Proving Q is a Lattice but Not a (sigma)-Lattice

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In summary, The conversation discusses the problem of proving that the field Q (rational numbers) is a lattice but not a (sigma)-lattice under the usual order. The individual initially questions the existence of a supremum for the subset [0,1] in Q, but realizes that 1 is indeed a least upper bound. They then question the existence of a supremum for the subset between 0 and a positive irrational number, but ultimately conclude that the lack of a supremum does not disprove Q as a lattice. The conversation ends with the confirmation that the individual's understanding is correct.
  • #1
beeftrax
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I'm reading "A Course in Advanced Calculus" by Robert Borden, and one of the problems begins as follows:

"Prove that the field Q is a lattice, but not a (sigma)-lattice, under the usual order" (pg.25)

Q is of course the rational numbers.

However, Q doesn't seem to be a lattice, since the supremum of, say, [0,1] doesn't exist, since given any upper bound eg 1.1, a smaller upper bound eg 1.01 that is still in Q can be found.

So is Q not in fact a lattice, or am I missing something?

I apologize if this is in the wrong forum.
 
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  • #2
1 is a least upper bound of [0, 1]
 
  • #3
It is, isn't it. I feel silly. At the risk of getting another simple answer to a stupid question, what about an interval between 0 and a positive irrational number, say sqrt(2). Does the supremum of such an interval lie within Q?
 
  • #4
On further thought, I'll answer my own question (or try to). The subset I described isn't finite, so it's lack of a supremum doesn't mean that Q isn't a lattice.
 
  • #5
Exactly right.
 

1. What is a lattice?

A lattice is a mathematical structure that consists of a set of elements and two binary operations, usually denoted as "∨" (join) and "∧" (meet). These operations allow for the combination of any two elements in the set to produce a unique result.

2. What does it mean for a lattice to be a (sigma)-lattice?

A (sigma)-lattice is a special type of lattice that also satisfies the countable distributive law. This means that for any countable set of elements in the lattice, their join and meet operations can be interchanged without changing the final result.

3. How can a lattice be proven to not be a (sigma)-lattice?

A lattice can be proven to not be a (sigma)-lattice by providing a counterexample. This means finding a set of elements in the lattice where the countable distributive law does not hold.

4. What is the significance of proving Q is a lattice but not a (sigma)-lattice?

Proving Q, which represents the rational numbers, is a lattice but not a (sigma)-lattice is significant because it shows that not all lattices are (sigma)-lattices. This challenges the idea that a lattice must also be a (sigma)-lattice and highlights the importance of counterexamples in mathematical proofs.

5. How can this proof be applied in other areas of science?

This proof can be applied in other areas of science, such as computer science and physics, where lattices are used to model various systems. By understanding the properties of lattices and their variations, scientists can better analyze and interpret data, as well as develop more accurate models and simulations.

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