Algebra uses a lot of binary operations - addition, multiplication, Lie brackets - but not so many ternary operations. Why is this? Is it just our difficulty in understanding anything more complicated than glomming together two things at a time?
There are, however, some important examples of gadgets with ternary operations. For example, suppose you take a group and define
t(a,b,c) = ab⁻¹c
You should think of this as answering the question
"a is to b as WHAT is to c?"
like "king is to queen as WHAT is to father?", but made mathematical using ratios. The reason is that if we define t(a,b,c) as above we have
ab⁻¹ = t(a,b,c)c⁻¹
We can figure out which equations this opration t obeys, and get the definition of a 'heap'. If you take a heap and declare any element to be the identity, it becomes a group! So we see a heap is "a group that has forgotten its identity".
Two other important gadgets with ternary operations are 'Lie triple systems' and 'Jordan triple systems'. I've been thinking about these a lot, and fixing up the Wikipedia article on them:
https://en.wikipedia.org/wiki/Triple_system
The famous algebraist Nathan Jacobson invented them in 1949. The idea is this. Suppose you take an associative algebra A and define the commutator [a,b] and anticommutator {a,b} by
[a,b] = ab - ba
{a,b} = ab + ba
Then define the Lie triple product by
[a,b,c] = [[a,b],c]
and the Jordan triple product by
{a,b,c} = {{a,b},c}
These obey some equations which become the axioms of Lie triple system and Jordan triple system. But their real importance shows up in geometry!
