Abstract
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 II(lambda) has size at least lambda; if lambda is a singular cardinal and mu(<kappa) < lambda <= mu(kappa) for sonic 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 II(A) 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 <= lambda; (V) all cardinals of the form lambda(kappa) with 0 <= kappa <= lambda occur as the cardinalities of sets of complements to some partition P is an element of II(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.