Editing Complete lattice (section)

=== Completion ===
<!-- This section is linked from [[Completely distributive lattice]]. See [[WP:MOS#Section management]] -->

If a complete lattice is freely generated from a given ''poset'' used in place of the set of generators considered above, then one speaks of a ''completion'' of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that are left adjoint to the forgetful functor in the converse direction.

As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called [[Dedekind–MacNeille completion]]. For this process, elements of the poset are mapped to (Dedekind-) ''cuts'', which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above.

The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set. Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above.