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.

翻译：自动机网络，特别是布尔网络，用于模拟多种相互作用实体的网络。自动机网络的交互图是其最重要的参数，因为它代表了网络的整体架构。大量工作致力于仅基于交互图来推断自动机网络的动力学性质。罗伯特定理是该领域的开创性成果；它指出具有无环交互图的自动机网络收敛到唯一不动点。反馈界可视为罗伯特定理的推广；它根据交互图的最小反馈顶点集的大小，给出了自动机网络不动点数量的上界。布尔网络可视为实体集合的幂集格上的自映射。本文考虑一般完全格上的自映射。我们提出两个概念性贡献。首先，我们将有向图视为幂集格上的剩余映射；由此，我们将完全格上的图定义为该格上的剩余映射。我们将有向图的一些结果推广到我们的设定中。其次，我们引入一种广义依赖概念，使得任意映射 $\phi$ 可以依赖于任意其他映射 $\alpha$。实际上，在此情况下我们能够给出四种依赖类型。进而，我们将罗伯特定理大幅扩展到一般完全格上的自映射；并类似地推广了反馈界。当格为完全布尔代数时，我们获得了更强的结果。最后，我们展示了如何应用这些结果证明自动机网络的收敛性。