On ultraproducts of Boolean algebras and irr

§ 1. Consistent inequality [We prove the consistency of irr(∏ _i<K B_i/D) < ∏_i<K irr(B _i)/D where D is an ultrafilter on K 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 K = א₁ (easily we cannot have it for K = א₀) and Boolean algebras B_i (i < K) of cardinality < _ω1.] This article continues Magidor Shelah [MgSh 433] and Shelah Spinas [ShSi 677], but does not rely on them: see [M2] for the background.