(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, $\mathcal L\left(M\right)$ 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 $sup\left\{$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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