Orders, lattices and well-orders

Which of these relations are partial orders (reflexive, antisymmetric, transitive)? (Choose all that apply.)

Consider $\{2, 3, \dots, 10\}$ ordered by divisibility. List all its maximal elements, separated by commas.

In the lattice $(\N, \mid)$, what is the join (least upper bound) $12 \vee 18$?

How many edges (covering pairs) does the Hasse diagram of $(\mathcal{P}(\{1, 2, 3, 4\}), \subseteq)$ have?

Theorem 2 says every finite non-empty poset has a maximal element. Which example shows that the word "finite" cannot simply be dropped?

Which of these totally ordered sets are well-ordered (every non-empty subset has a least element)? (Choose all that apply.)

Which Lean statement says correctly that every finite non-empty poset has a maximal element?

Knaster–Tarski: a monotone map on a complete lattice has a fixed point. Which examples show that one of the two hypotheses cannot be dropped? (Choose all that apply.)

Write linear_extension(elements, leq) that lists the elements of a finite poset in an order compatible with leq (a function returning True when $a \le b$): whenever $a \le b$ and $a \neq b$, $a$ must come before $b$. Use the idea of Corollary 3, or repeatedly take a minimal element.