Concepts as mathematical notions

Mathematical notions are concepts since they must be understood in order to be applied correctly. In this paper, the elementary concepts of service, replicate service, and distributed services are introduced. A notation to specify the introduced concepts is given, and both a centralised and a distributed system are specified. The elementary concepts are also exploited to specify a system architecture in which an open system is proposed. The system is specified first at the established time to and then at the generic time t. In the dynamic open system, further services can be added to/removed from the previous ones. Analogously, further servers can be added to/ removed from the network nodes. The notation is based on the three main constructs of CLASS service; CLASS replicate service; and CLASS distributed service. The first is a basic concept, the latter specialised concepts. A discussion to summarize the results of this paper is provided.