Binary operation $xy$, has identity, but not associativity. Is the inverse unique?

Question

Let $S$ be a set with a binary operation $xy$ defined on it, with a neutral element, but not satisfying associativity. I want to prove that the inverse isn't necessarily unique.

My attempt to answer this is in the answer below.

Answer 1

Let $S=\{1,2,3\}$ and define $\star$ so that $x\star 1 = 1\star x=x$ and $x\star y=1$ if $x,y\in\{2,3\}$. Then $(S,\star)$ has a neutral element, $1$, but $2\star 3=2\star 2=1$, so inverse elements are not unique.