For the problem domain of business process engineering we introduce, model, and formalize notions of business processes such as action, actor, event, business process, business transaction. In addition, for the solution domain of service-oriented architectures (SOA) we introduce, model, and formalize notions of service, service composition, service-oriented architecture, and layered SOA in a systematic way. We do that by a rigorous mathematical system model. For that purpose, we first develop a basic mathematical system model for formalizing fundamental concepts of processes and services. The goal is to provide a minimal set of formal modeling concepts, nevertheless expressive enough to formalize key notions and concepts in business process engineering and service-oriented architectures capturing also their mutual relationships. This way, the relationship between central notions in business process modeling is captured formally, which provides a basis for a methodology for deriving the systematic specification and design of service-oriented architectures from business process modeling. The purpose of the approach is manifold; one goal is a clear definition of terminology, concepts, terms, and models in business process modeling and SOA; another goal is a rigorous formal basis for the specification, design, and development of business processes and, in particular, SOAs. We end up with a strictly formal concept for the development steps from business process models to services as part of a SOA-oriented development process.