On extension to "other" iteration roots

[JmsNxn]

I'd love to just jump in here, and respond on Mphlee's behalf. Who designed how \(f \circ g = g \circ f\) means they belong to the same categorical group, but does not admit that we have the semi-group property. It simply means the preimages, and that the images, agree in a fairly good manner. But it does not mean the semi-group property is followed; at best it means the super function property is defined. I know this might not mean much, but if you read Mphlee's work this is very much his disagreement with some of the things Leo has said.

I think you are expressing Mphlee's concern analytically, bo. Again, though, this is just off of Mphlee and his description of the class of functions which are commutative--and their relationship to fixed points.