For the algebraic convergence λ s , which generates the well known sequential topology τ s on a complete Boolean algebra B, we have λ s = λ ls ∩ λ li , where the convergences λ ls and λ li are defined by λ ls (x) = {lim sup x}↑ and λ li (x) = {lim inf x} ↓ (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology O lsi extending the (unique) sequential topologies O λ ls (left) and O λ li (right) generated by the convergences λ ls and λ li and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences. In addition, we observe some special classes of algebras and, in particular, show that in (ω, 2)-distributive algebras we have lim O lsi = lim τs = λ s , while the equality O lsi = τ s holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity,