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Rational, Linear and Algebraic Process Languages and Iteration Lemmata

In this paper we define a certain class of process languages viewing processes as bipartite graphs with an associative operation (sequential composition) on them. They describe finite evolutions of Petri nets. When extended to sets, we get an ω-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.