Let P be a poset.

Then the least upper bound of {} is, naturally, the least of the upper bounds of {}. Every member of P is an upper bound for {}. So the least upper bound for {} is the least member of P, i.e., the minimum element of P -- if such an element exists.

By the same logic, the greatest lower bound of {} is the maximum element of P, if it exists. This is the same conclusion Dan reached.

Notes:

* Since we are in a poset, antisymmetry implies there can be at most one maximum, and one minimum element. For if there were two, \\(a\\) and \\(b\\), then we'd have both \\(a \le b\\) and \\(b \le a\\), which would imply they are the same.

* Minimal and maximal elements are a different, and weaker notion than the concepts of minimum and maximum, which are global. To be minimal means there's nothing less than it, and to be maximal means there's nothing greater than it. In the poset where the only relation is identity, every element is minimal, and maximal -- but there is no maximum or minimum for the whole set.

Then the least upper bound of {} is, naturally, the least of the upper bounds of {}. Every member of P is an upper bound for {}. So the least upper bound for {} is the least member of P, i.e., the minimum element of P -- if such an element exists.

By the same logic, the greatest lower bound of {} is the maximum element of P, if it exists. This is the same conclusion Dan reached.

Notes:

* Since we are in a poset, antisymmetry implies there can be at most one maximum, and one minimum element. For if there were two, \\(a\\) and \\(b\\), then we'd have both \\(a \le b\\) and \\(b \le a\\), which would imply they are the same.

* Minimal and maximal elements are a different, and weaker notion than the concepts of minimum and maximum, which are global. To be minimal means there's nothing less than it, and to be maximal means there's nothing greater than it. In the poset where the only relation is identity, every element is minimal, and maximal -- but there is no maximum or minimum for the whole set.