Distributivity in lattices
R. P. Dilworth, J. E. McLaughlin · Duke Mathematical Journal · 1952
A lattice L is infinitely (join) distributive if a ∩ ∪ B b = ∪ B (a ∩ b) whenever the indicated joins exist in L. Clearly infinite distributivity implies ordinary distributivity. On the other hand it is easy to give examples of distributive lattices which are not infinitely distributive. For example, the rational integers under the relation of division form a distributive lattice which is not infinitely distributive. However for complemented lattices, as observed by Tarski [8] and von Neumann [6], distributivity implies infinite distributivity.