Timeline for An extension of Lagrange's theorem to semigroups?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2012 at 15:35 | comment | added | Benjamin Steinberg | You have to interpret Lagrange as a statement about the automorphism group of a transitive G-set to get the right generalization. | |
| Jun 15, 2012 at 13:28 | comment | added | Gil Kalai | Dear Salvo, Please state clearly in the body of the question what IS your question. (And not in later comments which are hard to understand.) | |
| Jun 15, 2012 at 12:44 | comment | added | Tara Brough | (I was typing that comment quite slowly, so didn't see the two preceding comments until afterwards.) | |
| Jun 15, 2012 at 12:43 | comment | added | Salvo Tringali | As for the rest, I'm not asking for (possible) extensions to arbitrary semigroups. And I don't expect that, if any non-trivial extension is possible, it looks exactly like Lagrange's theorem for groups. I'm just asking for any possible non-trivial extension that is already there, in the literature. Say, for instance, an extension to some interesting classes of semigroups. Apart from the -cypa- ones where the theorem sounds true by definition. I know, non-trivial and interesting are not well-defined terms. But let me believe in your common sense. | |
| Jun 15, 2012 at 12:41 | comment | added | Tara Brough | That's a good example. It's already clear from considering finite monogenic semigroups that the order of a subsemigroup of a 'Smarandache Semigroup' doesn't have to divide the order of the semigroup, but this example shows that nothing I would call an 'analogue of Lagrange's theorem' holds. | |
| Jun 15, 2012 at 12:36 | comment | added | Vladimir Dotsenko | OK, what is your motivation for this question then? I think this example shows that there is no serious general result to hope for. If you are saying that in the only somewhat general result the theorem holds by definition, what makes you hope that something meaningful does exist? | |
| Jun 15, 2012 at 12:31 | comment | added | Salvo Tringali | Yes, I mean that theorem. But Lagrange's theorem for Smarandache semigroups is not exactly what you're thinking of. In particular, it is not a statement about all the subsemigroups of a given semigroup. Indeed, it is not even about arbitrary Smarandache semigroups. Instead, it concerns a special class of them, which somebody calls Smarandache lagrangian semigroups. Thus, what they refer to as the Lagrange's theorem for Smarachande semigroups is essentially true by definition. And yes, I agree with you that this may be a -cypa- thing, but I'm not here to quibble in the value of others' work. | |
| Jun 15, 2012 at 12:09 | history | answered | Vladimir Dotsenko | CC BY-SA 3.0 |