Timeline for Ordered groups - examples
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2020 at 12:20 | review | Close votes | |||
| Feb 17, 2020 at 3:01 | |||||
| Feb 9, 2020 at 11:58 | comment | added | YCor | (...) In particular, a group admits a nontrivial bi-invariant partial order iff it has a nonempty conjugation-invariant subsemigroup not containing $1$. This is obviously satisfied if it has a $\mathbf{Z}$ quotient, as Misha already said. On the other hand, this does not exist in the infinite dihedral group $D_\infty$. (Side remark: it also follows that a group admits a nontrivial left-invariant partial order iff it is not torsion.) | |
| Feb 9, 2020 at 11:58 | comment | added | YCor | In a group $G$, there is a canonical bijection between the set of left-invariant partial preorders $\le$ and the set of submonoids of $G$. It is given by $\le\mapsto S_\le$ and $S\mapsto \le_S$, where $S_\le=\{g:g\ge 1_G\}$ and $g\le_S h$ iff $g^{-1}h\in S$. Then (a) $\le$ is nontrivial iff $S_\le\neq\{1_G\}$; (b) $\le$ is a order (i.e., antissymetric) iff $S\cap S^{-1}=\{1_G\}$; (c) $\le$ is bi-invariant iff $S$ is conjugation-invariant; (d) $\le$ is total iff $S\cup S^{-1}=G$. (...) | |
| Feb 9, 2020 at 11:45 | history | edited | YCor | CC BY-SA 4.0 |
added tag, added missing information
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| May 7, 2013 at 14:21 | history | edited | Lee Mosher |
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| Apr 24, 2013 at 20:57 | comment | added | Misha | Every BS-group $G$ admits an epimorphism $f$ to ${\mathbb Z}$. Pull-back of the order on ${\mathbb Z}$ will give a partial order on $G$ after you declare that distinct elements $x, y\in G$ satisfying $f(x)=f(y)$ are not comparable. | |
| Apr 24, 2013 at 15:34 | comment | added | Alain Valette | Oops, thank you Yves, indeed my remark on $Homeo^+(\mathbb{R})$ is false. One day it will be possible to edit comments on MO. | |
| Apr 24, 2013 at 15:25 | comment | added | YCor | @Alain: a countable group admits a left-invariant ordering iff it embeds in $Homeo^+(\mathbf{R})$. This proves that the (positive!) affine group of $\mathbf{R}$ is left-orderable. But it is indeed bi-orderable by a general fact on group extensions, here using the fact that the action of the multiplicative group of positive reals on the additive group of reals preserves an ordering. | |
| Apr 24, 2013 at 14:39 | comment | added | YCor | It is safe to write bi-orderable group instead of ordered group. Because orderable groups denote, according to authors, either left-orderable or bi-orderable groups and is thus blatantly ambiguous. Also left/bi-ordered group is more suitable to mean a group endowed with a (left or bi...)-invariant total order. For instance, there is a Chabauty space of bi-ordered groups on $k$ generators, which surjects onto the Chabauty space of bi-orderable groups on $k$ generators, which is a closed subset of the space of marked groups on $k$ generators. | |
| Apr 24, 2013 at 14:12 | history | edited | Nick Gill |
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| Apr 24, 2013 at 13:01 | answer | added | Dietrich Burde | timeline score: 2 | |
| Apr 21, 2013 at 16:54 | comment | added | Alain Valette | Follow-up to Qiaochu's comment: two answers posted on math.stackexchange.com/questions/365660/ordered-groups-examples I found the one by Tournesol interesting. Recall that a group of orientation-preserving homeo's of $\mathbb{R}$ admits a total, bi-invariant ordering (with $f\leq g$ iff $f(x)\leq g(x)$ for every $x$). Since $B(m,1)$ embeds in the affine group of $\mathbb{R}$, it admits such an ordering. | |
| Apr 19, 2013 at 5:34 | comment | added | bsog | To Noah: Note that I assume that G is non-abelian. If m=n=1, then G is abelian and the question is trivial. | |
| Apr 19, 2013 at 5:19 | comment | added | bsog | Of course, I mean a non-trivial partial order. | |
| Apr 19, 2013 at 2:26 | comment | added | Fernando Muro | No exercises here, please. | |
| Apr 18, 2013 at 21:26 | comment | added | Noah Schweber | Taking $m, n=1$ has a natural non-trivial ordering. I think you should sharpen your question. | |
| Apr 18, 2013 at 20:34 | comment | added | Włodzimierz Holsztyński | Do you mean a non-trivial partial order? | |
| Apr 18, 2013 at 20:02 | comment | added | Qiaochu Yuan | Crossposted at math.SE: math.stackexchange.com/questions/365660/ordered-groups-examples | |
| Apr 18, 2013 at 18:39 | history | asked | bsog | CC BY-SA 3.0 |