On totalisation of computable functions in a distributive environment

Totalisation of operations and functions is a subject of great importance in the precise sciences at large, and in particular, in computer science, where computed functions must be totalised to account for edge cases and prevent system failures. In this paper, we investigate the totalisation of computable functions in the general and abstract setting of the axiomatic theory of computing systems, such as automata, computer networks, and algorithms, allowing us to define characteristics and discover relations which resonate true through many computational models – from individual automata through complex distributed systems with concurrent functioning. We explore the possible types of totalisations, the complexes of computing systems that do (or do not) have totalisations, and the behaviour of totalisation in various transformations and compositions. In addition, we point to a few interesting directions for future research.