Timeline for When does a power semigroup have a zero, and what can the zero be?
Current License: CC BY-SA 3.0
12 events
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| Oct 25, 2012 at 12:12 | comment | added | Michał Masny | @Ben, thank you. Yes, I knew that. From the above it also follows easily that the answer is "yes" if $S$ doesn't have any idempotents apart from $1$. I believe the answer is "yes" for any pair $G,S,$ but I still do not see a proof. | |
| Oct 24, 2012 at 13:27 | comment | added | user6976 | @Ben: You are correct. I thought the OP probably could find it out himself. | |
| Oct 24, 2012 at 3:09 | comment | added | Benjamin Steinberg | Note if S is a finite monoid with group of units H, then P(S) has group of units H. So if P(S) is isomorphic to P(G) then G is isomorphic to H and for cardinality reasons S=G. So the question is easy if G is finite. | |
| Oct 23, 2012 at 15:46 | vote | accept | Michał Masny | ||
| Oct 23, 2012 at 14:07 | comment | added | user6976 | I think that looking for more and more complicated examples is not the way to prove what you want. I think you know enough to be able to prove it (or find a relatively easy counterexample). | |
| Oct 23, 2012 at 13:32 | comment | added | Michał Masny | Oh, sorry. I should have asked about semigroups without zero. | |
| Oct 23, 2012 at 13:14 | comment | added | user6976 | Take any semigroup $S$ with 0 such that $S\setminus \{0\}$ is not a subsemigroup, then $S$ is an example. | |
| Oct 23, 2012 at 12:24 | comment | added | Michał Masny | Thank you, I see. Could you give an example of a semigroup that has a subgroup as an ideal but isn't of this kind? Perhaps I should explain why I'm asking these questions. I'm trying to prove that if $S$ is a semigroup and $G$ is a group, then $P(S)\cong P(G)$ implies $S\cong G$. Clearly, if $P(S)\cong P(G)$, then $S$ has a subgroup as an ideal, so if I knew what such semigroups are, my job would be easier. | |
| Oct 23, 2012 at 11:56 | comment | added | user6976 | Your examples are like that where $T$ is the chain of idempotents ($\{e_1,...,e_n\}$, $e_ie_j=e_je_i=e_i$ if $i\le j$),and $\phi$ is a homomorphism that maps everything to 1. | |
| Oct 23, 2012 at 11:50 | comment | added | user6976 | Here is a general construction. Take any semigroup $T$, any group $G$ and any homomorphism $\phi: T\to G$. Consider $S$ the disjoint union of $T$ and $G$ with operation * that coincides with the native operations on $T,G$ and $t*g=\phi(t)g$, $g*t=g\phi(t)$ for every $g\in G$, $t\in T$. Then $S$ is a semigroup (the semilattice of $T$ and $G$) and $G$ is an ideal of $S$. These are not all semigroups with subgroups which are ideals, but it gives a lot of examples. | |
| Oct 23, 2012 at 11:44 | comment | added | Michał Masny | Right, thanks a lot! But which semigroups have such an ideal? Are there any examples other than the ones I mentioned? Can we characterize such semigroups? | |
| Oct 23, 2012 at 11:35 | history | answered | user6976 | CC BY-SA 3.0 |