RUDIN-KEISLER POSETS OF COMPLETE BOOLEAN ALGEBRAS PETER JIPSEN, ALEXANDER PINUS, HENRY ROSE Abstract. The Rudin-Keisler ordering of ultrafilters is extended to com- plete Boolean algebras and characterised in terms of elementary embeddings of Boolean ultrapowers. The result is applied to show that the Rudin-Keisler poset of some atomless complete Boolean algebras is nontrivial. 1. Introduction All concepts and notations not defined below can be found in [3]. Let B be a Boolean algebra, and let P B denote the set of all partitions of B (i.e. maximal sets of pairwise disjoint elements). Note that P B is ordered by the refinement relation: τ ≤ σ if for all x ∈ τ there exists a y ∈ σ such that x ≤ y. Let ˆ σ =  {τ : τ ≤ σ} be the set of nonzero elements of B that are below some element of σ. Since σ is a partition, each x ∈ ˆ σ is less than or equal to a unique y ∈ σ, so there is a natural map j σ from ˆ σ to σ given by j σ (x)= y. For a map s : σ → Y we define ˆ s = s ◦ j σ , and occasionally we also abbreviate doms by s d . For σ ∈ P B we let P (σ) be the powerset Boolean algebra over the set σ. If all joins of subsets of σ exist in B (e.g. if B is |σ|-complete) then we identify P (σ) with the complete subalgebra of B that is completely generated by σ. For powerset Boolean algebras, the Rudin-Keisler ordering of ultrafilters is defined on D ∈ Uf(P (X )), E ∈ Uf(P (Y )) by D ≤ E if there exists a function f : Y → X such that for all S ∈P (X ), S ∈ D implies f −1 [S ] ∈ E. (∗) We also write D ≤ f E if (∗) holds. Note that this implication implies its converse, since S/ ∈ D implies X \ S ∈ D, hence f −1 [X \ S ]= Y \ f −1 [S ] ∈ E and therefore f −1 [S ] / ∈ E. The duality between sets and powerset Boolean algebras implies the following equivalent definition: D ≤ E iff there exists a complete homomorphism α : P (X ) →P (Y ) such that α[D] ⊆ E. Date : June 7, 2004. 1991 Mathematics Subject Classification. 06E10, 03E55, 04A10. Key words and phrases. Boolean algebras, partitions, Rudin-Keisler ordering. The first and third author were supported by grants from the National Research Foundation of South Africa and the University of Cape Town Research Committee. The second author was supported by the Russian Foundation for Fundamental Researchers 99-01-00571. 1