Decomposition-Closed Sublattices as Minimizer Sets of Modular Functions over Distributive Lattices

What the math gives to ML

The paper gives a constructive characterization of which subsets of a finite distributive lattice can be made exactly optimal by a modular score. Under Birkhoff's representation, every lattice element becomes a down-set of join-irreducibles, and modular scores become additive sums of per-join-irreducible weights. This suggests a principled way to design trainable discrete neural components whose complete set of zero-cost configurations is a prescribed decomposition-closed family, rather than relying on ad hoc penalties. The most plausible transfer is structured pruning or hierarchical MoE routing over partially ordered masks, where exact validity and closure of optimal configurations are useful.