Timeline for Closed submonoid of $(\mathbb{C}^*)^n$
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 12, 2020 at 16:56 | vote | accept | phdstud | ||
| Feb 21, 2020 at 3:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 21, 2020 at 23:38 | answer | added | user44191 | timeline score: 1 | |
| Jan 21, 2020 at 8:00 | comment | added | phdstud | @user44191:How do you transfer this to a torus as you mentionned earlier? | |
| Jan 20, 2020 at 22:09 | comment | added | user44191 | @MarkSapir Presumably, it means that the subset is closed under multiplication, i.e. it forms a monoid. | |
| Jan 20, 2020 at 20:43 | comment | added | phdstud | @Mark Sapir: I changed this mysterious set. Is it clearer now? | |
| Jan 20, 2020 at 20:42 | comment | added | phdstud | @YemonChoi: That's exactly what I meant, thank you! | |
| Jan 20, 2020 at 15:48 | comment | added | Yemon Choi | @user44191 if that is the case, then the answer to the problem stated in your comment is "yes" by invoking general macinery: such a submonoid would be a compact semigroup which is cancellative (ax=bx implies a=b and similarly on the left) and compact cancellative semigroups are known to be groups. In the case of tori this should be provable in a more direct way, I guess | |
| Jan 20, 2020 at 9:59 | comment | added | user44191 | With the new modification, I'm pretty sure the problem reduces to the question of whether every closed submonoid of a torus is a subgroup. | |
| Jan 20, 2020 at 9:37 | history | edited | phdstud | CC BY-SA 4.0 |
I tried to be clearer in the assumptions and I precised the multi-dimensional case
|
| Jan 19, 2020 at 22:41 | comment | added | Yemon Choi | @MarkSapir I agree that the OP should clarify some of his or her terminology, but I think a reasonable guess would be that he or she seeks a closed subset of $({\mathbb C}^*)^n$ with its usual topology, which is also a submonoid for the natural pointwise product on $({\mathbb C}^*)^n$ | |
| Jan 19, 2020 at 19:05 | review | Close votes | |||
| Jan 21, 2020 at 14:26 | |||||
| Jan 18, 2020 at 23:22 | comment | added | user44191 | @YCor Ah, I thought "closed" here was algebraic, not topological. That does make more sense. | |
| Jan 18, 2020 at 22:24 | comment | added | YCor | @user44191 This is not a closed subset. Indeed its log is the subsemigroup generated by $\log(2)$ and $-\log(3)$. But for any $u<0<v$ with $u/v$ irrational, the additive subsemigroup generated by $u$ and $v$ is not discrete (it's dense in $\mathbf{R}$). | |
| Jan 18, 2020 at 16:42 | history | edited | YCor | CC BY-SA 4.0 |
changed to standard terminology, changed tags
|
| Jan 18, 2020 at 12:06 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
missing dollars
|
| Jan 18, 2020 at 10:09 | comment | added | user44191 | How about $n = 1, S = 2^a 3^{-b}, (a, b) \in \mathbb{N}$? | |
| Jan 18, 2020 at 10:02 | history | asked | phdstud | CC BY-SA 4.0 |