Timeline for What is the bottom of a directed-complete partial order of groups?
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| S Jun 6, 2016 at 8:30 | history | suggested | Martin Sleziak |
changed (lattices) to (lattice-theory) (See the tag-info; the tag lattices is for lattices in number theory.)
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| Jun 6, 2016 at 7:58 | review | Suggested edits | |||
| S Jun 6, 2016 at 8:30 | |||||
| Jun 4, 2016 at 14:24 | comment | added | Sean Lawton | @YemonChoi Regardless, thank you for the good suggestions. They certainly do make the question better. I made your suggested edits. | |
| Jun 4, 2016 at 14:22 | history | edited | Sean Lawton | CC BY-SA 3.0 |
Made suggestions of another editor.
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| Jun 4, 2016 at 14:12 | comment | added | Sean Lawton | @YemonChoi Why not just further improve it? | |
| Jun 4, 2016 at 14:05 | history | rollback | Yemon Choi |
Rollback to Revision 1
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| Jun 4, 2016 at 14:03 | comment | added | Yemon Choi | Moreover, there seems to me to be no semantic rationale for putting one of the questions in blockquote format and the other one not. This is not just gratuitous editing, it is incosistent editing. I have rolled back to Ben Sprott's original. | |
| Jun 4, 2016 at 14:02 | comment | added | Yemon Choi | @SeanLawton your change to the title has actually made it incorrect. One should speak of a DCPO of groups, not of "DCPO groups". Moreover, why not insert the definite or indefinite article before the word "bottom" if you feel these changes are so necessary as to warrant bumping this question? | |
| Jun 4, 2016 at 13:36 | history | edited | Sean Lawton | CC BY-SA 3.0 |
Minor edits to formatting, explained acronym.
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| Feb 25, 2011 at 14:19 | comment | added | ADL | Is the Largeness order not a dcpo (search for `the concept of largeness in group theory'; bascially $H\leq G$ if there exists a finite-index subgroup of G which maps onto a finite-index subgroup of H, and the kernel is finite)? Here free groups (and groups with a finite index subgroup which map onto free groups) sit at the top, while the trivial group lies at the bottom. | |
| Feb 25, 2011 at 12:34 | comment | added | aaron | Yes, I'm confused. Do you say $G < H$ if $G$ is a subgroup of $H$? Or if $H$ is a subgroup of $G$? If it is the former then everything less than a free group will be free. If it is the latter then you can definitely have other interesting things that are less than a free group. | |
| Feb 25, 2011 at 6:01 | comment | added | HJRW | Well, it's maximal if you think that a subgroup is greater than the group... which seems like an odd convention. | |
| Feb 25, 2011 at 4:33 | comment | added | Ben Sprott | Subgroups of free groups are free, but a free group can be the subgroup of a non-free group. Thus, the free groups have to be at the top and free is kind of maximal. Freeness is maximal, not minimal. Is that a good statement? | |
| Feb 25, 2011 at 3:44 | comment | added | HJRW | Perhaps you're looking for the Nielsen--Schreier Theorem, which asserts that any subgroup of a free group is free? | |
| Feb 25, 2011 at 3:42 | comment | added | aaron | Well, subgroups of free groups are free, so if you have a minimal element lying below a free group then the minimum will be free. (I think I must be misunderstanding the question . . . I wouldn't expect there to be a minimum most of the time.) | |
| Feb 25, 2011 at 3:39 | comment | added | Ben Sprott | The order relation in the dcpo is usually defined by subgroup and I think free groups have no subgroups with added axioms (ie, added relations)...I guess I'm lost. Thank you for your help. | |
| Feb 25, 2011 at 3:32 | comment | added | Ben Sprott | Okay, take any dcpo of groups with a free group as an element and at least one other element related to the free group either by subgroup or inclusion. I am trying to decide if free groups end up as maximal or minimal. | |
| Feb 25, 2011 at 0:11 | comment | added | François G. Dorais | See en.wikipedia.org/wiki/Complete_partial_order | |
| Feb 25, 2011 at 0:08 | comment | added | aaron | What does "dcpo" mean? | |
| Feb 24, 2011 at 23:34 | comment | added | François G. Dorais | Also, signatures (and greetings) aren't really necessary on MO. A signature is automatically attached to your posts. | |
| Feb 24, 2011 at 23:32 | comment | added | François G. Dorais | You should probably clarify what you mean by a typical dcpo of groups. Otherwise the bottom can be anything you want: just take the dcpo {G}... | |
| Feb 24, 2011 at 23:27 | history | asked | Ben Sprott | CC BY-SA 2.5 |