Distributive Lattice - Characteristic Properties

Characteristic Properties

Various equivalent formulations to the above definition exist. For example, L is distributive if and only if the following holds for all elements x, y, z in L:

(xy)(yz)(zx) = (xy)(yz)(zx).

Similarly, L is distributive if and only if

xz = yz and xz = yz always imply x=y.

• Hasse diagrams of the two prototypical non-distributive lattices

• The diamond lattice M₃.

• The pentagon lattice N₅.

The simplest non-distributive lattices are M₃, the "diamond lattice", and N₅, the "pentagon lattice". A lattice is distributive if and only if none of its sublattices is isomorphic to M₃ or N₅; a sublattice is a subset that is closed under the meet and join operations of the original lattice. Note that this is not the same as being a subset that is a lattice under the original order (but possibly with different join and meet operations). Further characterizations derive from the representation theory in the next section.

Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements. If a lattice is distributive, its covering relation forms a median graph.

Furthermore, every distributive lattice is also modular.