TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 5, Pages 2023–2047 S 0002-9947(99)02270-9 Article electronically published on April 13, 1999 THE DISTRIBUTIVITY NUMBERS OF P (ω)/FIN AND ITS SQUARE SAHARON SHELAH AND OTMAR SPINAS Abstract. We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length ω 2 , the distributivity number of P(ω)/fin is ω 2 , whereas the distributivity number of r.o.(P(ω)/fin) 2 is ω 1 . This answers a problem of Balcar, Pelant and Simon, and others. Introduction A complete Boolean algebra (B, ≤) is called κ-distributive, where κ is a cardinal, if and only if for every family 〈u αi : i ∈ I α ,α<κ〉 of members of B  α<κ  i∈Iα u αi =  f ∈  α<κ Iα  α<κ u αf (α) holds. It is well-known (see [J, p.152]) that every partially ordered set (P, ≤) which is separative can be densely embedded in a unique complete Boolean algebra, which is usually denoted with r.o.(P ). The distributivity number of (P, ≤) is defined as the least κ such that r.o.(P ) is not κ-distributive. It is well-known (see [J, p.158]) that the following four statements are equivalent: (1) r.o.(P ) is κ-distributive. (2) The intersection of κ open dense sets in P is dense. (3) Every family of κ maximal antichains of P has a refinement. (4) Forcing with P does not add a new subset of κ. The distributivity number of the Boolean algebra P (ω)/fin is denoted with h. This cardinal was introduced in [BPS], where it has been shown that ω 1 ≤ h ≤ 2 ω and the axioms of ZFC do not decide where exactly h sits in this interval. For λ a cardinal let h(λ) be the distributivity number of (P (ω)/fin) λ , where by (P (ω)/fin) λ we mean the full λ-product of P (ω)/fin in the forcing sense. That is, p ∈ (P (ω)/fin) λ if and only if p : λ →P (ω)/fin \{0}. The ordering is coordinatewise. Trivially, h(λ) ≥ h(γ ) holds whenever λ<γ . In fact, if 〈D α : α< h(λ)〉 is a family of dense open subsets of (P (ω)/fin) λ whose intersection is not dense, then, letting D ′ α = {p ∈ (P (ω)/fin) γ : p↾λ ∈ D α }, clearly the D ′ α are dense open in (P (ω)/fin) γ and their intersection is not dense. Since h ≤ 2 ω , this implies that under CH the sequence 〈h(λ): λ ∈ Card〉 is constant with value ℵ 1 . In [BPS, 4.14(2)] we read: “We do not know of any further Received by the editors February 12, 1997 and, in revised form, November 5, 1997. 1991 Mathematics Subject Classification. Primary 03E05, 06E05. The first author is supported by the Basic Research Foundation of the Israel Academy of Sci- ences; publication 494. The second author is supported by the Swiss National Science Foundation. c 2000 American Mathematical Society 2023 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use