I’ve written about Möbius transformations many times because they’re simple functions that nevertheless have interesting properties.

A Möbius transformation is a function *f* : ℂ → ℂ of the form

*f*(*z*) = (*az* + *b*)/(*cz* + *d*)

where *ad* – *bc* ≠ 0. One of the basic properties of Möbius transformations is that they form a group. Except that’s not quite right if you want to be completely rigorous.

The problem is that a Möbius transformation isn’t a map from (all of) ℂ to ℂ unless *c* = 0 (which implies *d* cannot be 0). The usual way to fix this is to add a point at infinity, which makes things much simpler. Now we can say that the Möbius transformations form a group of automorphisms on the Riemann sphere *S*².

But if you insist on working in the finite complex plane, i.e. the complex plane ℂ with no point at infinity added, each Möbius transformations is actually a *partial function* on ℂ because a point may be missing from the domain. As detailed in [1], you technically do not have a group but rather an inverse monoid. (See the previous post on using inverse semigroups to think about floating point partial functions.)

You can make Möbius transformations into a group by *defining* the product of the Möbius transformation *f* above with

*g*(*z*) = (*Az* + *B*) / (*Cz* + *D*)

to be

(*aAz* + *bCz* + *aB* + *bD*) / (*Acz* + *Cdz* + *Bc* + *dD*),

which is what you’d get if you computed the composition *f* ∘ *g* as functions, ignoring any difficulties with domains.

The Möbius inverse monoid is surprisingly complex. Things are simpler if you compactify the complex plane by adding a point at infinity, or if you gloss over the fine points of function domains.

## Related posts

- Transformations of Olympic rings
- Curiously simple approximations
- Solving for Möbius transformation coefficients

[1] Mark V. Lawson. The Möbius Inverse Monoid. Journal of Algebra. 200, 428–438 (1998).

It is also noteworthy that the operation you defined induce a morphism to the algebra of 2×2 complex matrices… which comes with a bunch of nice surprises too.