Chains,antichains, and complements in infinite partition lattices

We consider the partition lattice (Pi (lambda )) on any set of transfinite cardinality (lambda ) and properties of (Pi (lambda )) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly (lambda ); (II) there are maximal chains in (Pi (lambda )) of cardinality (> lambda ); (III) a regular cardinal (lambda ) is strongly inaccessible if and only if every maximal chain in (Pi (lambda )) has size at least (lambda ); if (lambda ) is a singular cardinal and (mu ^{< kappa } < lambda le mu ^kappa ) for some cardinals (kappa ) and (possibly finite) (mu ), then there is a maximal chain of size (< lambda ) in (Pi (lambda )); (IV) every non-trivial maximal antichain in (Pi (lambda )) has cardinality between (lambda ) and (2^{lambda }), and these bounds are realised. Moreover, there are maximal antichains of cardinality (max (lambda , 2^{kappa })) for any (kappa le lambda ); (V) all cardinals of the form (lambda ^kappa ) with (0 le kappa le lambda ) occur as the cardinalities of sets of complements to some partition (mathcal {P} in Pi (lambda )), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.