# A "group" with two binary operations that inter-associate

I am working with a group, of sorts, that contains not one, but two, binary operations. Let's call these $+$ and $*$. For either operation, the group has a common identity, $e$, and every element has an inverse.

I am able to show that these operations are associative, not only within themselves, but with each other. For example, $$(a+b)*c=a+\left(b*c\right)$$.

If this is not a group, what is the name of the type of algebra I am dealing with? Why does it seem like such groups have not received much attention?

My question is related to the one here, except that my case has every property of a group (associativity, identity, and inverse). In other words, if I disallowed either $*$ or $+$, I would still have a group.

• For either operation, but $e$ must not be the same for both operations? Maybe we should call them $e_*$ and $e_+$ respectively. Commented Sep 21, 2017 at 18:05
• Please give an example of a set with the two operations. Preferably the smallest example where the two operations are distinct. Can you do that? Commented Sep 21, 2017 at 18:52

Let's call these $+$ and $*$. For either operation, the group has a common identity, $e$, and every element has an inverse.
Then $+=\ast$, because $a\ast c=(a+e)\ast c = a+(e\ast c)=a+c$ for every $a,c$.
• For "either" operation, not for "both". Just that there exists some element $e_1$ s.t. $e_1+g = g+e_1 = g, \forall g \in G$ and $e_2$ s.t. $e_2*g = g*e_2 = g, \forall g \in G$ Commented Sep 21, 2017 at 18:37