The Lattice, partially ordered sets, infimum and supremum.

[zaff9]

Hi,

So my question is this:

Let X = {1,2,3,12,18,36} be a set, and ≥ is a binary relation defined as follows: For any x,y∈X, x≥ y if x is a multiple of y. Is X a lattice?

Okay, so here is what I know.

I know that a lattice is a partially ordered set. So to show X is a lattice the first step I need is to show that the binary relation is reflexive, transitive, and anti-symmetric.

Then we define a lattice as being a partially ordered set where every pair of elements x,y have a least upper bound and a greatest lower bound. So I have to find both of these characteristics. Once I do this, I have successfully proven a lattice.

So my problem is that I don't know how to take the sequence of X and show where the upper and lower bounds of it are. I know I have to split it apart, and find every possible pair for the sequence to find their upper and lower bounds. But i'm not quite sure what to do there.. For instance, do I just take {1,2} and say okay, 1 is the greatest lower and 2 is the least upper and im done? How does the binary relation effect how I split my x,y pairs, or is the binary relation only there for me to show partial order first.

Thanks guys.

 

[pka]

So my question is this:
Let X = {1,2,3,12,18,36} be a set, and ≥ is a binary relation defined as follows: For any x,y∈X, x≥ y if x is a multiple of y. Is X a lattice?
Okay, so here is what I know.
I know that a lattice is a partially ordered set. So to show X is a lattice the first step I need is to show that the binary relation is reflexive, transitive, and anti-symmetric. Then we define a lattice as being a partially ordered set where every pair of elements x,y have a least upper bound and a greatest lower bound. So I have to find both of these characteristics. Once I do this, I have successfully proven a lattice.

Have you written out the partial order on that set? Have you drawn the Hasse diagram?

Once you do those two things, the answer should clear,

Note that there is a greatest term and a least term in this poset. That should do it for also.