# 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.