Zorn's Lemma ensures that every inductively ordered set has a maximal element. There is an obvious duality between the notion of a maximal and minimal element.
Question 1: Given this duality, could we also formulate a dual Lemma that gives the existence of a minimal element for any inductively ordered set?
Question 2: I suppose this fails, since i have never heard or seen of such a Lemma and there might be some good counterexamples. However, how does this failure can be traced back to the equivalence of Zorn's Lemma with the well-ordering principle of the integers? Why does the latter imply that we can say more about maximal elements rather than for minimal elements?