# Example of non-abelian partially ordered group

What is a simple example of a non-abelian partially ordered group?

• Question in the title and question in the question's body to not match. Apr 6 '13 at 18:20
• $S_3$ is nonabelian and if we give it the trivial ordering (every element is $\leq$ every other element) we're done. If you want a nontrivial ordering, let $x \leq y$ if $x$ and $y$ have the same sign. Apr 6 '13 at 18:23
• @Julien Clancy: This is not a partial order, because it is not an antisymmetric relation. Apr 8 '13 at 16:30
• @GejzaJenča See en.wikipedia.org/wiki/Partially_ordered_group Apr 8 '13 at 20:15
• @Julien Clancy I referenced to the second part of your comment. "To have the same sign" is clearly a nontrivial equivalence relation, so it is not antisymmetric. Apr 8 '13 at 20:43

All strictly increasing functions $\mathbb R\to \mathbb R$, equipped with composition as the group operation and with the usual partial order given by $f\leq g$ if and only if $f(x)\leq g(x)$ for every $x\in\mathbb R$. Note that this is a lattice ordered group.
Free group over two generators $F(a,b)$ is non-abelian and is poset w.r.t dictionary order induced by $a<b$.