Inverse element in a magma

Question

Given $(S,*)$ a magma and an identity element $e$. The inverse of $x\in S$ is $y$ such that $x*y=e=y*x$. Is it correct to say that if $x$ is the inverse of $y$ then $y$ is the inverse of $x$?

Answer 1

It is correct to say that, if $x$ is an inverse of $y$ then $y$ is an inverse of $x$, the property $x*y=e=y*x$ is symmetric, it remains the same after swapping $x$ and $y$ and getting $y*x=e=x*y$.

Without associativity you won't be able to prove that inverses are unique however.