Nondistributive

Nondistributive is a term used in lattice theory to describe lattices that do not satisfy the distributive laws for the join and meet operations. In a lattice, the join is typically denoted by ∨ and the meet by ∧. A lattice is distributive if, for all elements x, y, and z, the identities x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) hold. If a lattice fails to satisfy at least one of these identities, it is called nondistributive.

Nondistributive lattices often serve as fundamental counterexamples illustrating the limits of distributive behavior in algebraic structures. The two smallest and classic examples are the diamond-shaped lattice M3 and the pentagon-shaped lattice N5. M3, also known as the diamond lattice, has five elements: a bottom element, a top element, and three atoms arranged so that each atom lies below the top. N5, the pentagon lattice, also has five elements arranged in a pentagonal shape, and it demonstrates non-distributive behavior in a different structural form. These lattices are minimal witnesses to nondistributivity.

A central result in lattice theory states that a lattice is distributive if and only if it contains no sublattice isomorphic to M3 or N5. Consequently, the presence of either M3 or N5 as a sublattice guarantees nondistributivity. Nondistributive lattices arise in various mathematical contexts, including order theory, algebra, and logic, where the distributive property fails to hold globally.