Distributivity conditions and the order-skeleton of a lattice Jianning Su, Wu Feng, and Richard J. Greechie Abstract. 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 D 0π if a ∧ (b ∨ c)  (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  L := L/ ∼. We prove that the following are equivalent for a lattice L:(i) L satisfies D 0π ,(ii)  L satisfies any of the five π-versions of distributivity, (iii) the order-skeleton  L is distributive. 1. Introduction Distributive lattices are perhaps the most familiar class of lattices. They may be defined via any of the following ternary relations on L: D(a, b, c) means a ∧ (b ∨ c)=(a ∧ b) ∨ (a ∧ c), D ∗ (a, b, c) means a ∨ (b ∧ c)=(a ∨ b) ∧ (a ∨ c), D m (a, b, c) means (a ∧ b) ∨ (b ∧ c) ∨ (c ∧ a)=(a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a), and D 0 (a, b, c) means a ∧ (b ∨ c)  (a ∧ b) ∨ c. In fact, a lattice L is distributive in case any one, and hence all, of the follow- ing equivalent conditions hold: (i) D(a, b, c) for all a, b, c ∈ L, (iii) D m (a, b, c) for all a, b, c ∈ L, (ii) D ∗ (a, b, c) for all a, b, c ∈ L, (iv) D 0 (a, b, c) for all a, b, c ∈ L. Recall that elements a, b of a lattice L are incomparable, written as a ‖ b, if they are not comparable. An antichain in L is a subset of L in which any two distinct elements are incomparable. We denote by π L the set of antichains in L and by π n L the set of n-element antichains in L, where n ≥ 1. In [8], one of us found that a π-version of distributivity proved to be of some importance in the study of when certain mappings are residuated. This motivated us to consider the π-version of each of the properties (i)-(iv), by replacing, in each case, “for all a, b, c ∈ L” with “ for all {a, b, c}∈ π 3 L .” Presented by . . . Received . . . ; accepted in final form . . . 2010 Mathematics Subject Classification: Primary: 06D75. Key words and phrases : lattice, distributive lattice, π-distributive lattice, order-skeleton, residuated, exclusion systems.