In my Ph.D. thesis, I argue that lattice-based multi-agent systems constitute a broad class of networked multi-agent systems in which relational data is passed between nodes. Mathematically modeled as lattice-valued sheaves, I 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 lattice-valued sheaf.
I made a 10 minute video1 that provides a brief summary of my thesis.
In the video, I define lattice-valued sheaves with order-reversing restriction (i.e.~structure) maps. If you want sheaves with order-preserving restriction maps, you can simply replace the join in the definition of the Tarski Laplacian in the video with a meet. (This definition coincides with the definition of the Tarski Laplacian in my thesis.) My rationale for presenting the order-reversing notion of a Galois connection in the video is the connection to the formal concept analysis (FCA) literature. ↩