Editing Complete lattice (section)

=== Free "complete semilattices" ===
The construction of [[free object]]s depends on the chosen class of morphisms. Functions that preserve all joins (i.e. lower adjoints of Galois connections) are called ''free complete join-semilattices''.

The standard definition from [[universal algebra]] states that a free complete lattice over a generating set <math>S</math> is a complete lattice <math>L</math> together with a function <math>i: S \rightarrow L</math>, such that any function <math>f</math> from <math>S</math> to the underlying set of some complete lattice <math>M</math> can be ''factored uniquely'' through a morphism <math>f^ \circ</math> from <math>L</math> to <math>M</math>. This means that <math>f(s) = f^\circ(i(s))</math> for every element <math>s</math> of <math>S</math>, and that <math>f^\circ</math> is the only morphism with this property. Hence, there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is [[adjoint functors|left adjoint]] to the [[forgetful functor]] from complete lattices to their underlying sets.

Free complete lattices can thus be constructed such that the complete lattice generated by some set ''<math>S</math>'' is just the [[powerset]] <math>2^S</math>, the set of all subsets of ''<math>S</math>'' ordered by [[subset|subset inclusion]]. The required unit <math>i: S \rightarrow 2^S</math> maps any element <math>s</math> of <math>S</math> to the singleton set <math>\{s\}</math>. Given a mapping <math>f</math> as above, the function <math>f^\circ : 2^S \rightarrow M</math> is defined by
 
:<math>f^\circ (X) = \bigvee \{ f(s) | s \in X \}</math>.

Then <math>f^\circ</math> transforms unions into suprema and thus preserves joins.

These considerations also yield a free construction for morphisms that preserve meets instead of joins (i.e. upper adjoints of Galois connections). The above can be [[duality (order theory)|dualized]]: free objects are given as powersets ordered by reverse inclusion, such that set union provides the meet operation, and the function <math>f^\circ</math> is defined in terms of meets instead of joins. The result of this construction is known as a ''free complete meet-semilattice''. It can be noted that these free constructions extend those that are used to obtain [[semilattice|free semilattices]], where finite sets need to be considered.