Modular Lattice - Modular Pairs and Related Notions

Modular Pairs and Related Notions

In any lattice, a modular pair is a pair (a, b) of elements such that for all x satisfying a ∧ b ≤ x ≤ b, we have (x ∨ a) ∧ b = x, i.e. if one half of the diamond isomorphism theorem holds for the pair. An element b of a lattice is called a (right) modular element if (a, b) is a modular pair for all elements a.

A lattice with the property that if (a, b) is a modular pair, then (b, a) is also a modular pair is called an M-symmetric lattice. Since a lattice is modular if and only if all pairs of elements are modular, clearly every modular lattice is M-symmetric. In the lattice N₅ described above, the pair (b, a) is modular, but the pair (a, b) is not. Therefore N₅ is not M-symmetric. The centred hexagon lattice S₇ is M-symmetric but not modular. Since N₅ is a sublattice of S₇, it follows that the M-symmetric lattices do not form a subvariety of the variety of lattices.

M-symmetry is not a self-dual notion. A dual modular pair is a pair which is modular in the dual lattice, and a lattice is called dually M-symmetric or M*-symmetric if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M*-symmetric. The same equivalence holds for infinite lattices which satisfy the ascending chain condition (or the descending chain condition).

Several less important notions are 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.