Timeline for Indecomposable weirdos (cnt.)
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2020 at 1:00 | vote | accept | Wlod AA | ||
| May 20, 2020 at 10:48 | history | edited | Wlod AA | CC BY-SA 4.0 |
No removing
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| May 20, 2020 at 10:10 | history | edited | Wlod AA | CC BY-SA 4.0 |
(...)
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| May 20, 2020 at 8:50 | comment | added | YCor | Actually, if $X$ is a cancelative magma, and $u,v$ are permutations of $X$, the new magma with law $(x,y)\mapsto u(x)v(y)$ is cancelative as well, defining an E-structure. If $X$ is commutative and $u$ is a single permutation, then $(x,y)\mapsto u(x)u(y)$ defines a W-structure. Even if $X$ is associative, it's not associative in general. So the most obvious examples of W-structures are those for which there is a commutative semigroup $X$ and a permutation $u$ of $X$ such that the law is given by $(x,y)\mapsto u(x)u(y)$. This covers all the given examples. | |
| May 20, 2020 at 8:42 | comment | added | YCor | @WlodAA It's even simpler. In both examples you mention (abelian, bimodules+ pair of invertible), the left and right multiplications are bijective (which is automatic for finite E-structures). So any indecomposable W-structure that for which it's not true yields an example. So even the monoid $\mathbf{N}$ works itself. | |
| May 20, 2020 at 8:28 | comment | added | Wlod AA | @YCor, you could write this in an explicit and complete way right here (? I'd welcome it). | |
| May 20, 2020 at 6:47 | comment | added | YCor | Actually an answer to this question was suggested in my previous answer: take a cancelative magma with the quasimedial property, in which the right multiplications are not all surjective, and modify arbitrary $\lambda$ outside the image of $(x,y)\mapsto (xy,y)$ (and similarly $\mu$ outside the image of $(x,y)\mapsto (x,xy)$). The answer here just plays this game in the case of the additive law on $\mathbf{N}$. | |
| May 20, 2020 at 5:10 | answer | added | Keith Kearnes | timeline score: 2 | |
| May 20, 2020 at 4:46 | comment | added | Wlod AA | A good question! | |
| May 20, 2020 at 4:43 | comment | added | Gerhard Paseman | What about the free one generated weirdo? Gerhard "It's Weird Enough For Me" Paseman, 2020.05.19. | |
| May 20, 2020 at 3:25 | review | Close votes | |||
| May 22, 2020 at 5:12 | |||||
| May 20, 2020 at 2:40 | history | asked | Wlod AA | CC BY-SA 4.0 |