TITLE: About sets with two mutually associative group structures
QUESTION [6 upvotes]: 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.

REPLY [19 votes]: 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 $f$.