computing in a loop

Question

Consider a loop, i.e. a set $Q$ with operation $\cdot$ such that we have cancellation law and an identity element. For given $x,y\in Q$ consider the equation $$x=((xy)^{-1}_R\cdot z)^{-1}_L$$ where the unknown is $z\in Q$, and $x^{-1}_R$, and $x^{-1}_L$ denote the right inverse and left inverse of $x$ in $Q$, respectively. I need to express $z$, but I am always falling into the trap of the absence of the associativity law, so I am asking for help.

Answer 1

As this looks a bit like a homework assignment, I will only give a hint:

In a loop, you can construct partial or complete inverses of compound expressions (think "balanced cancellation") by composing the various laws.