Timeline for Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Current License: CC BY-SA 4.0
18 events
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| Aug 13, 2021 at 16:47 | history | edited | LSpice | CC BY-SA 4.0 |
Links to @RichardStanley's comment and @HarryWest's answer
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| Aug 13, 2021 at 14:06 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 29, 2020 at 5:36 | vote | accept | Daniel Li | ||
| Oct 17, 2020 at 10:01 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 17, 2020 at 5:13 | comment | added | Tony Huynh | @BrendanMcKay Nice observation. I noticed the same thing. Answer is now updated. | |
| Oct 17, 2020 at 5:11 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 17, 2020 at 3:38 | comment | added | Brendan McKay | At the least the simple case works if the function values are in a group, not necessarily abelian. Just put the inverse on the left side for $\phi_2(x_3,x_4)$. | |
| Oct 17, 2020 at 2:32 | comment | added | Brendan McKay | I don't think the condition that the meet is discrete is essential. The same logic says that if it factors with respect to two partitions then it factors with respect to their meet. | |
| Oct 16, 2020 at 15:03 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 16, 2020 at 14:54 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 16, 2020 at 14:49 | comment | added | Tony Huynh | @BrendanMcKay I edited my answer accordingly, since what I wrote earlier was a special case of what you suggested. | |
| Oct 16, 2020 at 14:48 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 16, 2020 at 14:22 | comment | added | Tony Huynh | @BrendanMcKay Yes, the proof seems to work for an arbitrary pair of partitions whose meet is $\{1\} \dots \{n\}$. | |
| Oct 16, 2020 at 14:21 | comment | added | Tony Huynh | @RichardStanley Good question! I have to think about it. | |
| Oct 16, 2020 at 14:13 | comment | added | Brendan McKay | In the lattice of set partitions {1}{2}{3}{4} is the meet of {1,2}{3,4} and {1,3}{2,4}. So does it generalise to an arbitrary pair of partitions and their meet? | |
| Oct 16, 2020 at 14:03 | comment | added | Richard Stanley | Does the result hold in any commutative monoid (so inverses may not exist)? | |
| Oct 16, 2020 at 8:59 | history | edited | Tony Huynh | CC BY-SA 4.0 |
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| Oct 16, 2020 at 6:54 | history | answered | Tony Huynh | CC BY-SA 4.0 |