Order Theory/Lattices

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Definition and characterisation[edit]

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 lattices[edit]

Definition (complete lattice):

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