Chains, antichains, and complements in infinite partition lattices

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$, and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is between the cofinality $\mathrm{cf}(\kappa)$ and $\kappa$, and $\kappa$ always occurs as the cardinality of a maximal well-ordered chain; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every ordinal $\delta$ with $|\delta| $ 2$.