Order Theory/Lattices

Definition and characterisation

Definition (lattice):

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

Definition (algebraic lattice):

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

1. ${\displaystyle }$

Special types of lattices

Definition (complete lattice):

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