Chains, Antichains, and Complements in Infinite Partition Lattices

Abstract: 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 always exactly $\kappa$; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $&gt; \kappa$; (III) if, for every cardinal $\lambda &lt; \kappa$, we have $2^{\lambda} &lt; 2<sup>\kappa$, there exists a maximal chain of cardinality $&lt; 2<sup>{\kappa}$ (but $\ge \kappa$) in $\Pi_{2<sup>\kappa}$; (IV) every non-trivial maximal antichain in $\Pi_\kappa$ has cardinality between $\kappa$ and $2<sup>{\kappa}$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(\kappa, 2<sup>{\lambda})$ for any $\lambda \le \kappa$; (V) all cardinals of the form $\kappa<sup>\lambda$ with $0 \le \lambda \le \kappa$ occur as the number of complements to some partition $\mathcal{P} \in \Pi_\kappa$, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.