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_{*},
t_{0},
t_{1}
Î \mathcal L(M) $.
We write $\$$t_{1} =
t_{*}
\sqcup
t_{0} $
or say that $$t_{1}
is a disjoint sum of
$$t_{*}
and $$t_{0} if $$t_{1} =
inf{t_{0},
t_{*}} and
$sup\{$t_{0}, t_{*}} is the
discrete topology.

Given $\$$t_{1},
t_{0}
Î
\mathcal L(M) $, we say that $$t_{0}
is a disjoint summand
of $$t_{1} if
$\$$t_{1} =
t_{*}
\sqcup
t_{0}
$ for a certain $\$$t_{*}
Î
\mathcal L(M) $.
Some necessary and some sufficient conditions are proved
for $$t_{0} to be
a disjoint summand of $$t_{1}.

Location: http://mech.math.msu.su/~fpm/eng/k03/k031/k03101h.htm

Last modified: April 4, 2004.