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1. Consistent inequality [We prove the consistency of irr$(\displaystyle\prod_{i B i)/ D where D is an ultrafilter on ? 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 ?=? 1 (easily we cannot have it for ?=? 0) and Boolean algebras B i ( i<?) 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.

On ultraproducts of Boolean algebras and irr