meetdistributesoverjoin

Meetdistributesoverjoin, in lattice theory, refers to the distributive law where the meet operation distributes over the join operation. In a lattice L with binary operations meet (∧) and join (∨), the property states that for all elements a, b, and c in L, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). This law is one half of the standard distributivity conditions that define a distributive lattice.

A lattice is called distributive when both distributive laws hold: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). The first law is the “meet distributes over join,” while the second is its dual. In distributive lattices, both laws hold for all elements.

Examples where the law holds include Boolean algebras and the lattice of subsets of a set under intersection and union, since A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) for all sets A, B, C. Conversely, many non-distributive lattices fail the law. Classic counterexamples are the nondistributive lattices M3 (the diamond) and N5 (the pentagon), which do not satisfy a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for some elements.

In practice, meet distributing over join is fundamental to simplifying expressions in logic, set theory, and computer science, and it serves as a key criterion for identifying distributive (and thus more tractable) lattices and Boolean algebras.