Ultrafilter semigroups
Date
2011-06-21
Authors
Shuungula, Onesmus
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Abstract
(1) Given a local left topological group X with a distinguished element 0,
denote by Ult(X) the subsemigroup of X consisting of all nonprinci-
pal ultra lters on X converging to 0. Any two countable nondiscrete
zero-dimensional local left topological groups X and Y with count-
able bases are locally isomorphic and, consequently, the subsemigroups
Ult(X) X and Ult(Y ) Y are isomorphic. However, not every
two homeomorphic zero-dimensional local left topological groups are
locally isomorphic. In the rst result of this thesis it is shown that for
any two homeomorphic direct sums X and Y , the semigroups Ult(X)
and Ult(Y ) are isomorphic.
(2) Let S be a discrete semigroup, let S be the Stone- Cech compacti-
cation of S, and let T be a closed subsemigroup of S. The second
and main result of this thesis consists of characterizing ultra lters from
the smallest ideal K(T) of T and from its closure c` K(T), and show-
ing that for a large class of closed subsemigroups T of S, c` K(T)
is not an ideal. This class includes the subsemigroups 0+ Rd and
H (
L
Z2).
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