# Order Theory/Lattices

## Definition and characterisationEdit

**Definition (lattice)**:

Let be an ordered set. is called a **lattice** if and only if any two elements have both a join and a meet.

**Definition (algebraic lattice)**:

Let be any set, and let and be two functions. is called an **algebraic lattice** if and only if the functions and satisfy the following: For all

## Special types of latticesEdit

**Definition (complete lattice)**:

A **complete lattice** is an ordered set such that whenever is a family of elements of , both and exist.