Artificial intelligence (AI) in its various forms finds more and more its way into complex distributed systems. For instance, it is used locally, as part of a sensor system, on the edge for low-latency high-performance inference, or in the cloud, e.g. for data mining. Modern complex systems, such as connected vehicles, are often part of an Internet of Things (IoT). To manage complexity, architectures are described with architecture frameworks, which are composed of a number of architectural views connected through correspondence rules. Despite some attempts, the definition of a mathematical foundation for architecture frameworks that are suitable for the development of distributed AI systems still requires investigation and study. In this paper, we propose to extend the state of the art on architecture framework by providing a mathematical model for system architectures, which is scalable and supports co-evolution of different aspects for example of an AI system. Based on Design Science Research, this study starts by identifying the challenges with architectural frameworks. Then, we derive from the identified challenges four rules and we formulate them by exploiting concepts from category theory. We show how compositional thinking can provide rules for the creation and management of architectural frameworks for complex systems, for example distributed systems with AI. The aim of the paper is not to provide viewpoints or architecture models specific to AI systems, but instead to provide guidelines based on a mathematical formulation on how a consistent framework can be built up with existing, or newly created, viewpoints. To put in practice and test the approach, the identified and formulated rules are applied to derive an architectural framework for the EU Horizon 2020 project ``Very efficient deep learning in the IoT" (VEDLIoT) in the form of a case study.