Name of this homomorphism like mapping edited:
I have a problem in which, a map $f: (S, \star)\rightarrow   (S', \cdot)$ such that $f(e)=e'$, where $e$ and $e'$ are identities of $S$ and $S'$, respectively,  and for all $x, y\in S$, we get $f(x\star y)=f(p)\cdot f(q)$, where $p, q\in S$ and $f(p), f(q)\in S'$. Can such mappings be called a homomorphism? If not, is there any name for such mappings?

Note $p$ and $q$ may be distinct from $x$ and $y$, respectively. Whereas, in standard definition of a monoid homomorphism, we have for all $x, y\in S$ implies that $f(x\star y)=f(x)\cdot f(y)$.

 A: [Posting more or less what I said in the comment thread, plus a bit extra.]
Any homomorphism $f : (S, \star) \to (S', \cdot)$ satisfies the property in your question: just take $p=x$ and $q=y$.
When $f$ is not injective, it's possible to have $f(x \star y) = f(p) \cdot f(q)$ with $p \ne x$ and $q \ne y$—as long as $f(x)=f(p)$ and $f(y)=f(q)$ you're good to go. For example, for example the homomorphism
$$f : (\mathbb{Z}, +) \to (\mathbb{Z}_2, +_2)$$
given by $f(n) = (n \bmod 2)$ for all $n \in \mathbb{Z}$ satisfies $f(1+5) = f(3) +_2 f(7)$.
However, if your definition is:

A function $f : (S, \star) \to (S', \cdot)$ is a [insert name of notion here] if, for all $x,y \in S$, there exist $p,q \in S$ such that $f(x \star y) = f(p) \cdot f(q)$.

Then such functions might not be homomorphisms. As a trivial example, define $f : (\mathbb{Z}_3, +_3) \to (\mathbb{Z}, +)$ by $f(0)=f(1)=0$ and $f(2)=1$. This is most certainly not a homomorphism, since for example
$$f(2 +_3 2) = f(1) = 0 \quad \text{but} \quad f(2) + f(2) = 1+1 = 2$$
But it satisfies the property in your question, since the only values of $f$ are $0=f(0)+f(0)$ and $1 = f(0)+f(2)$.
I can't see a reason why such functions would be useful, so I very much doubt they have a name.
