# About sets with two mutually associative group structures

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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Not a direct hit, but the following might be useful. Consider the universally quantified second order statement F(a,G(b,c))=G(F(a,b),c), where a,b,and c are elements and G and F range over a chosen set of binary operations. Your S satisfies a statement like this when the set has two operations. Belousov and Movsisyan studied algebraic systems satisfying schema like these, which they call hyperidentities. Perhaps they named S when the operations were group operations, but I think they were more interested in quasi or semigroups. – The Masked Avenger Oct 8 '13 at 7:17

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 $f$.