Timeline for A non-matroidal notion of dependence on a set of ideals
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2012 at 0:41 | comment | added | Hailong Dao | Ah, sorry, I was thinking union. | |
| Oct 16, 2012 at 21:18 | comment | added | Thomas Kahle | @Hailong + Patricia: It is the ideal generated by the union, so prime avoidance can't be applied necessarily. As in the example: $(x+y) \subset (x) + (y)$. | |
| Oct 16, 2012 at 19:39 | comment | added | Patricia Hersh | @Hailong: I think the sum which Thomas is using is not the same thing as "union". | |
| Oct 16, 2012 at 3:30 | history | edited | Noah Stein | CC BY-SA 3.0 |
Changed "well-defined" to "well-studied" because it seems the technical definition of well-defined was not intended.
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| Oct 16, 2012 at 3:08 | comment | added | Hailong Dao | Silly comment: Prime Avoidance states that if the $I_i$s are prime ideals, then they form an independent set if and only if they are pairwise incomparable. I only mentioned it because Prime Avoidance is somewhat subtle and widely used in commutative algebra. | |
| Oct 16, 2012 at 2:38 | comment | added | Patricia Hersh | Indeed I was thinking of having a generator for each poset element and then associating to each poset element the ideal generated by its generator as well as those below it. I didn't think this was what you were actually trying to get at though and can see how the pair wise in comparability assumption might be handy. On the other hand, in the poset case I mention, your independent sets would be the poset anti chains, which certainly are something that some people study -- e.g. Trying to determine size of largest antichain. | |
| Oct 16, 2012 at 2:08 | comment | added | Thomas Kahle | Yes that would be possible using ideals generated by variables, I think. It's probably unrelated, though. For simplicity we may assume that the original ideals are all pairwise containment incomparable. (that is not the case in the non-matroid example, but one could come up with another example). Then what is the poset you are thinking of? Is it the poset of all possible sums of those ideals? | |
| Oct 16, 2012 at 0:24 | comment | added | Patricia Hersh | It seems like for every finite poset $P$, you could cook up a family of ideals with $P$ giving exactly the containment relations among your ideals and with no other dependencies. Is that right or am I misunderstanding your definition? Thanks. | |
| Oct 15, 2012 at 23:55 | history | asked | Thomas Kahle | CC BY-SA 3.0 |