of ${\displaystyle p(x)=p_{n}x^{n}+\dots +p_{0}}$ and ${\displaystyle q(x)=q_{m}x^{m}+\dots +q_{0}}$, respectively. Equivalently: for rational functions ${\displaystyle f(x),g(x)\in \mathbb {R} (x)}$ we have ${\displaystyle f(x)<g(x)}$ if and only if ${\displaystyle f(t)<g(t)}$ for all sufficiently large ${\displaystyle t\in \mathbb {R} }$. In this ordered field the polynomial ${\displaystyle p(x)=x}$ is greater than any constant polynomial and the ordered field is not Archimedean.

rational functions

The field ${\displaystyle \mathbb {R} ((x))}$ of with real coefficients, where x is taken to be infinitesimal and positive

formal Laurent series

the

transseries

real closed fields

the

superreal numbers

the

hyperreal numbers

Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.

One can "add inequalities": if a ≤ b and c ≤ d, then a + c ≤ b + d.

One can "multiply inequalities with positive elements": if a ≤ b and 0 ≤ c, then ac ≤ bc.

"Multiplying with negatives flips an inequality": if a ≤ b and c ≤ 0, then ac ≥ bc.

If a < b and a, b > 0, then 1/b < 1/a.

Squares are non-negative: 0 ≤ a² for all a in F. In particular, since 1=1², it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 < 1.

An ordered field has 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc., and no finite sum of ones can equal zero.) In particular, finite fields cannot be ordered.

characteristic

Every non-trivial sum of squares is nonzero. Equivalently: ${\displaystyle \textstyle \sum _{k=1}^{n}a_{k}^{2}=0\;\Longrightarrow \;\forall k\;\colon a_{k}=0.}$^[3]

[2]

 – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb

Linearly ordered group

 – Group with a compatible partial orderPages displaying short descriptions of redirect targets

Ordered group

 – ring with a compatible total orderPages displaying wikidata descriptions as a fallback

Ordered ring

Ordered topological vector space

 – Vector space with a partial order

Ordered vector space

 – Ring with a compatible partial order

Partially ordered ring

 – Partially ordered topological space

Partially ordered space

 – Algebraic concept in measure theory, also referred to as an algebra of setsPages displaying short descriptions of redirect targets

Preorder field

 – Partially ordered vector space, ordered as a lattice

Riesz space

(1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001

Lam, T. Y.

(2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.

Lam, Tsit-Yuen

(1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001

Lang, Serge