Complete Valuations on Finite Distributive Lattices

TL;DR

This paper characterizes finite distributive lattices that admit a complete valuation, showing they are downset lattices of posets with dimension at most two and providing a recursive method to determine such valuations.

Contribution

It establishes a precise characterization of lattices with complete valuations as downset lattices of certain posets and introduces a recursive relation for weights based on chain counting.

Findings

Finite distributive lattices with complete valuations are downset lattices of posets of dimension at most two.

A recursive relation between weights on the poset is used to determine valuations.

Valuations based on chain counting in the complementary poset are complete for these lattices.

Abstract

We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are downset lattices of finite posets of dimension at most two, and determine a realizer through a recursive relation between weights on the poset associated to valuation. The relation shows that the weights count chains in the complementary poset. Conversely, we prove that a valuation defined on a poset of dimension at most two, through the weight function which counts chains in the complementary poset, is complete.