1. ## Well-ordered Relation Help!

Suppose every nonempty subset of a partially ordered set has a least element. Does it follow that this set is well-ordered? [A well-order relation is a linear order with the property that every non-empty subset S has a least element]

2. Originally Posted by emlevy
Suppose every nonempty subset of a partially ordered set has a least element. Does it follow that this set is well-ordered? [A well-order relation is a linear order with the property that every non-empty subset S has a least element]
If a & b are any two points in S, by the given the subset {a,b} has a least element.
It is either a or b. So a<b or b<a.
Therefore, S is fully ordered.

3. fully ordered?

4. Originally Posted by emlevy
fully ordered?
$\text{linearly ordered }=\text{fully ordered }=\text{completely ordered }=\text{totally ordered}.$