Thread: 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]

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.

fully ordered?

Originally Posted by emlevy

fully ordered?