We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω1 together with ¬CH and Martin's axiom for countable p.o.-sets.

Abstract A Boolean algebra B is said to be openly generated if {A : A ≤rcB, ∣A∣ = ℵ0} includes a club subset of . We show: (V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1). (MA+(σ-closed)). Every -free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Ščepin [15] (Proposition 4.3 and Theorem 4.4).