Ordered sets model signals such as binary relations, concepts, partitions, rankings, matchings, events, as well as other taxa of information, temporal, hierarchical, relational, or, in general, logical in nature. We argue that (order-) lattice-based (networked) multi-agent systems constitute a broad class of systems in which data fusion, consensus, synchronization, and other collaborative tasks are described with lattices and Galois connections (maps between lattices that preserve structure). Mathematically speaking, these systems are network sheaves. Motivated by analogous vector diffusion and consensus algorithms, we initiate a discrete Hodge theory with the Tarski Laplacian, a diffusion operator—analogous to the graph Laplacian and the graph connection Laplacian—acting on assignments of lattice-valued data to the nodes of a network. The Hodge-Tarski theorem (Main Theorem) relates the fixed point theory of the Tarski Laplacian to the global sections (consistent signals, equilibria, if you will) of the sheaf. We present novel applications in signal processing and multi-modal semantics where we design a consensus algorithm on statements such as “I know that she knows that he doesn’t know that I’m defending my thesis.”