Timeline for Non-free projective pearls (general and Abelian)
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2020 at 1:17 | history | edited | YCor |
edited tags
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| May 13, 2017 at 16:34 | answer | added | Keith Kearnes | timeline score: 3 | |
| Mar 28, 2017 at 22:48 | vote | accept | Włodzimierz Holsztyński | ||
| Mar 28, 2017 at 17:03 | history | edited | Arturo Magidin | CC BY-SA 3.0 |
restored the original "Obviously" to better context (it's not immediately clear from the final edit where the "obviously" was).
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| Mar 27, 2017 at 23:35 | answer | added | Arturo Magidin | timeline score: 5 | |
| Mar 26, 2017 at 1:57 | comment | added | Arturo Magidin | I haven't thought it through, but perhaps one way to get around the examples I have would be to require $f(S)=T$ rather than simple containment. It might destroy other desirable properties, though. | |
| Mar 25, 2017 at 22:03 | comment | added | Włodzimierz Holsztyński | The notion is simple. The potential for this to be known or approximately known is high. Nevertheless, I've never seen anything like this (until I have written it myself). Sorry. | |
| Mar 25, 2017 at 21:04 | comment | added | Qfwfq | Any reference to some literature in which this notion (pearl) or related notions appear? | |
| Mar 25, 2017 at 19:46 | comment | added | Arturo Magidin | Sure, I can do that; it'll be later today, though. | |
| S Mar 25, 2017 at 11:13 | history | suggested | Rahman. M |
tag editing...
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| Mar 25, 2017 at 10:56 | review | Suggested edits | |||
| S Mar 25, 2017 at 11:13 | |||||
| Mar 25, 2017 at 8:42 | comment | added | Włodzimierz Holsztyński | @ArturoMagidin, I appreciate your answers. I have a very hard time reading mathematics, and especially comments. I'd like you to present your material in one or more Answers--it'd help me a lot. | |
| Mar 25, 2017 at 8:40 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
a clearer statement
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| Mar 25, 2017 at 7:07 | comment | added | Arturo Magidin | Actually, I'm not sure that free pearls are projective. Consider the free pearl $(F_2,\{x\})$, where $F_2$ is the free (abelian) group on $x$ and $y$. Now consider the epimorphism $f\colon (F_2,\{y\})\to(F_2,\{x,y\})$ induced by the identity map on $F_2$, and likewise $g\colon (F_2,\{x\})\to (F_2,\{x,y\})$. If $(F_2,\{x\})$ were projective, then there would be a homomorphism $h\colon (F_2,\{x\}) \to (F_2,\{y\})$ such that $fh = g$. But then the map on underlying sets satisfies $\mathrm{id}\circ h = \mathrm{id}$, so $h=\mathrm{id}$; yet we also need $h(x) = y$, which is clearly impossible. | |
| Mar 25, 2017 at 6:29 | answer | added | Arturo Magidin | timeline score: 6 | |
| Mar 25, 2017 at 6:15 | comment | added | Arturo Magidin | On the other hand, the underlying group homomorphism must be surjective: suppose that $f\colon (G,S)\to (H,T)$ is a morphism and $f(G)\neq H$. Then there exists a group $K$ and embeddings $g,h\colon H\to K$ such that $gf=hf$ but $g\neq h$ (e.g., the free amalgamated product, or the construction in Linderholm's 1970 paper in the Monthly); in particular, neither $g(T)$ nor $h(T)$ contain the identity, so we can take the pearl $(K,g(T)\cup h(T))$ to get pearl maps $g=h$ such that $gf=hf$ but $g\neq h$. Hence, epimorphisms must be surjective on the group but not on the set. | |
| Mar 25, 2017 at 6:09 | comment | added | Arturo Magidin | I don't think the induced function on the sets must be a surjection. Consider a group $G$ and sets $S\subsetneq T$ such that both $(G,S)$ and $(G,T)$ are pearls. Then $i\colon (G,S)\to(G,T)$ induced by the identity on $G$ is an epimorphism: if $f,g\colon (G,T)\to(K,V)$ are pearl morphisms with $fi=gi$, then for all $x\in G$ we have $f(x) = fi(x) = gi(x) = g(x)$, hence $f=g$. But the induced function $i\colon S\to T$ is not onto. | |
| Mar 24, 2017 at 22:52 | comment | added | Włodzimierz Holsztyński | I've coined name pearl because a group is like gem, while its subset $\ S\ $ is like impurity. (BTW, profound mathematics always has this combination of a gem and, necessarily, of certain impurity, or else it's too sweet). | |
| Mar 24, 2017 at 22:46 | comment | added | Włodzimierz Holsztyński | "Possibly, the induced function S→T always must be a surjection."--I've meant a function induced by a pearl epimorphism. | |
| Mar 24, 2017 at 22:34 | comment | added | Włodzimierz Holsztyński | I mean epimorphism in the general abstract sense of the theory of categories, i.e. right cancellable morphism. My projective object definition here also follows the classical definition of the (abstract) theory of categories. *** I can only guess that in general pearl epimorphisms don't induce surjections (i.e. onto) of the sets of elements of the respective groups. Possibly, the induced function $\ S\rightarrow T\ $ always must be a surjection. I am not a specialist hence my guesses do not have much weight. | |
| Mar 24, 2017 at 22:19 | comment | added | Arturo Magidin | Do you know if pearl epimorphisms are surjective on underlying sets (on the group, or on either component)? (I take "epimorphism" to refer to "right cancellable morphism", and projective to be relative to epimorphisms rather than strong epimorphisms or surjections). | |
| Mar 24, 2017 at 22:12 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
motivation
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| Mar 24, 2017 at 21:59 | comment | added | Włodzimierz Holsztyński | @DavidHandelman, I'll expand the Question above. | |
| Mar 24, 2017 at 21:47 | comment | added | David Handelman | What is the motivation for studying pearls? | |
| Mar 24, 2017 at 21:23 | history | edited | Włodzimierz Holsztyński | CC BY-SA 3.0 |
epimorphism remark
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| Mar 24, 2017 at 21:19 | history | undeleted | Włodzimierz Holsztyński | ||
| Mar 24, 2017 at 21:10 | history | deleted | Włodzimierz Holsztyński | via Vote | |
| Mar 24, 2017 at 21:03 | history | asked | Włodzimierz Holsztyński | CC BY-SA 3.0 |