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 ${\displaystyle x<y\Rightarrow a+x<a+y}$

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. ${\displaystyle a,b\in K}$ ${\displaystyle a\leq b}$ ${\displaystyle b-a\in P}$

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 ${\displaystyle a>0\land x<y\Rightarrow ax<ay}$

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 . ${\displaystyle a\in K}$ ${\displaystyle a\neq 0}$ ${\displaystyle -a<0<a}$ ${\displaystyle a<0<-a}$
• 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. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \{x\in K\mid x<a\}}$ ${\displaystyle \{x\in K\mid x>a\}}$

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. ${\displaystyle 0\leq a^{2}}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$

• More generally and for the same reason, an algebraically closed field can never be ordered.

• The -adic numbers cannot be arranged because they contain for a square root of and for a square root of . ${\displaystyle p}$ ${\displaystyle p>2}$ ${\displaystyle 1-p}$ ${\displaystyle p=2}$ ${\displaystyle -7}$

See also

In synthetic geometry , in the context of determining possible side divisions of the affine plane over a formally real body, all conceivable arrangements of such bodies are also classified by certain nontrivial square characters of the body. → See division of pages .

Web links

Wikibooks: Math for Non-Freaks: Axioms of Arrangement  - Learning and Teaching Materials

Wikibooks: Math for Non-Freaks: Conclusions of the Axioms of Arrangement  - Learning and Teaching Materials

Individual evidence

1. ↑ Manfred Knebusch, Klaus Schneiderer, Introduction to Real Algebra , Vieweg, 1989, ISBN 3-528-07263-6
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. ${\displaystyle K:=\mathbb {Q} ({\sqrt {2}})}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x\in K}$ ${\displaystyle x^{2}={\sqrt {2}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle \mathbb {A} \cap \mathbb {R} }$
3. ↑ Alexander Prestel, Charles N. Delzell, Positive Polynomials. From Hilbert's 17th Problem to Real Algebra , Springer, 2001