Group ordered

In general Algebra, a group ordered is the data of a unit $\backslash \; mathcal\; \{G\}$, provided with a Law of composition interns (noted $\backslash \; star$ in the article) conferring to him a structure of group, and a Relation of order (noted $\backslash \; leq$ in the article) compatible with the law of group.

Definition of the compatibility of the order with the law of group

More precisely, with the preceding notations, one says that the relation of order $\backslash \; leq$ is compatible with the law $\backslash \; star$ if, for all elements $x$, $y$ and $g$ of the group $\backslash \; mathcal\; \{G\}$, the relation $x\; \backslash \; Leq\; y$ involves the relations $g\; \backslash \; star\; X\; \backslash \; Leq\; G\; \backslash \; star\; y$ and $x\; \backslash \; star\; G\; \backslash \; Leq\; there\; \backslash \; star\; g$.

Properties

• In a group ordered $\backslash \; mathcal\; \{G\}$, for all elements $x$, $y$, $x\text{'}$ and $y\text{'}$, the inequalities $x\; \backslash \; Leq\; y$ and $x\text{'}\; \backslash \; Leq\; y\text{'}$ involve the inequality $x\; \backslash \; star\; x\text{'}\; \backslash \; Leq\; there\; \backslash \; star\; y\text{'}$.

In light, one can compose member with member of the of the same inequalities direction . Indeed, according to the definition, the inequality $x\; \backslash \; Leq\; y$ involves $x\; \backslash \; star\; x\text{'}\; \backslash \; Leq\; there\; \backslash \; star\; x\text{'}$. In the same way, the inequality $x\text{'}\; \backslash \; Leq\; y\text{'}$ involves $y\; \backslash \; star\; x\text{'}\; \backslash \; Leq\; there\; \backslash \; star\; y\text{'}$. One concludes by transitivity from the relation from order.

• In a group ordered $\backslash \; mathcal\; \{G\}$, for all elements respective opposite $x$ and $y$, for the law $\backslash \; star$, $x^\; \{-\; 1\}$ and $y^\; \{-\; 1\}$, the inequality $x\; \backslash \; Leq\; y$ involves the inequality $y^\; \{-\; 1\}\; \backslash \; Leq\; x^\; \{-\; 1\}$.

In light, one can pass contrary in an inequality by changing some the direction . To see it, it is enough, in the inequality $x\; \backslash \; Leq\; y$, to compose by $y^\; \{-\; 1\}$ on the left and by $x^\; \{-\; 1\}$ on the right.

Group completely ordered

One calls group completely ordered to an ordered group whose relation of order is total.

Examples

• the additive group $(\backslash \; mathbb\; \{Z\},\; +)$ of the relative entireties, provided with the usual relation of order, is a completely ordered abelian Groupe.

• the multiplicative group $\backslash \; left\; (\backslash \; mathbb\; \{R\}\; \_+^*,\; \backslash \; times\; \backslash \; right)$ of strictly positive realities is another completely ordered abelian Groupe.

• But the multiplicative group $\backslash \; left\; (\backslash \; mathbb\; \{R\}\; ^*,\; \backslash \; times\; \backslash \; right)$ of real nonnull is not an ordered group. Indeed, there is for example $-2\; \backslash \; Leq\; 2$, but while passing contrary, there is $\backslash \; frac\; \{1\}\; \{2\}\; >\; \backslash \; frac\; \{1\}\; \{2\}$. That is to be connected to the fact that the Fonction reverses is decreasing on $\backslash \; mathbb\; \{R\}\; \_+^*$, but not on $\backslash \; mathbb\; \{R\}\; entire\; ^*$.

See too

• Group

• Relation of order
• total Order
• Body ordered
• Axiom of Archimedes

Random links:

Kipawa | Gothic jargon | Iaşi | Blackville (parish) | Genji era | Le_théorème_de_Rado

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org