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 of all, 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 the paper, we introduce two indices that count how many of the triads are distributive. In an extensive experiment with over 2 million lattices (all up to size 13), we study the distributions of these indices and 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''.