We have seen that lattices are nicer structures than general posets because they allow us to take the meet and the join for any pair of elements in the set. What if we wanted to take the join and the meet of arbitrary subsets? Complete lattices allow us to do exactly that. All finite lattices are complete, so the concept of complete lattices is important only for infinite lattices. In this chapter, we first discuss complete lattices and show many ways in which complete lattices arise in mathematics and computer science. In particular, topped –structures and closure operators give us complete lattices.
Next we consider the question: What if the given poset is not a complete lattice or even a lattice? Can we embed it into a complete lattice? This brings us to the notion of lattice completion which is useful for both finite and infinite posets.
Recall the definition of a complete lattice.
Finite lattices are always ...