Free Lattice - The Complete Free Lattice

The Complete Free Lattice

Another corollary is that the complete free lattice "does not exist", in the sense that it is instead a proper class. The proof of this follows from the word problem as well. To define a complete lattice in terms of relations, it does not suffice to use the finitary relations of meet and join; one must also have infinitary relations defining the meet and join of infinite subsets. For example, the infinitary relation corresponding to "join" may be defined as

Here, f is a map from the elements of a cardinal N to FX; the operator denotes the supremum, in that it takes the image of f to its join. This is, of course, identical to "join" when N is a finite number; the point of this definition is to define join as a relation, even when N is an infinite cardinal.

The axioms of the pre-ordering of the word problem may be adjoined by the two infinitary operators corresponding to meet and join. After doing so, one then extends the definition of to an ordinally indexed given by

when is a limit ordinal. Then, as before, one may show that is strictly greater than . Thus, there are at least as many elements in the complete free lattice as there are ordinals, and thus, the complete free lattice cannot exist as a set, and must therefore be a proper class.