Partially additive categories and flow-diagram semantics

AbstractThe “semantics of flow diagrams” are used to motivate the notion of partially additive monoids and of partially additive categories as those based on the category of partially additive monoids. We show that such categories support a notion of iteration; and then axiomatize iteration in a fashion which yields other approaches as a special case. The partially additive categories generalize semiadditive categories, and we provide an alternative characterization based on the fact that coproducts +̌ Ai in a partially additive category are equipped with morphisms (prj: +̌ Ai → Aj) which enjoy many of the properties of products. A number of other approaches to flow-diagram semantics have used either the concept of partial order or of algebraic theory—we provide a systematic overview of these approaches from the perspective afforded by partially additive categories