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  1. For a partial-ordered set, I was wondering if the least-upper-bound/greatest-lower-bound property means that any nonempty subset that has an upper/lower bound has a least-upper/greatest-lower bound, or any subset that has an upper/lower bound has a least-upper/greatest-lower bound? Is least-upper-bound property also called Dedekind completeness?
  2. Why is the following statement true:

    A partial ordered set has the least upper bound property if and only if it has the greatest lower bound property.

Thanks and regards!

2 Answers 2

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"Least upper bound" property is that every nonempty set that is bounded above has a least upper bound; dually for "greatest lower bound", so it is only required that nonempty sets have the property.

(For example, the real numbers have the least upper bound property; if you also required the empty set to have a least upper bound, this would require the reals to have a least element).

Yes, Dedekind completeness is the same thing as the least upper bound property.

For 2: If $S$ is a nonempty set that is bounded below, let $B$ be the set of lower bounds of $S$. Show that $B$ is (i) nonempty; and (ii) bounded above. Conclude that $B$ has a least upper bound. Show that the least upper bound of $B$ is also the greatest lower bound of $S$. The converse is proven dually: the least upper bound of a nonempty set that is bounded above is equal to the greatest lower bound of the set of upper bounds.

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    @Tim: the supremum of $X$ is the infimum of the set of upper bounds for $X$, when either of them exists.2011-05-27
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Arturo has answered your questions quite well; I would just like to mention that for a general poset, the property without the non-empty requirement is referred to as bounded completeness; the version with the non-empty requirement is called conditional completeness. Because a least upper bound for the empty set is necessarily a minimum element and a greatest lower bound for the empty set is a maximum element, that explains the claim in the article that

A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element 1, its minimum element 0, or both.

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    @Tim: Apologies, you are correct. I've edited. And yes, your statement in (2) is correct; one direction is clear, and a conditionally complete lattice has a least upper bound and greatest lower bound for every non-empty subset, which combined with the minimum and maximum elements (l.u.b. and g.l.b. of $\varnothing$, respectively), means every subset has a l.u.b. and g.l.b., hence it is complete.2011-05-27