Timeline for Karolyi's theorem for finite groups and its extensions
Current License: CC BY-SA 3.0
9 events
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| Jun 12, 2013 at 8:33 | comment | added | Salvo Tringali | [...] $\mathbb A=(A,\ast)$ is a (commutative) magma, and $Y\ast Y=\{z_0\}$, but $p(\mathbb A)\ge n$, so the general "conjecture" is disproved for $n \ge 2$ and $|Y| \ge 2$. Notice that $\mathbb A$ is non-associative for $n\ge 2$, since then $(z_{n-2}\ast z_{n-2})\ast z_{n-1}=z_0$, while $z_{n-2}\ast(z_{n-2}\ast z_{n-1})=z_{n-1}$. It is then natural to ask whether the general "conjecture" holds true if the ambient magma is associative. I will update the OP to summarize this exchange for others. | |
| Jun 12, 2013 at 8:16 | comment | added | Salvo Tringali | So your 2nd sentence was referring to $(2,2)$-deficient magmas. Désolé, je ne l'avais pas entendu ! And thank you for the elegant example, which in turn implies a counterexample to the general "conjecture". For, let $\{X,Y\}$ be a non-trivial partition of $A$ with $n:=|X|<\infty$, and $(z_i)_{i\in\alpha}$ a numbering of $A$ s.t. $X=\{z_0,\ldots,z_{n-1}\}$, where $\alpha$ is the ordinal of $A$. We define a binary operation $\ast$ on $A$ as follows: For $i,j\in \alpha$ we take $z_i\ast z_j:=z_0$ if $\max(i,j)\ge n$ and $z_i\ast z_j:=z_{(\max(i,j)+1)\,\bmod\,n}$ otherwise. Then, [...] | |
| Jun 11, 2013 at 23:53 | comment | added | Gerhard Paseman | Let A(n) be the number of all magmas on an n element set, D(n) the number of deficient magmas, and T(n) the number of those having a two element subset B such that BB has two or fewer elements. D(n)/A(n) goes to 0 as n grows. T(n)/A(n) does not go to zero as n grows. For a class of deficient magmas with no elements e such that ee=e, pick a large subset B of A that does not include z in A, let BB be the singleton set z, fill the rest of the diagonal so that ee = e does not occur for any e, and then complete the table as you wish. Gerhard "That Is One Classy Example" Paseman, 2013.06.11 | |
| Jun 11, 2013 at 22:51 | comment | added | Salvo Tringali | Then thanks for the hints; now, your point is clear to me too. So, let me ask a couple more of questions. You wrote that "[...] being deficient turns out to be relatively rare", but one line later I read that you "[...] discovered those magmas are not rare". Should I consider as correct the 1st or the 2nd sentence? If the 2nd sentence is the right one, then it's likely that you had found out a way to construct a lot of deficient magmas, didn't you? Then, let me repeat my question from the above: Would you mention a class of examples of deficient magmas with no non-trivial idempotents? | |
| Jun 11, 2013 at 22:21 | comment | added | Gerhard Paseman | Also, my first sentence might be throwing you off. Let us postulate an answer Phi to your second question. Phi might look like "finite magmas with the Cauchy-Davenport property are abundant in the following probabilistic sense...", and may help you with you quest, as well as being a partial result. My answer above does not talk about Phi. My answer talks about something weakly related, which may or may not help you find Phi in the general algebra literature. Gerhard "Good Luck In Your Search" Paseman, 2013.06.11 | |
| Jun 11, 2013 at 22:10 | comment | added | Gerhard Paseman | To personalize your Q2 and say what do I know about partial results regarding your question in the case of general algebraic structures, the answer is nothing. The closest I can come to is this notion of deficiency, which has ties with probability and equational logic. The hope is that someone wrote on your concept and studied some of what I mentioned and is citing Murskii or Quackenbush. Then a citation index might aid you. It is slim, but it is what I have to offer. Gerhard "My Purse Is Not Trash" Paseman, 2013.06.11 | |
| Jun 11, 2013 at 22:02 | comment | added | Salvo Tringali | [...] "conjecture", but this is somewhat trivial. So let us focus on the case of $(m,n)$-deficient magmas with no non-trivial idempotents: Are you suggesting that these could verify/provide a counterexample for the general "conjecture"? Would you mention a class of examples of deficient magmas with no non-trivial idempotents? Also, in which sense the work that you're referring to at the end of your answer should hopefully fully address Q2, considering that it's not even known, for what I can say, whether or not it holds for infinite non-commutative groups? | |
| Jun 11, 2013 at 21:49 | comment | added | Salvo Tringali | I don't quite understand what exactly you're claiming. If I take it correctly, I'd say that, given $m,n\in\mathbb N$ with $m\le n$, an $(m,n)$-deficient magma is a magma $\mathbb A=(A,\ast)$ for which there exists $X\subseteq A$ with $m\le |X|\le n$ s.t. $|X|\ge |X\ast X|$; in particular, you're looking at the case where $3\le m$, $n<|A|<\infty$. So, e.g., if $\mathbb A$ is a zero-left or zero-right sgrp (and hence a band), then it's $(m,n)$-deficient as long as $n\le |A|$. And it's clear that if $\mathbb A$ has at least one non-trivial idempotent then it does certainly satisfy the [...] | |
| Jun 11, 2013 at 20:35 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |