Consider the following: A set $S$ with two operations, $\ast$ and $\times,$ such that both $(S,\ast)$ and $(S,\times)$ are groups, and such that whenever one writes down a mixed product, e.g. $a\ast b\times c\times b \ast d \times a$ it doesn't matter where one puts the parentheses. My question is whether this structure has a name and whether it has been studied.
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Let $e$ be an identity for $\cdot$ and $f$ for $*$. Then $e*e\cdot f=e$, i.e. $f^{1}=e*e$. Further, $$a*b=ae*(bb^{1})b=a(e*e)b=af^{1}b,$$ so the second operation ($*$) is defined by the first one and by fixing an arbitrary element $f$. 

