Subgroup wth morphism into itself

[mnb96]

Hello,
given a (semi)group $$A$$ and a sub-(semi)group $$S\leq A$$, I want to define a morphism $$f:A\rightarrow A$$ such that $$f(s)\in S$$, for every $$s \in S$$.
Essentially it is an ordinary morphism, but for the elements in $$S$$ 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?

 

[guildmage]

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?

 

[mnb96]

...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 $$f:S\rightarrow S$$ on a semigroup $$(S,*)$$ such that for a given sub-semigroup K of S, one has $$x*f(x)=k$$ (for every $$x\in K$$) where k is a fixed element (not necessarily the identity). Note that if k was the identity f would be the inversion operator.