FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2003, VOLUME 9, NUMBER 1, PAGES 3-18

On disjoint sums in the lattice of linear topologies

V. I. Arnautov
K. M. Filippov

Abstract

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Let $M$ be a vector space over a skew-field equipped with the discrete topology, $\$\; \backslash mathcal\; L(M)\; \$$ be the lattice of all linear topologies on $M$ ordered by inclusion, and $\$$t_*, t₀, t₁ Î \mathcal L(M) $. We write $\$$t₁ = t_* \sqcup t₀ $ or say that $$t₁ is a disjoint sum of $$t_* and $$t₀ if $$t₁ = inf{t₀, t_*} and $sup\{$t₀, t_*} is the discrete topology.

Given $\$$t₁, t₀ Î \mathcal L(M) $, we say that $$t₀ is a disjoint summand of $$t₁ if $\$$t₁ = t_* \sqcup t₀ $ for a certain $\$$t_* Î \mathcal L(M) $. Some necessary and some sufficient conditions are proved for $$t₀ to be a disjoint summand of $$t₁.

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Last modified: April 4, 2004.