Mixed Relations as Enriched Semiringal Categories

A study of the classes of finite relations as enriched strict monoidal categories is presented in [CaS91]. The relations there are interpreted as connections in flowchart schemes, hence an angelic theory of relations is used. Finite relations may be used to model the connections between the components of dataflow networks [BeS98, BrS96], as well. The corresponding algebras are slightly different enriched strict monoidal categories modeling a forward-demonic theory of relations. In order to obtain a full model for parallel programs one needs to mix control and reactive parts, hence a richer theory of finite relations is needed. In this paper we (1) define a model of such mixed finite relations, (2) introduce enriched (weak) semiringal categories as an abstract algebraic model for these relations, and (3) show that the initial model of the axiomatization (it always exists) is isomorphic to the defined one of mixed relations. Hence the axioms gives a sound and complete axiomatization for the these relations. 1 C.S.Calude and G.Stefanescu (eds.). Automata, Logic, and Computability. Special issue dedicated to Professor Sergiu Rudeanu Festschrift