Then f° transforms unions into suprema and thus preserves joins. Our considerations also yield a free construction for morphisms that do preserve meets instead of joins (i.e. upper adjoints of Galois connections). In fact, we merely have to dualize what was said above: free objects are given as powersets ordered by reverse inclusion, such that set union provides the meet operation, and the function f° is defined in terms of meets instead of joins. The result of this construction could be called a free complete meet-semilattice. One should also note how these free constructions extend those that are used to obtain free semilattices, where we only need to consider finite sets. Free complete lattices The situation for complete lattices with complete homomorphisms obviously is more intricate. In fact, free complete lattices do generally not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument-sets of every cardinality. This property in itself is not a problem: as the case of free complete semilattices above shows, it can well be that the solution of the word problem leaves only a set of equivalence classes. In other words, it is possible that proper classes of the class of all terms have the same meaning and are thus identified in the free construction. However, the equivalence classes for the word problem of complete lattices are "too small", such that the free complete lattice would still be a proper class, which is not allowed. Now one might still hope that there are some useful cases where the set of generators is sufficiently small for a free complete lattice to exist. Unfortunately, the size limit is very low and we have the following theorem: The free complete lattice on three generators does not exist; it is a proper class. A proof of this statement is given by Johnstone; the original argument is attributed to Alfred W. Hales; see also the article on free lattices. Completion 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,