Algebra groups

\displaystyle [br]\begin{equation*}(a * b) * c = \left(a + (-1)^a b\right) * c = a + (-1)^a b + (-1)^{a + (-1)^a b}c\end{equation*}

\displaystyle [br]\begin{equation*}a * (b*c) = a * \left(b + (-1)^b c\right) = a + (-1)^{a} \left(b + (-1)^{b}c\right)\end{equation*}