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Subgroup wth morphism into itself

  1. Jul 4, 2009 #1
    given a (semi)group [tex]A[/tex] and a sub-(semi)group [tex]S\leq A[/tex], I want to define a morphism [tex]f:A\rightarrow A[/tex] such that [tex]f(s)\in S[/tex], for every [tex]s \in S[/tex].
    Essentially it is an ordinary morphism, but for the elements in [tex]S[/tex] it has to behave as an endomorphism.
    Is this a known concept? does it have already a name? or can it be expressed more compactly?
    Last edited: Jul 4, 2009
  2. jcsd
  3. Jul 5, 2009 #2
    I've not heard of such a morphism. But note that I'm not a seasoned mathematician. I'm just curious about what you would like to show. Of course, the identity mapping restricted to S would be an example of the kind of mapping that you want to construct.

    Are you trying to make an analogue of ideals for rings?
  4. Jul 6, 2009 #3
    ...it seems, the example you gave of a "homomorphism on S which behaves as an identity-mapping on an ideal K" has in fact a name: retract homomorphism

    see: http://books.google.fi/books?id=Bmy...o7jJBQ&sa=X&oi=book_result&ct=result&resnum=4

    What I want to achieve is slightly weaker:
    I want to define a homomorphism [tex]f:S\rightarrow S[/tex] on a semigroup [tex](S,*)[/tex] such that for a given sub-semigroup K of S, one has [tex]x*f(x)=k[/tex] (for every [tex]x\in K[/tex]) where k is a fixed element (not necessarily the identity). Note that if k was the identity f would be the inversion operator.
    Last edited: Jul 6, 2009
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