Robert's theorem and graphs on complete lattices

Abstract: Automata networks, and in particular Boolean networks, are used to model diverse networks of interacting entities. The interaction graph of an automata network is its most important parameter, as it represents the overall architecture of the network. A continuous amount of work has been devoted to infer dynamical properties of the automata network based on its interaction graph only. Robert's theorem is the seminal result in this area; it states that automata networks with an acyclic interaction graph converge to a unique fixed point. The feedback bound can be viewed as an extension of Robert's theorem; it gives an upper bound on the number of fixed points of an automata network based on the size of a minimum feedback vertex set of its interaction graph. Boolean networks can be viewed as self-mappings on the power set lattice of the set of entities. In this paper, we consider self-mappings on a general complete lattice. We make two conceptual contributions. Firstly, we can view a digraph as a residuated mapping on the power set lattice; as such, we define a graph on a complete lattice as a residuated mapping on that lattice. We extend and generalise some results on digraphs to our setting. Secondly, we introduce a generalised notion of dependency whereby any mapping $\phi$ can depend on any other mapping $\alpha$. In fact, we are able to give four kinds of dependency in this case. We can then vastly expand Robert's theorem to self-mappings on general complete lattices; we similarly generalise the feedback bound. We then obtain stronger results in the case where the lattice is a complete Boolean algebra. We finally show how our results can be applied to prove the convergence of automata networks.