5
$\begingroup$

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.

  • 0
    What's wrong with *non-injective*?2015-06-08

3 Answers 3

8

n-1 + onto is sometimes called n-fold cover (by analogy).

7

IMHO, an n to 1 function should be called "an n to 1 function." Simple, decriptive, to the point. Adding a new term in this case just muddies the waters.

  • 2
    An injective function injects one set into another, and a surjective function... doesn't.2010-11-24
0

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:

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

  2. We can't say much about their composites:

Composite