Chains, Antichains, and Complements in Infinite Partition Lattices

Abstract

We consider the partition lattice $Π_κ$ on any set of transfinite cardinality $κ$ and properties of $Π_κ$ 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 $κ$; (II) there are maximal chains in $Π_κ$ of cardinality $> κ$; (III) if, for every cardinal $λ< κ$, we have $2^λ < 2^κ$, there exists a maximal chain of cardinality $< 2^κ$ (but $\ge κ$) in $Π_{2^κ}$; (IV) every non-trivial maximal antichain in $Π_κ$ has cardinality between $κ$ and $2^κ$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(κ, 2^λ)$ for any $λ\le κ$; (V) all cardinals of the form $κ^λ$ with $0 \le λ\le κ$ occur as the number of complements to some partition $\mathcal{P} \in Π_κ$, 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.