Timeline for A family of subsets with a "gluing" property
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| Jun 14, 2010 at 9:37 | comment | added | Charles Matthews | A way of thinking is in terms of equations a = b, or to be clearer F(x) = F(y). If you give me as axioms such equalities for all pairs of elements of your connected sets, the question for me becomes the scope of what can be proved about F. So my description is: use the partition induced by the values of F being provably equal. Your remark is to do with the sets of axioms. So this is a kind of series of remarks about "equational logic", in my view. The gluing on overlaps is accurately modelled by the conjunction of F(x) = F(y) and F(y) = F(z), which gives F(x) = F(z). | |
| Jun 14, 2010 at 9:27 | comment | added | joshuahhh | (continuation) Any given connected set can either be built up from smaller sets or not, and these smaller sets don't have to be pairs. The obligation runs in the other direction: if smaller connected sets are arranged in the correct way, they entail the existence of larger connected sets. (Thanks a lot for taking the time to think about this stuff, by the way!) | |
| Jun 14, 2010 at 9:27 | comment | added | joshuahhh | Re: your edit -- "transitive closure" isn't quite an appropriate term to use. There is no relation here, other than perhaps the relation "a~b if {a,b} is connected". And that relation is not at all fundamental, since we can have interesting families without any pairs present. Take F2={singletons, {1,2,3}, {2,3,4}, {1,2,3,4}}. And if you mean to say that each maximal connected set can be built from smaller sets via gluing, that's not true either... Take the example I called F above. (continued) | |
| Jun 14, 2010 at 8:12 | history | edited | Charles Matthews | CC BY-SA 2.5 |
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| Jun 14, 2010 at 8:01 | comment | added | joshuahhh | Oh, and also note that if {a,b} and {b,c} are connected, that doesn't imply that {a,c} is connected (as you may have assumed, based on your mention of transitivity?). All that glueability implies is that {a,b,c} is connected. | |
| Jun 14, 2010 at 7:58 | comment | added | joshuahhh | I'm not sure I follow you, but if you're saying that the only possibility is for the set S to be partitioned into subsets and a given set to be "connected" iff it is/(is a subset of) one of those subsets, that's not right. For instance, we might have S={1,2,3}, F={{1},{2},{3},{1,2},{1,2,3}}... But please let me know if I am misunderstanding your point. | |
| Jun 14, 2010 at 7:44 | history | answered | Charles Matthews | CC BY-SA 2.5 |