## 6

## Lattices

A lattice is a special kind of an ordered set with two binary operations. Lattice structures are used in computing and mathematical applications.

##### 6.1 LATTICE

**Definition 6.1**

A **lattice** is a partially ordered set (*L*, ≤) in which every subset {*a*, *b*} consisting of two elements has a least upper bound and a greatest lower bound.

We denote LUB({*a*, *b*}) by *a* ∨ *b* and call it **join** or **sum of** *a* **and** *b*. Similarly, we denote
GLB ({*a*, *b*}) by *a* ∧ *b* and call it **meet** or **product of** *a* **and** *b*.

Other symbols used are

Thus **Lattice is** a mathematical structure with **two binary operations, join and meet.**

A totally ordered set is obviously a lattice but not all partially ordered sets are lattices.

**EXAMPLE 6.1**

Let *A* be any set and ...

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