# Ordered body

In algebra , a sub-discipline of mathematics , an ordered body (also called an arranged body ) is a body together with a total order " " that is compatible with addition and multiplication. The best known example is the field of real numbers . Bodies of the characteristic cannot be arranged in a structurally compatible manner. An important example of a field of characteristic 0, which cannot be arranged in a structurally compatible way, is the field of complex numbers . ${\ displaystyle \ leq}$ ${\ displaystyle p> 0}$

## definition

The property ${\ displaystyle x

A body on which a total order is defined is called an ordered body (or arranged body ) if the order is compatible with body operations, i.e. H. if the following axioms of arrangement apply to all of them : ${\ displaystyle (K, +, \ cdot)}$ ${\ displaystyle \ leq}$${\ displaystyle a, b, c}$${\ displaystyle K}$

• From follows .${\ displaystyle a \ leq b}$${\ displaystyle a + c \ leq b + c}$
• Out and follows .${\ displaystyle 0 \ leq a}$${\ displaystyle 0 \ leq b}$${\ displaystyle 0 \ leq a \ cdot b}$

Instead of the second condition, the following can also be required:

• Out and follows .${\ displaystyle a \ leq b}$${\ displaystyle 0 \ leq c}$${\ displaystyle a \ cdot c \ leq b \ cdot c}$

Elements that are greater than or equal to are called positive , elements that are less than or equal to are called negative . ${\ displaystyle 0}$${\ displaystyle 0}$

The positive area is then defined as the set of all positive elements, i.e. h .: . ${\ displaystyle P}$${\ displaystyle P = \ left \ {k \ in K \ mid k \ geq 0 \ right \}}$

One can show that for is equivalent to , that is , the arrangement is clearly determined by its positive range. ${\ displaystyle a, b \ in K}$ ${\ displaystyle a \ leq b}$${\ displaystyle ba \ in P}$

A positive area fulfills the properties

• ${\ displaystyle P + P \ subseteq P}$, (Closeness with regard to addition and multiplication)${\ displaystyle P \ cdot P \ subseteq P}$
• ${\ displaystyle P \ cap \ left (-P \ right) = \ left \ {0 \ right \}}$
• ${\ displaystyle P \ cup \ left (-P \ right) = K}$

A preorder is a subset , which ${\ displaystyle T}$${\ displaystyle T \ subseteq K}$

• ${\ displaystyle T + T \ subseteq T}$, (Closeness with regard to addition and multiplication)${\ displaystyle T \ cdot T \ subseteq T}$
• ${\ displaystyle T \ cap \ left (-T \ right) = \ left \ {0 \ right \}}$
• ${\ displaystyle K ^ {2} \ subseteq T}$

Fulfills.

A preorder is weaker than an order and only defines a partial relation on the body.

## properties

The property ${\ displaystyle a> 0 \ land x

These properties follow from the axioms (for all ): ${\ displaystyle a, b, c, d \ in K}$

• The negative of a positive element is negative and the negative of a negative element is positive: For each with either or .${\ displaystyle a \ in K}$${\ displaystyle a \ neq 0}$${\ displaystyle -a <0 ${\ displaystyle a <0 <-a}$
• Inequalities can be added: From and it follows .${\ displaystyle a \ leq b}$${\ displaystyle c \ leq d}$${\ displaystyle a + c \ leq b + d}$
• One can multiply inequalities with positive elements: From and it follows . (Alternatively, as shown above, this can also be required as an axiom.)${\ displaystyle a \ leq b}$${\ displaystyle 0 \ leq c}$${\ displaystyle ac \ leq bc}$
• Squares are not negative: . Likewise, every finite sum of squares is nonnegative. In particular is .${\ displaystyle 0 \ leq a ^ {2}}$${\ displaystyle 0 <1}$
• By induction we can conclude that any finite sum of ones is positive .${\ displaystyle 0 <1 + 1 + \ cdots +1}$

## Structural statements

Every ordered body has the characteristic . This follows directly from the latter property . ${\ displaystyle 0}$${\ displaystyle 0 <1 + 1 + \ cdots +1}$

Every part of an ordered body is ordered. As for every field of characteristic 0, the smallest contained field is isomorphic to the rational numbers , and the order on this sub-field is the same as the natural order on . ${\ displaystyle \ mathbb {Q}}$

If every element of an arranged field lies between two rational numbers, then the field is called Archimedean ordered (i.e. if there is a larger and a smaller rational number for each element). For example, the real numbers are Archimedean, but the hyper real numbers are non-Archimedean. The property of an ordered body to be ordered Archimedes is also called Archimedes axiom .

### Ordered solids and real numbers

Every Archimedean ordered body is (as an ordered body) isomorphic to a clearly defined sub-body . In this sense, the real numbers form the "largest" Archimedean ordered body. ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$

The order on a parent body induces a topology , the order topology to that through the open intervals and as sub-base is generated, and addition and multiplication are continuous with respect to this topology. ${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle \ {x \ in K \ mid x ${\ displaystyle \ {x \ in K \ mid x> a \}}$

An ordered body is called orderly complete when every limited subset of the body has an infimum and a supremum .

The field of real numbers can be characterized (except for isomorphism) by the following property:

${\ displaystyle \ mathbb {R}}$ is a fully ordered body.

Since exactly the nonnegative numbers are squares in the field of real numbers (it applies there if and only if a real number exists with ) the set of positive real numbers and thus the arrangement of all real numbers is algebraically determined (namely by means of the ring operations ). The rational numbers, which form a partial field and the prime field of the real numbers, do not allow any automorphism other than identity. It is said: the rational numbers are a rigid body . Also is rigid. So there is always exactly one ring isomorphism between two models of the real numbers and this is always an order-preserving body automorphism. The article " Real number " describes different ways of constructing such models. ${\ displaystyle x \ geq 0}$${\ displaystyle y}$${\ displaystyle y ^ {2} = x}$${\ displaystyle +, \ cdot}$${\ displaystyle \ mathbb {R}}$

→ More generally, bodies that only allow one body order for the reason given here are Euclidean bodies .

## Formally real bodies

A body is called formally real (or only real ) if it cannot be written as a finite sum of squares. One can show that this is the case if and only if the 0 can only be represented in a trivial way as a finite sum of squares. ${\ displaystyle -1}$

Every arranged body is therefore a formally real body. Conversely, an order can be introduced on every formally real body that makes it an arranged body. Formally real bodies can be expanded into real closed bodies.

## Examples and counterexamples

• The whole numbers and the natural numbers satisfy the axioms of arrangement, but not the axioms of the body . The whole numbers simply form an ordered integrity ring .
• The rational numbers form the smallest arranged body in the sense that they are part of every ordered body and do not contain any real part.${\ displaystyle \ mathbb {Q}}$
• The real numbers and each subfield of are arranged fields.${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$
• Every real closed field and, more generally, every Euclidean field , like the real numbers, only allows an arrangement that is uniquely determined by its algebraic structure.
• The hyper-real numbers are real closed and thus an arranged body that allows only one arrangement.
• The surreal numbers form a real class and not a set , but otherwise fulfill all the axioms of an ordered field. Any arranged body can be embedded in the surreal numbers.
• Finite bodies cannot be arranged.
• The complex numbers can not be arranged because the property by the imaginary unit because is violated.${\ displaystyle 0 \ leq a ^ {2}}$ ${\ displaystyle \ mathrm {i}}$${\ displaystyle \ mathrm {i} ^ {2} = - 1}$
• The -adic numbers cannot be arranged because they contain for a square root of and for a square root of .${\ displaystyle p}$${\ displaystyle p> 2}$${\ displaystyle 1-p}$${\ displaystyle p = 2}$${\ displaystyle -7}$

2. ↑ But not, for example, the body that lies between and (also close) and knows a non-trivial conjugation map. There is (in contrast to ) none here , so that , as a result, the positivity of cannot be proven with ring-theoretical means. The Euclidean solids are also rigid, e.g. B. the real closed field of algebraic real numbers.${\ displaystyle K: = \ mathbb {Q} ({\ sqrt {2}})}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle x \ in K}$${\ displaystyle x ^ {2} = {\ sqrt {2}}}$${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle \ mathbb {A} \ cap \ mathbb {R}}$