Timeline for Residual finiteness of groups versus residual finiteness of semigroups
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2010 at 2:22 | vote | accept | dave | ||
| Aug 2, 2010 at 23:01 | comment | added | Tsuyoshi Ito | Thanks for the comment! I agree that this fact is easy to prove, and there is no real need to consult a textbook on semigroups. However, because the questioner asked for a good reference, I wanted to make clear that I did not provide any reference in my answer. | |
| Aug 2, 2010 at 22:21 | comment | added | Pete L. Clark | I don't think a reference to the literature on semigroups is required here. By definition, a semigroup homomorphism is a map such that $\varphi(xy) = \varphi(x) \varphi(y)$ for all $x,y \in G$. But this is also the definition of a group homomorphism! The point is that, for groups, that this implies that the identity gets sent to the identity and that inverses get sent to inverses. If one has not done so before, it is a good exercise to think about why for ring homomorphisms, in contrast, one must require explicitly that $1 \mapsto 1$. | |
| Aug 2, 2010 at 20:49 | history | edited | Tsuyoshi Ito | CC BY-SA 2.5 |
fixed typo
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| Aug 2, 2010 at 20:26 | history | answered | Tsuyoshi Ito | CC BY-SA 2.5 |