predistributive

Predistributive is a term used in lattice theory and universal algebra to describe structures in which the distributive law between the join and meet operations is weakened or constrained. In a distributive lattice, for all elements a, b, and c, the usual distributive identities hold: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). A predistributive lattice, by contrast, is defined in several ways in the literature, with no single universally adopted definition. The common theme is that some distributive behavior is present, but not necessarily the full set of distributive identities.

One frequently encountered approach is to require one or both distributive laws to hold under restricted conditions or to demand weaker, one-sided versions of distributivity. Another common formulation describes predistributivity as a conjunction of semidistributive properties: meet-semidistributivity and join-semidistributivity. Under such interpretations, every distributive lattice is predistributive, but some predistributive lattices need not be distributive.

Predistributivity also appears in other algebraic contexts where distributive behavior is desirable but not guaranteed. In these settings, the term signals a weakening of distributivity that may still yield useful structural consequences or facilitate the study of related varieties.

Because definitions vary, readers encountering predistributive in research or surveys should check the precise formulation used in that work. Related concepts include distributive lattices, semidistributive lattices, and modular lattices.

See also: distributive lattice, semidistributive lattice, modular lattice, lattice theory, universal algebra.