Free Lattice - Formal Definition

Formal Definition

Any set X may be used to generate the free semilattice FX. The free semilattice is defined to consist of all of the finite subsets of X, with the semilattice operation given by ordinary set union. The free semilattice has the universal property. The universal morphism is, where the unit map which takes to the singleton set . The universal property is then as follows: given any map from X to some arbitrary semilattice L, there exists a unique semilattice homomorphism such that . The map may be explicitly written down; it is given by

Here, denotes the semilattice operation in L. This construction may be promoted from semilattices to lattices; by construction the map will have the same properties as the lattice.

The symbol F is then a functor from the category of sets to the category of lattices and lattice homomorphisms. The functor F is left adjoint to the forgetful functor from lattices to their underlying sets. The free lattice is a free object.