Timeline for Enumerate all valid orders of subset sums
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| Aug 11, 2022 at 18:18 | comment | added | 喜欢算法和数学 | @xskxzr Let order $o$ start with $\emptyset$. A trivial condition for $o$ to be a valid order is for any $4$ sets $A, B, C, D$ in $o$ such that $A\cap B=C\cap D=\emptyset$, $A\prec_o B$ and $C\preceq_o D$, we also have $A\cup C\prec_o B\cup D$. Are you aware of an invalid order $o$ that satisfies that trivial condition? | |
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| Feb 24, 2020 at 15:14 | comment | added | HEKTO | Chain is a totally ordered subset of a poset. Maximum chain is a chain, which is not contained in any other chain. Totally ordered set consists of a single maximum chain. | |
| Feb 24, 2020 at 5:47 | comment | added | xskxzr | @HEKTO I now understand the relation between subsets and Boolean algebra, but how can this help solve the problem? I don't know how to clarify since I don't know the definition of "chain" and I have a sense that your "chain" is not equivalent to my "valid order". Maybe it is more proper to say that I'm essentially seeking for a method to enumerate all possible total orders as the values of $x_1,\ldots,x_n$ vary. I think the question statement is clear though it does not mention any concept about boolean lattice. | |
| Feb 24, 2020 at 4:04 | comment | added | HEKTO | If you want to know what happens with chains when the vector $X$ varies - then you should clarify your question. The set of subsets of a finite set is Boolean algebra and Boolean lattice (= $n$-dimensional hypercube), where each node is a Boolean $n$-tuple | |
| Feb 24, 2020 at 1:51 | comment | added | xskxzr | @HEKTO Yes, I'm essentially seeking for a method to enumerate all possible chains as the values of $x_1,\ldots,x_n$ vary. However, I don't understand how this problem is related to the Boolean algebra since this problem talks about orders of subsets. | |
| Feb 24, 2020 at 1:44 | history | edited | xskxzr | CC BY-SA 4.0 | added 204 characters in body |
| Feb 23, 2020 at 23:10 | comment | added | HEKTO | As far as I understand, you are asking about Boolean Lattice (please see: en.wikipedia.org/wiki/Boolean_algebra_(structure)), where you introduce a total order using an additional vector $X=(x_1,x_2,...,x_n)$. Your "valid order" is used to be called a chain. It's unclear why you define this total order using existence of $X$ only - it will depend on values in $X$. Also for totally ordered set there will be only one chain (provided that there are no pairwise equal elements). | |
| Feb 22, 2020 at 14:46 | history | edited | xskxzr | CC BY-SA 4.0 | edited body |
| Feb 22, 2020 at 5:59 | history | edited | xskxzr | CC BY-SA 4.0 | edited body |
| Feb 22, 2020 at 5:23 | answer | added | D.W.♦ | timeline score: 0 | |
| Feb 22, 2020 at 2:16 | history | asked | xskxzr | CC BY-SA 4.0 |