An **ordered field** is a field *F* together with a positive cone *P*.

The preorderings on *F* are precisely the intersections of families of positive cones on *F*. The positive cones are the maximal preorderings.

Read more about Ordered Field: Properties of Ordered Fields, Examples of Ordered Fields, Which Fields Can Be Ordered?, Topology Induced By The Order, Harrison Topology

### Other articles related to "ordered field, field, ordered, fields":

**Ordered Field**

... For an example of an

**ordered field**that is not Archimedean, take the

**field**of rational functions with real coefficients ... that the leading coefficient of the denominator is positive.) To make this an

**ordered field**, one must assign an ordering compatible with the addition and multiplication operations ... Therefore, 1/x is an infinitesimal in this

**field**...

**Ordered Field**- Harrison Topology

... The Harrison topology is a topology on the set of orderings XF of a formally real field F ... Each order can be regarded as a multiplicative group homomorphism from F* onto ±1 ...

... See, for example, algebraically closed

**field**or compactification ... either the completeness of R as metric space or as a partially

**ordered**set (see below) ... In order theory and related

**fields**such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially

**ordered**set ...

... A real closed

**field**is a

**field**F in which any of the following equivalent conditions are true F is elementarily equivalent to the real numbers ... as the reals any sentence in the first-order language of

**fields**is true in F if and only if it is true in the reals ... There is a total order on F making it an

**ordered field**such that, in this ordering, every positive element of F is a square in F and any polynomial ...

**Ordered Field**

... Every linearly

**ordered field**K contains (an isomorphic copy of) the rationals as an

**ordered**subfield, namely the subfield generated by the multiplicative unit 1 of K, which in turn contains the ... The following are equivalent characterizations of Archimedean

**fields**in terms of these substructures ... is not the case when there exist infinite elements.) Thus an Archimedean

**field**is one whose natural numbers grow without bound ...

### Famous quotes containing the words field and/or ordered:

“What though the *field* be lost?

All is not lost; the unconquerable Will,

And study of revenge, immortal hate,

And courage never to submit or yield:

And what is else not to be overcome?”

—John Milton (1608–1674)

“The case of Andrews is really a very bad one, as appears by the record already before me. Yet before receiving this I had *ordered* his punishment commuted to imprisonment ... and had so telegraphed. I did this, not on any merit in the case, but because I am trying to evade the butchering business lately.”

—Abraham Lincoln (1809–1865)