recently I have been considering the definition of a group and a ring. As you will recall a group is a set $G$ with a binary operation $\circ :G \times G \rightarrow G$, this is subject to certain conditions, such as the existance of a unit element etc.
I have always thought of a ring $R$ as a generalisation of the concept of group, where instead of a single operation we have two binary operations $R \times R \rightarrow R$. Both of these operations satisfy their own sets of conditions and the condition of distributivity determines how the operations interact with each other.
My question is, is it worth considering a set $X$ which has three binary operations $X \times X \rightarrow X$, each with its own set of algebraic conditions and a new distributivity condition determining how all three interact with each other?
This seems like the natural next step for structures to consider in algebra - the next stage after this would be to consider $n$ binary operations of the set.
Has this area of algebra been explored? Is there a reason why it is not well known or studied if it has?
I am sorry if my question is unclear, I hope the general idea of what I am trying to convey has come across.