We describe a special class of lattices called modular lattices. Modular lattices are numerous in mathematics; for example, the lattice of normal subgroups of a group is modular, the lattice of ideals of a ring is modular, and so is the finite-dimensional vector space lattice. Distributive lattices are a special class of modular lattices. The set of all consistent global states in a distributed computation forms a distributive lattice and is therefore a modular lattice.
In this chapter, we first introduce both modular and distributive lattices to show the relationship between them. Later, we focus on modular lattices. Distributive lattices are considered in detail in Chapter 9.
The definition says that if , then one can bracket the expression either way.
We will show that all distributive lattices are modular. Recall that a lattice is distributive if .
In this definition, ...