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.