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arXiv · 2608.10026

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

Ahmet Alkan·Kemal Yildiz
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Abstract

We characterize the subsets of a finite distributive lattice that arise as the minimizer sets of modular functions. It is immediate that the minimizer set of any modular function is a decomposition-closed sublattice. Our main theorem establishes the converse: every decomposition-closed sublattice is the minimizer set of a modular function. Our proof is constructive. Using Birkhoff's representation, we decompose the problem along the intervals determined by an arbitrary maximal chain of the given sublattice. The key ingredient is a weight assignment on connected difference posets that yields local modular functions vanishing precisely at the endpoints of each interval. We also provide an example showing that this characterization fails for nondistributive lattices.

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BibTeXRIS

Ahmet Alkan, Kemal Yildiz. 2026-08-09. Decomposition-Closed Sublattices as Minimizer Sets of Modular Functions over Distributive Lattices. https://arxiv.org/abs/2608.10026

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