On distributive triads

Distributivity is a well-established and extensively studied notion in lattice theory. In Formal Concept Analysis, it allows for powerful methods to decompose or factorize concept lattices. However, most lattices are not distributive. As a result, many properties weaker than distributivity have been introduced, such as modularity and semi-distributivity. Yet even these properties hold only for a relatively small number of lattices. To address this, we adopt a local perspective rather than weakening the definition of distributivity. When viewed as a computational check, the law of distributivity considers only three elements and their local interaction. However, in non-distributive lattices, not all three-element subsets satisfy the law. Most intriguingly, the manner in which they fail to do so varies. This motivates the notion that gives this work its title: A triad is a three-element subset of the lattice. We call a triad distributive if it generates a distributive sublattice. In this paper, we introduce two indices that count how many of the triads are distributive. In an extensive experiment with over 19 million lattices (all up to size 14), we study the distributions of these indices. We observe that even though distributive lattices are very rare, lattices that are “highly distributive” are very frequent. We also analyze which lattice structures generate low and which generate high indices. In particular, we throw a glance at the “maximally non-distributive lattices” and at the “maximally distributive non-distributive lattices”.