Properties of Ordered Group
Jump to navigation
Jump to search
Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group with identity $e$.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
The following properties hold:
Ordering Compatible with Group Operation is Strongly Compatible
| \(\ds \forall x, y, z \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ z \preccurlyeq y \circ z\) | |||||||||||
| \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds z \circ x \preccurlyeq z \circ y\) | ||||||||||||
| \(\ds x \prec y\) | \(\iff\) | \(\ds x \circ z \prec y \circ z\) | ||||||||||||
| \(\ds x \preceq y\) | \(\iff\) | \(\ds z \circ x \prec z \circ y\) |
Corollary
| \(\ds \forall x, y \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq y \circ x^{-1}\) | |||||||||||
| \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq x^{-1} \circ y\) | ||||||||||||
| \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ y^{-1} \preccurlyeq e\) | ||||||||||||
| \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds y^{-1} \circ x \preccurlyeq e\) |
| \(\ds \forall x, y \in G: \, \) | \(\ds x \prec y\) | \(\iff\) | \(\ds e \prec y \circ x^{-1}\) | |||||||||||
| \(\ds x \prec y\) | \(\iff\) | \(\ds e \prec x^{-1} \circ y\) | ||||||||||||
| \(\ds x \prec y\) | \(\iff\) | \(\ds x \circ y^{-1} \prec e\) | ||||||||||||
| \(\ds x \prec y\) | \(\iff\) | \(\ds y^{-1} \circ x \prec e\) |
Inversion Mapping Reverses Ordering in Ordered Group
| \(\ds \forall x, y \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \prec y^{-1} \preccurlyeq x^{-1}\) | |||||||||||
| \(\ds \forall x, y \in S: \, \) | \(\ds x \prec y\) | \(\iff\) | \(\ds y^{-1} \prec x^{-1}\) |
Corollary
| \(\ds \forall x \in G: \, \) | \(\ds x \preccurlyeq e\) | \(\iff\) | \(\ds e \preccurlyeq x^{-1}\) | |||||||||||
| \(\ds e \preccurlyeq x\) | \(\iff\) | \(\ds x^{-1} \preccurlyeq e\) | ||||||||||||
| \(\ds x \prec e\) | \(\iff\) | \(\ds e \prec x^{-1}\) | ||||||||||||
| \(\ds e \prec x\) | \(\iff\) | \(\ds x^{-1} \prec e\) |
Operating on Ordered Group Inequalities
If $x \prec y$ and $z \prec w$, then $x \circ z \prec y \circ w$.
If $x \prec y$ and $z \preccurlyeq w$, then $x \circ z \prec y \circ w$.
If $x \preccurlyeq y$ and $z \prec w$, then $x \circ z \prec y \circ w$.
If $x \preccurlyeq y$ and $z \preccurlyeq w$, then $x \circ z \preccurlyeq y \circ w$.
Power Function Preserves Ordering in Ordered Group
Let $n \in \N_{>0}$ be a strictly positive integer.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Then the following hold:
| \(\ds \forall x, y \in S: \, \) | \(\ds x \preccurlyeq y\) | \(\implies\) | \(\ds x^n \preccurlyeq y^n\) | |||||||||||
| \(\ds \forall x, y \in S: \, \) | \(\ds x \prec y\) | \(\implies\) | \(\ds x^n \prec y^n\) |
where $x^n$ denotes the $n$th power of $x$.