In this thesis, we argue that (order-) lattice-based multi-agent information systems constitute a broad class of networked multi-agent systems in which relational data is passed between nodes. Mathematically modeled as lattice-valued sheaves, we initiate a discrete Hodge theory with a Laplace operator, analogous to the graph Laplacian and the graph connection Laplacian, acting on assignments of data to the nodes of a Tarski sheaf. The Hodge-Tarski theorem (the main theorem) relates the fixed point theory of this operator, called the Tarski Laplacian in deference to the Tarski Fixed Point Theorem, to the global sections (consistent global states) of the sheaf. We present novel applications to signal processing and multi-agent semantics and supply a plethora of examples throughout.

翻译：本文论证了基于（序）格的多智能体信息系统构成了一类广泛的网络化多智能体系统，其中关系型数据在各节点间传递。通过数学建模为格值层，我们提出了一种离散霍奇理论，该理论包含一个拉普拉斯算子（类似于图拉普拉斯算子和图连接拉普拉斯算子），作用于Tarski层节点上的数据分配。霍奇-塔尔斯基定理（主要定理）将该算子（为致敬塔尔斯基不动点定理而命名为塔尔斯基拉普拉斯算子）的不动点理论与层的全局截面（一致的全局状态）联系起来。我们展示了其在信号处理和多智能体语义方面的创新应用，并在全文提供了大量实例。