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- W2893643274 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." @default.
- W2893643274 created "2018-10-05" @default.
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- W2893643274 date "2018-09-26" @default.
- W2893643274 modified "2023-09-27" @default.
- W2893643274 title "The Left, the Right and the Sequential Topology on Boolean Algebras." @default.
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