Free lattice

In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property.

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 (FX,η), where η is the unit map η:X→FX which takes x∈X to the singleton set {x}. The universal property is then as follows: given any map f:X→L from X to some arbitrary semilattice L, there exists a unique semilattice homomorphism ${\displaystyle {\tilde {f}}:FX\to L}$ such that ${\displaystyle f={\tilde {f}}\circ \eta }$. The map ${\displaystyle {\tilde {f}}}$ may be explicitly written down; it is given by

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