least-upper-bound and greatest-lower-bound properties 
*

*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?

*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!
 A: "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.
A: 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.

