also closely related. A lattice is cross-symmetric if for every modular pair (a, b) the pair (b, a) is dually modular. Cross-symmetry implies M-symmetry but not M*-symmetry. Therefore, cross-symmetry is not equivalent to dual cross-symmetry. A lattice with a least element 0 is ⊥-symmetric if for every modular pair (a, b) satisfying a ∧ b = 0 the pair (b, a) is also modular. History The definition of modularity is due to Richard Dedekind, who published most of the relevant papers after his retirement. In a paper published in 1894 he studied lattices, which he called dual groups () as part of his "algebra of modules" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual. In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility. In a digression he introduced and studied lattices formally in a general context. He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices dual groups of module type (). He also proved that the modular identity and its dual are equivalent. In the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual: (x ∧ b) ∨ (a ∧ b) = [x ∨ a] ∧ b. He called lattices that satisfy this identity dual groups of ideal type (). In modern literature, they are more commonly referred to as distributive lattices. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type. A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture). See also Modular graph, a class of graphs that includes the Hasse diagrams of modular lattices Young–Fibonacci lattice, an infinite modular lattice defined on strings of the digits 1 and 2 Orthomodular lattice Iwasawa group Notes References External links Free Modular Lattice Generator An open-source browser-based web application that can generate and visualize some free modular lattices. Lattice theory