Distributivity conditions and the order-skeleton of a lattice

We introduce “π-versions” of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π if a∧(b∨c) 6 (a∧b)∨c for all 3-element antichains {a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be L̃ := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D0π , (ii) L̃ satisfies any of the five π-versions of distributivity, (iii) the order-skeleton L̃ is distributive.