Rational, Linear and algebraic process languages and iteration lemmata

In this paper we define a certain class of process languages viewing processes as bi-partite graphs with an associative operation (sequential composition) on them. They describe finite evolutions of Petri nets. When extended to sets, we get an w-complete semiring such that rational, linear, and algebraic sets of such processes can be defined as least fixed points of systems of equations. With a norm of processes also iteration lemmata can be obtained. Finally, we also present a related structure of directed acyclic graphs.