Definitions Ordered field

1 definitions

1.1 total order
1.2 positive cone
1.3 equivalence of 2 definitions

definitions

there 2 equivalent common definitions of ordered field. definition of total order appeared first historically , first-order axiomatization of ordering ≤ binary predicate. artin , schreier gave definition in terms of positive cone in 1926, axiomatizes subcollection of nonnegative elements. although latter higher-order, viewing positive cones maximal prepositive cones provides larger context in field orderings extremal partial orderings.

total order

a field (f, +, ×) total order ≤ on f ordered field if order satisfies following properties a, b , c in f:

if ≤ b + c ≤ b + c, and
if 0 ≤ , 0 ≤ b 0 ≤ × b.

the symbol multiplication henceforth omitted.

positive cone

a prepositive cone or preordering of field f subset p ⊂ f has following properties:

for x , y in p, both x + y , xy in p.
if x in f, x in p.
the element −1 not in p.

a preordered field field equipped preordering p. non-zero elements p form subgroup of multiplicative group of f.

if in addition, set f union of p , −p, call p positive cone of f. non-zero elements of p called positive elements of f.

an ordered field field f positive cone p.

the preorderings on f precisely intersections of families of positive cones on f. positive cones maximal preorderings.

equivalence of 2 definitions

let f field. there bijection between field orderings of f , positive cones of f.

given field ordering ≤ in first definition, set of elements such x ≥ 0 forms positive cone of f. conversely, given positive cone p of f in second definition, 1 can associate total ordering ≤p on f setting x ≤p y mean y − x ∈ p. total ordering ≤p satisfies properties of first definition.