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!

"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.
