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.
Other articles related to "ordered field, field, ordered, fields":
... 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 ...
... 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 ...
... 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 (16081674)
“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 (18091865)