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.