A posteriori convergence in complete Boolean algebras with the sequential topology

A sequence x=〈xn:n∈ω〉 of elements of a complete Boolean algebra (briefly c.B.a.) B converges to b∈B a priori (in notation x→b) if lim infx=lim supx=b. The sequential topology τs on B is the maximal topology on B such that x→b implies x→τsb, where →τs denotes the convergence in the space 〈B,τs〉 — the a posteriori convergence.These two forms of convergence, as well as the properties of the sequential topology related to forcing, are investigated. So, the a posteriori convergence is described in terms of killing of tall ideals on ω, and it is shown that the a posteriori convergence is equivalent to the a priori convergence iff forcing by B does not produce new reals. A property (ħ) of c.B.a.’s, satisfying t-cc ⇒(ħ)⇒s-cc and providing an explicit (algebraic) definition of the a posteriori convergence, is isolated. Finally, it is shown that, for an arbitrary c.B.a. B, the space 〈B,τs〉 is sequentially compact iff the algebra B has the property (ħ) and does not produce independent reals by forcing, and that s=ω1 implies P(ω) is the unique sequentially compact c.B.a. in the class of Suslin forcing notions.