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

TL;DR

This paper explores various sequential topologies on Boolean algebras, analyzing their relationships, special cases, and conditions under which certain topologies coincide or differ, thus advancing understanding of convergence structures in algebraic topology.

Contribution

It introduces the minimal topology extending left and right sequential topologies on Boolean algebras and establishes a hierarchy among these topologies and convergences.

Findings

In $( ext{omega},2)$-distributive algebras, certain topologies coincide with the sequential topology.

In Maharam algebras, the minimal topology equals the sequential topology.

Some collapsing algebras exhibit maximal diversity in topological structures.

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.