Arigatoni is a lightweight communication model that deploys the Global Computing Paradigm over the Internet. Communications over the behavioral units of the model are performed by a simple Global Internet Protocol (GIP) on top of TCP or UDP protocol. Basic Global Computers Units (GCU) can communicate by first registering to a brokering service and then by mutually asking and offering services. Colonies and Communities are the main entities in the model. A Colony is a simple virtual organization composed by exactly one leader and some set (possibly empty) of individuals. A Community is a raw set of colonies and global computers (think it as a soup of colonies and global computer without a leader). We present an operational semantics via a labeled transition system, that describes the main operations necessary in the Arigatoni model to perform leader negotiation, joining/leaving a colony, linking two colonies and moving one GCU from one colony to another. Our formalization results to be adequate w.r.t. the algorithm performing peer logging/delogging and colony aggregation.