Given a groupoid $ \left( M, * \right) $ with an neutral element $0$ and $\setminus$ being an inverse operator of $*$, what are the groupoid's properties for this predicate to be true?
$$ \forall x, y \in M: x \setminus y = x * \left( 0 \setminus y \right) $$
It works for $\left(\mathbb{Z},+\right)$ and $\left(\mathbb{R},\cdot\right)$ but I didn't find any specific rule.
I assume that "$\setminus$ is an inverse operator of $*$" means that $(x*y)\setminus y=(x\setminus y)*y=x$ for all $x,y$. Then
$$ x*(0\setminus y)\stackrel I=((x*(0\setminus y))*y)\setminus y\stackrel A=(x*((0\setminus y)*y))\setminus y\stackrel I=(x*0)\setminus y\stackrel N=x\setminus y$$
by applying $I$ the inverse property, $A$ associativity, and $N$ the property of the neutral element.