Inductive sets with chains of finite elements.

Question

An inductive set is said to be a partially ordered set P that satisfies the hypothesis of Zorn's lemma when nonempty (that is every chain in P has a least upper bound).

1)Now is it true that if a poset P satisfies the condition that every chain in P is finite , then P is inductive;

2)As a result every finite poset is inductive and the discrete poset is inductive ;

Answer 1

If a partial order is finite, then every chain is finite, and therefore has a maximum, which is indeed the least upper bound of that chain.

As for the partial order given by the $=$ relation, indeed, it satisfies the hypothesis of Zorn's lemma, and indeed, every element is maximal.

Answer 2

The hypothesis of Zorn's lemma isn't that every chain should have a least upper bound, but rather that every chain has an upper bound.

Every finite non-empty chain has a maximum, which is an upper bound.