(sub)semigroup with an endomorphism

[mnb96]

Let's have a semigroup S and a proper sub-semigroup B of S.
If we have also an endomorphism $$f:S\rightarrow S$$, is it possible that the subset $$B'=\{f(b)|b\in B \}$$ is not a sub-semigroup anymore?

 

[HallsofIvy]

The only requirement in order that a set, A, be a "sub-semigroup" of a semigroup S is that it be closed under the group operation. If x and y are in B', then x= f(a) and y= f(b) for some a, b in B'. But xy= f(a)f(b)= f(ab) is in B' because ab is in B.