Proving Complete Lattice from Lower-Bounded & Maximal Conditions

[Aryth1]

I need some help figuring out a problem dealing with lattices. The problem is this:

Prove that any lower-bounded lattice satisfying the maximal condition is a complete lattice.

I've been able to figure out some things so far. I know that a lattice is a meet- and join-semilattice, which requires that all pairs of elements have a meet and a join. I used induction to show that this would then be true for all finite subsets as well.

On a side note I proved that if every subset has a join, then they also have a meet. So I really only need to show that every subset of this lower-bounded lattice has a join or that every subset has a meet. The other would then follow.

I thought that the maximal condition would help with proving that every subset has a join, but it is proving difficult since maximal elements are obviously not maximum elements.

One other method I've thought of was to try to prove that all directed subsets of the lattice have a join. That is equivalent to saying that every subset has a join (when you add-in the fact that we supposed a lower-bounded join-semilattice). I got a little farther with this by gathering the family of all maximal elements in the lattice that we gather by taking the subsets. I noticed that this set was necessarily an anti-chain. I didn't go much farther, hopefully someone else can shed some light on this.

Any help would be greatly appreciated!

 

[mmwave]

Hello there,

First of all, great job on using induction to prove that the lattice satisfies the meet and join-semilattice properties for finite subsets. That is definitely a good starting point.

To prove that the lattice is complete, we need to show that every subset has a join and a meet. Since you have already proved that if every subset has a join, then they also have a meet, we just need to focus on proving that every subset has a join.

As you mentioned, the maximal condition can be helpful here. Let's start by defining what a maximal element is in a lattice. A maximal element is an element that is not smaller than any other element in the lattice. In other words, there is no element in the lattice that is strictly greater than the maximal element.

Now, let's consider a subset of the lattice that does not have a join. This means that there is no element in the lattice that is an upper bound for this subset. In other words, there is no element in the lattice that is greater than or equal to all the elements in this subset.

But wait, if there is no upper bound for this subset, then there must be an element in the subset itself that is not smaller than any other element in the subset. This element is a maximal element in the subset.

Now, let's consider the family of all maximal elements in the lattice that can be obtained by taking subsets. This family is an anti-chain, as you have noted. This means that there is no element in this family that is greater than any other element in this family.

But wait again, if this family is an anti-chain, then there must be an element in this family that is not smaller than any other element in this family. This element is a maximal element in the lattice.

So, we have shown that every subset of the lattice has a maximal element. And since every maximal element is also an upper bound, every subset has an upper bound, and thus, a join.

I hope this helps to shed some light on the problem. If you have any further questions, please feel free to ask. Keep up the good work!