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.