  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.
