The Left, the Right and the Sequential Topology on Boolean Algebras

Abstract

For the algebraic convergence $λ_{\mathrm{s}}$, which generates the well known sequential topology $τ_s$ on a complete Boolean algebra ${\mathbb B}$, we have $λ_{\mathrm{s}}=λ_{\mathrm{ls}}\cap λ_{\mathrm{li}}$, where the convergences $λ_{\mathrm{ls}}$ and $λ_{\mathrm{li}}$ are defined by $λ_{\mathrm{ls}}(x)=\{ \limsup x\}\!\uparrow$ and $λ_{\mathrm{li}}(x)=\{ \liminf x\}\!\downarrow$ (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology $\mathcal{O}_{\mathrm{lsi}}$ extending the (unique) sequential topologies $\mathcal{O}_{λ_{\mathrm{ls}}}$ (left) and $\mathcal{O}_{λ_{\mathrm{li}}}$ (right) generated by the convergences $λ_{\mathrm{ls}}$ and $λ_{\mathrm{li}}$ 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 $\lim_{{\mathcal O}_{\mathrm{lsi}}}=\lim_{τ_{\mathrm{s}} }=λ_{\mathrm{s}}$, while the equality $\mathcal{O}_{\mathrm{lsi}}=τ_s$ holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.