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.