Do there exist algebras of more directions of operation than left-right?

Question

Again I am kind of new to most things algebraic, only having learned the very basics about groups.

As little I have learned about groups and their operations is that an operation has two arguments, one from the left and one from the right.

$$a\circ b = c$$

But is it meaningful and possible to define algebraic structures with operations with more than two operands? Like this three operand one for example (replacing $\circ$ with curlybraces to clarify direction of operation):

$$\underset{o_2}{\underbrace{o_1}} \} o_3 = c$$

or four:

$$o_4\{\underset{o_2}{\underbrace{o_1}} \} o_3 = c$$

This notation will make it get crowded on a 2D paper once we have a chain of many operations, but I hope you get the idea. Do you think such a thing would be possible to combine with the demands on a group in some sense? What meaning would "inverse" have for such a construction? Just pairs of elements as each others inverses would no longer make sense, right?

Answer 1

A general algebraic object is a set $X$ together with some specified

• relations $R_i\subseteq X^n,i\in I$,
• functions $f_j:X^n\to X,j\in J$ and
• constants $c_k\in X,k\in K$.

As you can see, the functions can be of any arity. Such structures are mostly discussed in abstract setting as in model theory. But there are examples which are used in applications, e.g. planar ternary rings are used in incidence geometry. Another example which uses no functions but a ternary relation is a cyclic order.

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Related:

• Why don't we study algebraic objects with more than two operations?
• Is anybody researching "ternary" groups?

Answer 2

If we do this in two dimensions, like you suggest, then we get something that starts looking like a $2$-category (possibly with just one object, i.e. a monoidal category).

A common way to describe "nice" monoidal categories is in terms of diagrams, and these diagrams can then be composed both horizontally, by putting them next to each other (this is the "tensor product" in the category), or vertically, by placing one on top of the other (the "usual composition" in the category). Of course, this has the issue of which objects can be composed, since not all of them need to be composable.

For examples of this, see for example the paper "Soergel Calculus" by Elias and Williamson (this forms the basis of their Annals paper in which they prove the Soergel conjecture).