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Introduction to Lattice Theory with Computer Science Applications by Vijay K. Garg

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Chapter 8Modular Lattices

8.1 INTRODUCTION

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.

8.2 MODULAR LATTICE

The definition says that if c08-math-0003, then one can bracket the expression c08-math-0004 either way.

We will show that all distributive lattices are modular. Recall that a lattice c08-math-0005 is distributive if .

In this definition, ...

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