There are several answers regarding this on the site but none of them seem to be in consensus. Some call it a function, other unoid and some people something else. So I decided to ask it with a example.
Consider the set
$S_1 = \{ 2, 5, 42, 10, 45, 4, 12, 60, 93\}$
and the unary operation
$f_1 : S_1 \to S_1, x \mapsto 26(x^2 +8x)$$\mod 103 $
in other words,
$f_1 (x) = 26(x^2+8x) \mod 103$
As we can see $S_1$ is closed under unary operation $f_1$. So like groups (set closed under binary operation satisfying certain axioms) is there a name for this structure?
Addittionally, let us consider the set
$S_2 = \{2, 4, 8, 16, 32, 64, 55, 37, 1\}$
and the following unary operation:
$f_2 : S_2 \to S_2, x \mapsto 2x$$\mod 73 $,
in other words,
$f_2 (x) = 2x \mod 73$
Since cardinality of both structures are $9$ is it possible that they are isomorphic to each other. If so then how could such isomorphism be found?
This corresponds to a directed graph with outdegree $1$. (I don't assume the operation to be surjective and/or bijective because the question doesn't say so.) Such graphs are called functional graphs because, yes, this is just a set and a function from that set to that set. Accordingly, you can apply tools for detecting graph isomorphisms to this problem.
Related questions: Is there a name for a directed graph in which every vertex has outdegree one? , What's the meaning of functional graph?