Timeline for Karolyi's theorem for finite groups and its extensions
Current License: CC BY-SA 3.0
16 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 13, 2013 at 23:41 | comment | added | Salvo Tringali | Here is another class for which the general "conjecture" holds: Strictly totally orderable magmas, where we say that $\mathbb A$ is strictly totally orderable if there exists a total order $\preceq$ on $A$ such that $a + c \prec b + c$ and $c + a \prec c + b$ for all $a,b,c \in A$ with $a \prec b$. Then, it seems reasonable to ask: What about torsion-free magmas? | |
| Jun 13, 2013 at 23:00 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Follow-ups to GH's 1st comment to the OP; deleted 5 characters in body; deleted 4 characters in body
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| Jun 13, 2013 at 22:28 | comment | added | Salvo Tringali | @GH. You were right. I followed your advice and wrote to Ruzsa. As far as groups are concerned, he has a theorem, whose proof doesn't rely on the Feit-Thompson theorem, which implies at once that Károlyi's theorem extends to arbitrary groups. So then, let me update the OP. | |
| Jun 13, 2013 at 8:41 | comment | added | GH from MO | Regarding Károlyi's result [1], I heard that Imre Ruzsa found a proof without the Feit-Thompson theorem. Of course I might be wrong, ask either Károlyi or Ruzsa if you are interested. | |
| Jun 12, 2013 at 20:52 | comment | added | Salvo Tringali | OK, but Smith's definition of a quasigroup is the same as the definition given on Wiki.en (en.wikipedia.org/wiki/Quasigroup). That is, a quasigroup (in the sense of Smith) is a magma in which every element is split, while it doesn't need to be cancellative. In any case, you're right, and my example is a non-unital, commutative, cancellative quasigroup (again, in the sense of Smith). | |
| Jun 12, 2013 at 19:49 | comment | added | Gerhard Paseman | Thank you for your three element example (which I believe is a quasigroup and not a loop). I was unsure about the notion of cancellative and thought it might relate to their being a one sided identity (sorry for the identity-unit confusion on my part). Since I am misremembering details, I'll say I think quasigroups and quasigroups with identity (loops) are cancellative magmas, or groups without necessarily being associative, and that I am using it as JDH Smith uses the term quasigroup, which should be standard usage somewhat. Gerhard "Needs Another Cup Of Coffee" Paseman, 2013.06.12 | |
| Jun 12, 2013 at 18:55 | comment | added | Salvo Tringali | As per your question, the answer is no. For, consider a set $S$ with three elements $a$, $b$ and $c$, and let $\diamond$ be the binary operation on $S$ given by the following (Cayley) table: $$\begin{array}{c|ccc} \diamond & a & b & c \\ \hline a & b & a & c \\ b & a & c & b \\ c & c & b & a \end{array}$$ Now, a magma $\mathbb A=(A,\ast)$ is cancellative if no row or column of its table contains repetitions, while an element $e \in A$ is a left (resp., right) identity (for $\mathbb A$) if it leaves unchanged its own row (resp., column) in the table of $\mathbb A$. | |
| Jun 12, 2013 at 17:40 | comment | added | Salvo Tringali | Thanks for the references. Just to be sure that we're speaking the same language: What do you mean by a quasigroup? Also, why do you mention that a finite cancellative magma should have a one sided unit? To wit, what's the link with the questions in the OP? Let it be as it may, if my guess is correct and you're using "unit" for "identity" (what else otherwise?), then I don't know the answer to your question. | |
| Jun 12, 2013 at 16:03 | comment | added | Gerhard Paseman | Also, I would like to point out that algebras that have "large" (i.e. more than 2 elements") subalgebras are deficient, and so algebras that are not deficient are a version of being "simple" in a subalgebra sense, which for groups relates to (but is not the same as) simple in the normal or congruential sense. If finite algebras interests you, reading the classic text on tame congruence theory by Hobby and McKenzie might inspire you. They use polynomial selfmaps to identify and classify minimal algebras. These might be C-D. Gerhard "Ask Me About System Design" Paseman, 2013.06.12 | |
| Jun 12, 2013 at 15:39 | comment | added | Gerhard Paseman | Based on our exchange, I have a bit more clarity about your questions. You might ask quasigroup theorists about a corresponding theorem. Also, for finite magmas, shouldn't cancellative imply at least a one sided unit? Gerhard "Ask Me About Structure Design" Paseman, 2013.06.12 | |
| Jun 12, 2013 at 8:47 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Update based on Gerhard Paseman's answer
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| Jun 11, 2013 at 21:03 | history | edited | GH from MO |
edited tags
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| Jun 11, 2013 at 20:35 | answer | added | Gerhard Paseman | timeline score: 1 | |
| Jun 11, 2013 at 18:10 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Forgot the attribute "cancellative"
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| Jun 11, 2013 at 18:00 | history | asked | Salvo Tringali | CC BY-SA 3.0 |