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

Complete Valuations on Finite Distributive Lattices