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positive cone
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(Definition)
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Let $R$ be a commutative ring with 1. A subset $P$ of $R$ is called a pre-positive cone of $R$ provided that
- $P+P\subseteq P$ ($P$ is additively closed)
- $P\cdot P\subseteq P$ ($P$ is multiplicatively closed)
- $-1\notin P$
- $\operatorname{sqr}(R):=\lbrace r^2\mid r\in R\rbrace \subseteq P.$
As it turns out, a field endowed with a pre-positive cone has an order structure. The field is called a formally real, orderable, or ordered field. Before defining what this ``order'' is, let's do some preliminary work. Let $P_0$ be a pre-positive cone of a field $F$ . By Zorn's Lemma, the set of pre-positive cones extending $P_0$ has a maximal element $P$ . It can be shown that $P$ has two additional properties:
- 5.
- $P\cup (-P)=F$
- 6.
- $P\cap (-P)=(0).$
Proof. First, suppose there is $a\in F-(P\cup (-P))$ . Let $\overline{P}=P+Pa$ . Then $a\in\overline{P}$ and so $P$ is strictly contained in $\overline{P}$ . Clearly, $\operatorname{sqr}(F)\subseteq \overline{P}$ and $\overline{P}$ is easily seen to be additively closed. Also, $\overline{P}$ is multiplicatively closed as the equation $(p_1+q_1a)(p_2+q_2a)=(p_1p_2+q_1q_2a^2)+(p_1q_2+q_1p_2)a$ demonstrates. Since $P$ is a maximal and
$\overline{P}$ properly contains $P$ , $\overline{P}$ is not a pre-positive cone, which means $-1\in \overline{P}$ . Write $-1=p+qa$ . Then $q(-a)=p+1\in P$ . Since $q\in P$ , $1/q=q(1/q)^2\in P$ , $-a=(1/q)(p+1)\in P$ , contradicting the assumption that $a\notin -P$ . Therefore, $P\cup (-P)=F$ .
For the second part, suppose $a\in P\cap (-P)$ . Since $a\in -P$ , $-a\in P$ . If $a\neq 0$ , then $-1=a(-a)(1/a)^2\in P$ , a contradiction. 
A subset $P$ of a field $F$ satisfying conditions 1, 2, 5 and 6 is called a positive cone of $F$ . A positive cone is a pre-positive cone. If $a\in F$ , then either $a\in P$ or $-a\in P$ . In either case, $a^2\in P$ . Next, if $-1\in P$ , then $1\in -P$ . But $1=1^2\in P$ , we have $1\in P\cap (-P)$ , contradicting Condition 6 of $P$ .
Now, define a binary relation $\leq$ , on $F$ by: $$a\leq b\Longleftrightarrow b-a\in P$$ It is not hard to see that $\leq$ is a total order on $F$ . In addition, with the additive and multiplicative structures on $F$ , we also have the following two rules:
- $a\leq b \Rightarrow a+c\leq b+c$
- $0\leq a$ and $0\leq b\Rightarrow 0\leq ab$ .
Thus, $F$ is a field ordered by $\leq$ .
Remark. Positive cones may be defined for more general ordered algebraic structures, such as partially ordered groups, or partially ordered rings.
- 1
- A. Prestel, Lectures on Formally Real Fields, Springer, 1984
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"positive cone" is owned by CWoo. [ full author list (2) ]
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Cross-references: partially ordered rings, partially ordered groups, algebraic structures, multiplicative, additive, addition, total order, binary relation, contradiction, contains, equation, contained, strictly, properties, maximal element, Zorn's lemma, ordered field, structure, order, field, multiplicatively closed, subset, commutative ring
This is version 7 of positive cone, born on 2004-10-29, modified 2009-03-21.
Object id is 6430, canonical name is PositiveCone.
Accessed 3197 times total.
Classification:
| AMS MSC: | 12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares ) | | | 13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings) |
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Pending Errata and Addenda
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