Arthur-Merlin protocol

I recently learned about the Arthur-Merlin protocol, and we defined the complexity classes $ AM,MA$ .

We have also seen that there exists a theorem stating that $ AMAMAM…AM=AM$ , however we have not formally defined $ AMAMAMAM…AM$ .

We did give a brief definition for it as a protocol, but i would like to see a formal definition without protocols, just like $ AM$ and $ MA$ (and $ MAM$ )

A definition for $ AMA$ should suffice for me, just to get the hang of how it should look.