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Is Q a lattice?

  1. Sep 24, 2003 #1
    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.
  2. jcsd
  3. Sep 24, 2003 #2


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    1 is a least upper bound of [0, 1]
  4. Sep 25, 2003 #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?
  5. Sep 25, 2003 #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.
  6. Sep 25, 2003 #5


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    Exactly right.
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