Thread: Partially ordered set with no maximal element

Could you please help with this?

Let be a non-empty partially ordered set and assume that no element of is maximal. Use the Axiom of Choice to show that there exists a function such that

Here's my idea. We're going to construct a relation that you can apply Axiom of Choice to for obtaining the function you desire.

Pick some arbitrary

Construct a set of ordered pairs R s.t. it has two properties.

(1)

(2) if for some and <x^{+}, y'> s.t. y P_{r} y'

You know that there will always exist greater y's because P has no maximal unit.

Then from axiom of choice you can create a function from this relation and I believe it should have all the properties you want.

Thank you a lot for your help; could you please just edit the equations, as they're given as "Latex Error: Syntax Error"'s now?

Thank a lot,

Mimi

I've edited my eqns. Also P_{r} represents the partial order relationship that is assumed in P.