# 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 ${\displaystyle x,y,z\in X}$

1. ${\displaystyle }$

## 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.