Abstract

We develop the theory of saturated transfer systems on modular lattices, ultimately producing a “matchstick game” that puts saturated transfer systems in bijection with certain structured subsets of covering relations. We also prove that Hill’s characteristic function $\chi$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$, and moreover, the fibers of $\chi$ have unique maxima which are exactly the saturated transfer systems. Lastly, after an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary abelian groups.

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