A $1-1$ function is called injective. What is an $n-1$ function called? A $1-1$ function is called injective. What is an $n-1$ function called ?
I'm thinking about homomorphisms. So perhaps homojective ?
Onto is surjective. $1-1$ and onto is bijective.
What about n-1 and onto ? Projective ? Polyjective ?
I think $n-m$ and onto should be hyperjective as in hypergroups.

I will:


*

*suggest some terminology for three related concepts, and

*suggest that $n$-to-$1$ functions probably aren't very interesting.


Terminology.
Let $f : X \rightarrow Y$ denote a function. Recall that $f$ is called a bijection iff for all $y \in Y$, the set $f^{-1}(y)$ has precisely $1$ element. So define that $f$ is a $k$-bijection iff for all $y \in Y$, the set $f^{-1}(y)$ has precisely $k$ elements.
We have:

The composite of a $j$-bijection and a $k$-bijection is a $(j \times k)$-bijection.

There is also a sensible notion of $k$-injection. Recall that $f$ is called an injection iff for all $y \in Y$, the set $f^{-1}(y)$ has at most $1$ element. So define that $f$ is a $k$-injection iff for all $y \in Y$, the set $f^{-1}(y)$ has at most $k$ elements.
We have:

The composite of a $j$-injection and a $k$-injection is a $(j \times k)$-injection.

There is also a sensible notion of $k$-subjection, obtained by replacing "at most" with "at least."
We have:

The composite of a $j$-surjection and a $k$-surjection is a $(j \times k)$-surjection.

A criticism.
I wouldn't advise thinking too hard about "$k$ to $1$ functions." There's a couple of reasons for this:


*

*Their definition is kind of arbitrary: we require that $f^{-1}(y)$ has either $k$ elements, or $0$ elements. Um, what?

*We can't say much about their composites:

