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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 $f$.

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In particular, this means that the two groups must be isomorphic. – Anton Klyachko Oct 8 '13 at 13:34
@Anton Klyachko: Oh course, thank you. – Boris Novikov Oct 8 '13 at 13:37
Thank you Boris for your answer, it explains why I hadn't heard of this. In fact, if I'm not mistaken your argument works even when instead of groups you have one monoid and one semigroup, since you only need to encapsulate the star by e:s. – David Witt Nyström Oct 9 '13 at 14:44
@David Witt Nyström: Yes, certainly. – Boris Novikov Oct 9 '13 at 15:51

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