# Order Theory/Lattices

## Definition and characterisation

Definition (lattice):

Let ${\displaystyle (X,\leq )}$ be an ordered set. ${\displaystyle X}$ is called a lattice if and only if any two elements ${\displaystyle x,y\in X}$ have both a join and a meet.

Definition (algebraic lattice):

Let ${\displaystyle L}$ be any set, and let ${\displaystyle \vee :L\times L\to L}$ and ${\displaystyle \wedge :L\times L\to L}$ be two functions. ${\displaystyle L}$ is called an algebraic lattice if and only if the functions ${\displaystyle \vee }$ and ${\displaystyle \wedge }$ satisfy the following: For all ${\displaystyle x,y,z\in X}$

## Special types of lattices

Definition (complete lattice):

A complete lattice is an ordered set ${\displaystyle (X,\leq )}$ such that whenever ${\displaystyle (x_{i})_{i\in I}}$ is a family of elements of ${\displaystyle X}$, both ${\displaystyle \bigvee _{i\in I}x_{i}}$ and ${\displaystyle \bigwedge _{i\in I}x_{i}}$ exist.