Lattices, mathematical economics

[ktheo]

Homework Statement

Let ≥ be a binary relation, defined on the set X as follows: for any x,y$\in$X, x≥y if x-y≥0 and x-y is even.

Determine if the following are lattices:

X={2,4,8}
X={1,4,8,9}

Homework Equations

Note that ≥ in this case is just the symbol for relation as it relates to economics where ≥ is the sign of weak preference. Sorry if that seems obvious I just don't know how familiar people are with economics math or if it's used elsewhere.

I know that the lattice requires that there is an least upper bound, supA and a greatest lower bound, infA.

The Attempt at a Solution

So I really don't know how to solve these tbh. I did a full group theory course but we never covered the lattice and this professor is AWFUL and the material isn't in the book he's taking it from "self notes" for this chapter. From what I can tell, I have to solve some least upper and greatest lower bounds by finding numbers from those sequences that meet the constraint. But I'm not sure how. Could someone just set me on the path and tell me if I at least have the right idea?

 

[Office_Shredder]

I did a full group theory course but we never covered the lattice

Probably because this has nothing to do with group theory? Anyway, as far as the problem goes let's start with your smaller set {2,4,8}. For every pair of numbers, of which there are three, you have to check if there is a lower bound and an upper bound. So you have to check six things in total.

 

[Mark44]

What's a lattice? You defined ≥, but didn't define lattice. Does a lattice use the ≥ relation somehow?

 

[ktheo]

Probably because this has nothing to do with group theory? Anyway, as far as the problem goes let's start with your smaller set {2,4,8}. For every pair of numbers, of which there are three, you have to check if there is a lower bound and an upper bound. So you have to check six things in total.

My group theory was technically "algebraic structures"... so started with set theory etc and we just skipped lattices there then moved into groups.

So take... {2,4}{2,8}{4,8}? and find uppers and lowers of those, considering its a relation if x-y≥0?

I take it it's not so simple as to say for {2,4} 4 is the greatest upper bound.

 

[ktheo]

What's a lattice? You defined ≥, but didn't define lattice. Does a lattice use the ≥ relation somehow?

From what I understand a lattice is a partially ordered set where every pair of elements x and y have a least upper bound and greatest lower bound denoted xVy and x$\wedge$y respectively. I don't quite understand your second question.

 

[Mark44]

You gave a definition of a relation, ≥, but your problem asks whether the given sets are lattices, which is not a mathematical term I'm familiar with. As you have presented the problem, the relation and lattices are two unconnected concepts.

If I were to give you a definition of the word axolotl, and then started talking about slide rules, wouldn't you wonder how these concepts are related? (They're not, of course.) That's why I asked.

 

[ktheo]

You gave a definition of a relation, ≥, but your problem asks whether the given sets are lattices, which is not a mathematical term I'm familiar with. As you have presented the problem, the relation and lattices are two unconnected concepts.

If I were to give you a definition of the word axolotl, and then started talking about slide rules, wouldn't you wonder how these concepts are related? (They're not, of course.) That's why I asked.

I believe that the definition of the relation ≥ has to do with the numbers I can choose for upper or lower bounds in my lattice. Lattices are ordered sets with a "supremum" and a "infimum". But I directly word-for-word wrote the problem out as per the hmwk question.. so I'm not sure heh

 

[Mark44]

Lattices are ordered sets with a "supremum" and a "infimum".

That's what I was looking for.

But I directly word-for-word wrote the problem out as per the hmwk question.. so I'm not sure heh

But you should have include a definition of "lattice" in your post. I'm reasonably sure its definition is given in your book or notes.

 

[Office_Shredder]

I take it it's not so simple as to say for {2,4} 4 is the greatest upper bound.

Mostly because 4 is the least upper bound, not the greatest upper bound (which would be 8, but is not particularly relevant to the problem). Other than that, yup that's pretty much all you have to do

 

[ktheo]

So am I just breaking down each set and describing that for instance for {2,4} we have a greatest lower bound and least upper bound therefore the set is a lattice. Or is it linked together per say and I take those 3 subsets and find 1 upper bound 1 lower bound from the whole set.

 

[ktheo]

Actually Office Shreddder,

Can I just build a table with an x and y, with {2,4,8} on both axis, and then match up according to the chart showing lower bounds and upper bounds, picking the lowest number depending on what bound I am finding?

 

[micromass]

I think it would be easier to first draw a Hasse diagram.

 

[ktheo]

I think it would be easier to first draw a Hasse diagram.

This is what I was thinking essentially
http://math.stackexchange.com/quest...se-diagram-tell-if-the-structure-is-a-lattice

except with numbers.. I see he shows a hasse diagram as well. I understand now that I need to break down the subsets of the set and show that each one has an infimum and supremum, I'm still a little hazy on how this changes my work: x-y≥0 and x-y is even. I understand the definition clear enough.. but it seems like it changes nothing for my work given my set X

 

[ktheo]

Mostly because 4 is the least upper bound, not the greatest upper bound (which would be 8, but is not particularly relevant to the problem). Other than that, yup that's pretty much all you have to do

Going on what you're saying here, my instinct is to say that...

For the first set X:

{2,4,8} is a lattice because:

{2,4} has a least upper bound of 4, and a greatest lower bound of 2.
{2,8} '' '' 8, '' '' 2
{4,8} '' '' 8, '' '' 4

A lattice is a partially ordered set in which every pair of elements x,y has an infimum and supremum, therefore I have shown it's a lattice.

Edit: Wait! I think I understand this now lol... the purpose for defining any x,y∈X, x≥y if x-y≥0 and x-y is even is that to show a lattice I have to show TWO separate things.

1. I have to show that there is a partial order, so I must show that it is reflexive/transitive/anti-symmetric.

2. show that once I know it's a partial order, I show it's a lattice by finding the bounds of each pair of sub elements in X. Am I on the right track?

 

[ktheo]

Ok I realize I am kinda talking to myself here haha but I find it helpful to write this out and finish my problem and if someone could tell me if this seems right I'd appreciate it.

So:

Let ≥ be a binary relation, defined on the set X as follows: for any x,y∈X, x≥y if x-y≥0 and x-y is even.

Determine if the following are lattices:

X={2,4,8}
X={1,4,8,9}

For the first one, I would prove the 3 conditions of the binary relation and then look at the set, and then I say that YES this is indeed a Lattice. The least upper bound of the set X exists and it is is 8. For any subset x,y, there is a common multiple (2). For instance {4,8} have common multiple 2 from the set.

The greatest lower bound of this lattice is 2. For any subset x,y, there is a common divisor. Essentially, 2 divides by anything in any subset that I can come up with.

For the second set X={1,4,8,9} I can immediately say no, this is not a lattice. The reason for this is that there is no common multiple with any combination of 9 in the set. For instance {4,9} has no common multiple from the set.