Division automorphism

Question

Let S be a set and A be a set of automorphisms of S such

$\forall\ x,y\in S, \exists!\ a\in A \ |\ ax=y$

What's the name of this structure/property, that's similar to a "division algebra" using morphisms?

Answer 1

If $S$ is not empty then this structure is a Latin square: https://en.wikipedia.org/wiki/Latin_square

The elements of $A$ are the symbols and we have a map $S \times S \to A$, sending $(x,y)$ to the unique $a\in A$ which maps $x\mapsto y$.