Timeline for What are the monoids in which every globally idempotent subsemigroup contains the identity element?
Current License: CC BY-SA 3.0
11 events
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| Nov 3, 2012 at 23:12 | vote | accept | Michał Masny | ||
| Nov 2, 2012 at 20:24 | history | edited | Michał Masny | CC BY-SA 3.0 |
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| Nov 2, 2012 at 20:12 | history | edited | Michał Masny | CC BY-SA 3.0 |
added 267 characters in body
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| Nov 2, 2012 at 2:48 | answer | added | Victor | timeline score: 5 | |
| Oct 30, 2012 at 0:08 | answer | added | Benjamin Steinberg | timeline score: 2 | |
| Oct 29, 2012 at 23:23 | comment | added | Gerhard Paseman | As a semigroup under addition, Z is not one generated. And it is true that some of what you are interested in lacks torsion. However, if none of the one generated subsemigroups is globally idempotent, then you have no torsion at all, and that may tell you something about your picture. For general semigroups, I don't know what the answer is, but for commutative finitely generated examples, you may have something like a classification of f.g. Abelian groups possible. That last is a guess, but it is worthy of consideration. Gerhard "Also, Could Be Globally Wrong" Paseman, 2012.10.29 | |
| Oct 29, 2012 at 22:41 | comment | added | Michał Masny | @Gerhard Oh, sorry. You're right. Something went wrong in my brain. But still, the second example proves they don't have to be torsion, right? (Sorry, all my confidence has just disappeared.) | |
| Oct 29, 2012 at 22:08 | comment | added | Gerhard Paseman | In your first example, what two elements add to 1? Gerhard "Ask Me About Adding Ones" Paseman, 2012.10.29 | |
| Oct 29, 2012 at 21:05 | comment | added | Michał Masny | @Gerhard I don't understand. Isn't $(\mathbb N\setminus\lbrace0\rbrace,+)$ both one generated and globally idemopotent? In fact, $(\mathbb Z,+)$ isn't torsion, and if $S$ is its globally idempotent subsemigroup, then if all elements of $S$ have the same sign, $S$ contains $0$, by the same argument as for $\mathbb N$. If there are $x,y\in S$ with $xy<0$ then $0=|y|x+|x|y\in S.$ So $\mathbb Z$ also has this property. | |
| Oct 29, 2012 at 20:15 | comment | added | Gerhard Paseman | Probably torsion, since every one generated subsemigroup which is globally idempotent is finite. Of course, the situation isn't quite that simplistic, but that seems to be a good starting point. Gerhard "Ask Me About System Design" Paseman, 2012.10.29 | |
| Oct 29, 2012 at 20:07 | history | asked | Michał Masny | CC BY-SA 3.0 |