Subgroup wth morphism into itself

1. Jul 4, 2009

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?

Last edited: Jul 4, 2009
2. Jul 5, 2009

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?

3. Jul 6, 2009

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

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.