Multiplicative groups

When using this notation, I always write multiplication explicitly. This makes the descriptor “multiplicative” make no sense in describing structures where the binary operator is not written.

The implicit operator is application. Therefore, we may want to make applicative groups, where applying one element to another produces a new element. So, we have elements of type \(t\), where \(t = t\rightarrow t\). But this is bad. It means that our elements have infinite type, and that opens us up to Curry's paradox.

Instead, we give all elements type \(a\rightarrow a\), and make a compositional group. For example, let's look at \(C_3\), with elements \(\{e,f,g\} : \{G\rightarrow G\}\). The group table gives us ways to rewrite any composition of these functions, like \(f\circ g = e\).

So, what happens when we don't write the \(\circ\) operator? Implicit mapping comes into play. In \(fg\), \(f : G\rightarrow G\) expects a \(G\), but gets \(g : G\rightarrow G\). (It's important to note that the \(G\)s in each type signature are the same. \(G\) can't match any type expression non-trivially containing \(G\).) Fortunately, \((G\rightarrow)\) is a functor, so \(fg\) becomes \(\text{fmap}\ f\ g\), which is \(f\circ g\).

This way of describing group-like structures happens to be very similar to the way of defining monoids (and hence groups) in category theory. A monoid is defined as a category with a single object and some arrows, one of which is the identity arrow. Necessarily, all of the arrows go from and to the object. In my example, the object corresponds to type \(G\), and the elements each correspond to an arrow.

(I wish I had a digit negative 1 right now!)

One thing to note is that one should not use this notation unless the structure has been proven associative. Function composition is associative, so using it as the operation in a non-associative structure would be wrong.