Timeline for are endomorphisms "small" compared to the full transformations?
Current License: CC BY-SA 4.0
24 events
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| May 13, 2023 at 15:38 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 7, 2019 at 4:01 | vote | accept | T. Amdeberhan | ||
| Feb 4, 2017 at 5:01 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 5, 2017 at 16:01 | comment | added | Jan-Christoph Schlage-Puchta | @Benjamin Steinberg: The question by Schein and Teclezghi is about monotonic convergence, not just convergence. Using analytic estimates monotonicty can often only be shown for large $n$, and dealing with the smaller values can be a serious pain. | |
| Jan 4, 2017 at 17:40 | answer | added | Jan-Christoph Schlage-Puchta | timeline score: 2 | |
| Jan 1, 2017 at 22:57 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 1, 2017 at 18:53 | comment | added | Benjamin Steinberg | @YCor, the second interpretation makes no sense. Every automorphism of $T_n$ is inner. This must have led the OP to speculate that the endomorphism monoid would also embed in the full transformation semigroup, but that doesn't make sense. | |
| Jan 1, 2017 at 18:16 | comment | added | YCor | I think I have no idea what you're asking. Your answer to my comment seems to suggest you also have no idea what I'm asking. I understood the first interpretation in Steven's comment, and don't get what the second interpretation is. Since some other people seems to have understood the question, they might clarify the question in more universal mathematical language? | |
| Jan 1, 2017 at 18:15 | comment | added | Benjamin Steinberg | The paper gives the values for small n and the sequence appears to go to zero. | |
| Jan 1, 2017 at 18:09 | comment | added | Benjamin Steinberg | The fact that $S_n$ is the automorphism group is trivial because automorphisms preserve constant maps. Schein et al give a description that would not seem to give an embedding. | |
| Jan 1, 2017 at 18:05 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 1, 2017 at 18:04 | comment | added | Benjamin Steinberg | This count is not requiring the identity to be preserved. | |
| Jan 1, 2017 at 18:04 | comment | added | Benjamin Steinberg | This is a question of Schein and TECLEZGHI who computed the endomorpism monoid ams.org/journals/proc/1998-126-09/S0002-9939-98-04764-9/… | |
| Jan 1, 2017 at 17:03 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 1, 2017 at 17:03 | comment | added | T. Amdeberhan | It's funny you said that, I'd be amusing with the $\infty$. | |
| Jan 1, 2017 at 16:55 | comment | added | Steven Stadnicki | I'm prepared to be surprised, but I would naively expect it to be exactly the opposite, that $\#End(T_n)/\#T_n\to\infty$. | |
| Jan 1, 2017 at 16:53 | comment | added | T. Amdeberhan | @YCor: For example, $\# End(T_2)=7, \# End(T_3)=40$. | |
| Jan 1, 2017 at 16:52 | comment | added | T. Amdeberhan | @StevenStadnicki: No, you're not mistaken. I fixed the title. | |
| Jan 1, 2017 at 16:50 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 1, 2017 at 16:30 | comment | added | Steven Stadnicki | I'm still a little confused here - do you mean the endomorphisms of $T_n$, or the endomorphisms within $T_n$? Your question makes it seem like the latter (you ask about the 'ratio' of endomorphisms to all transformations) but you haven't imposed any structure on $[n]$ that you're morphing. Otherwise, perhaps it's the hour but I'm just not seeing how you have $End(T_n)\subseteq T_n$ in the first place. | |
| Jan 1, 2017 at 16:29 | comment | added | YCor | Do you require $1\mapsto 1$? when not required, it's not true for endomorphisms of arbitrary monoids. | |
| Jan 1, 2017 at 16:17 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
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| Jan 1, 2017 at 16:14 | comment | added | YCor | "Let $T_n$ be the monoid/semigroup of self-maps of $[n]$". Just to make clear you mean endomorphisms of $T_n$ as a semigroup (still there is an ambiguity on whether you require endomorphisms to map identity to itself). | |
| Jan 1, 2017 at 16:10 | history | asked | T. Amdeberhan | CC BY-SA 3.0 |