On ultraproducts of Boolean algebras and irr

Abstract.

1. Consistent inequality [We prove the consistency of irr$(\displaystyle\prod_{i < \kappa} B_i/D) < \displaystyle\prod_{i < \kappa}$irr(B _i )/D where D is an ultrafilter on κ and each B _i is a Boolean algebra and irr(B) is the maximal size of irredundant subsets of a Boolean algebra B, see full definition in the text. This solves the last problem, 35, of this form from Monk's list of problems in [M2]. The solution applies to many other properties, e.g. Souslinity.] 2. Consistency for small cardinals [We get similar results with κ=ℵ₁ (easily we cannot have it for κ=ℵ₀) and Boolean algebras B _i (i<κ) of cardinality $< \beth_{\omega_1}$.] This article continues Magidor Shelah [MgSh:433] and Shelah Spinas [ShSi:677], but does not rely on them: see [M2] for the background.