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.

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    $\begingroup$ Sometimes, generalization for the sake of generalization doesn't happen to be useful. Examples of groups and rings were known before their definitions were coined. Rings aren't really a generalization of groups, rather, they are a system with a richer algebraic structure: they always contain an abelian group, it's additive subgroup. And this might be a bit misleading too: abelian groups (at least finitely generated abelian groups) happen to be rather boring compared to other groups, so rings are maybe a different creature altogether. $\endgroup$ – Pedro Tamaroff Mar 14 '14 at 19:45
  • $\begingroup$ Try this question and its answers. $\endgroup$ – John Habert Mar 14 '14 at 19:49

Sometimes it's convenient to study and work with algebraic structures having more than two binary operations. An example are boolean algebras $(\mathcal{A}, \land,\lor,\lnot)$ with two binary operators and one unary for logical and, logical or and negation satisfying the Boolean laws. If you take e.g. a nonempty set $X$, the powerset $(\mathcal{P}(X),\cap,\cup,\mathsf{^c})$ with union, intersection and complement forms a boolean algebra. Another example are Lie rings, rings which have additionally defined a bracket operator $[x,y]=xy-yx$.


I am not sure whether this is what you are looking for, but there is a rather general notion of universal algebra where you can have arbitrarily many operations. (And you can use unary, binary, ternary, $n$-ary operations.)

If you add some questions which they have to fulfill, you get variety of algebras.


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