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

Abstract

For the algebraic convergence $\lambda_{\mathrm{s}}$, which generates the well known sequential topology $\tau_s$ on a complete Boolean algebra ${\mathbb B}$, we have $\lambda_{\mathrm{s}}=\lambda_{\mathrm{ls}}\cap \lambda_{\mathrm{li}}$, where the convergences $\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$ are defined by $\lambda_{\mathrm{ls}}(x)=\{ \limsup x\}\!\uparrow$ and $\lambda_{\mathrm{li}}(x)=\{ \liminf x\}\!\downarrow$ (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 $\mathcal{O}_{\lambda_{\mathrm{ls}}}$ (left) and $\mathcal{O}_{\lambda_{\mathrm{li}}}$ (right) generated by the convergences $\lambda_{\mathrm{ls}}$ and $\lambda_{\mathrm{li}}$ 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 $(\omega,2)$-distributive algebras we have $\lim_{{\mathcal O}_{\mathrm{lsi}}}=\lim_{\tau _{\mathrm{s}} }=\lambda _{\mathrm{s}}$, while the equality $\mathcal{O}_{\mathrm{lsi}}=\tau_s$ holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.