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Ideals in Stone-Cech compactificationsToko, Wilson Bombe 04 April 2013 (has links)
A thesis submitted in ful llment of the
requirements for the degree of Doctor of Philosophy
in Mathematics
School of Mathematics
University of the Witwatersrand
Johannesburg
October, 2012 / Let S be an in nite discrete semigroup and S the Stone- Cech compacti
cation of S. The operation of S naturally extends to S and makes S
a compact right topological semigroup with S contained in the topological
center of S. The aim of this thesis is to present the following new
results.
1. If S embeddable in a group, then S contains 22jSj pairwise incomparable
semiprincipal closed two-sided ideals.
2. Let S be an in nite cancellative semigroup of cardinality and
U(S) the set of uniform ultra lters on S. If > !, then there is a
closed left ideal decomposition of U(S) such that the corresponding
quotient space is homeomorphic to U( ). If = !, then for
any connected compact metric space X, there is a closed left ideal
decomposition of U(S) with the quotient space homeomorphic to
X.
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Ações de semigrupos : recorrencia por cadeias em fibrados e compactificações de Ellis / Semigroup actions : Chan recurrence in fiber Bundles and Ellis compactificationsSouza, Josiney Alves de 15 July 2008 (has links)
Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T09:58:09Z (GMT). No. of bitstreams: 1
Souza_JosineyAlvesde_D.pdf: 1777083 bytes, checksum: 43943f3a9ea228d0eb5fbe6c6906cd93 (MD5)
Previous issue date: 2008 / Resumo: Um semigrupo de transformação consiste de um semigrupo de aplicações contínuas definidas num espaço topológico. A hipótese sobre o semigrupo é a propriedade de reversibilidade, isto é, que a coleção das translações do semigrupo satisfaz a propriedade de intersecção finita. A idéia central é de dinamizar um semigrupo de transformação, sendo isto realizado pela introdução dos correspondentes objetos dinâmicos elementares da teoria de semifluxos, ou seja, os conjuntos limites, atratores e repulsores. O conceito de recorrência por cadeias é abordado de uma forma generalizada, sobre espaços paracompactos, tendo como fundamento certas famílias especiais de coberturas abertas do espaço base chamadas famílias admissíveis. Estudamos também ações de grupos de homeomorfismos sobre espaços compactos. Neste caso, a hipótese sobre o grupo é que ele seja gerado por um subsemigrupo reversível, a partir do qual são definidos todos os objetos dinâmicos elementares. Estudamos dois casos específicos de semigrupos de transformações. No primeiro caso, abordamos semigrupos de transformações em fibrados topológicos, especialmente em fibrados flag, e enfatizamos o estudo sobre transitividade por cadeias fibra a fibra. No segundo caso, estudamos ações de grupos sobre compactificações de Ellis, onde apresentamos uma relação entre o conceito de subsemigrupo semitotal e a transitividade por cadeias. Por último, introduzimos o conceito de função recorrente por cadeias, generalizando o conceito de função recorrente. / Abstract: Transformation semigroups are actions of semigroups of continuous maps on topological spaces. We consider reversible semigroups and study dynamics behaviors by introducing the elementary dynamic objects, originals of the semiflows theory, that is, the limit sets, attractors and repellers. We present the concept of chain recurrence for admissible families on paracompact spaces. We also study homeomorphism group action on compact spaces. In this case, the hypothesis on the group is the Ore's condictions. The elementary dynamics objects are defined from the action of the generator reversible subsemigroup. Then we study two specific cases of transformation semigroups. In the first case, we present results on the actions of endomorphism in flag bundles by emphasizing the chain transitivity in the fibres. Next, we study group actions in Ellis compactifications and relate the concept of semitotal subsemigroup to the chain transitivity. Finally, we introduce the concept of chain recurrent function and generalize the concept of recurrent function. / Doutorado / Geometria / Doutor em Matemática
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Ordered spaces of continuous functions and bitopological spacesNailana, Koena Rufus 11 1900 (has links)
This thesis is divided into two parts: Ordered spaces of Continuous Functions and
the algebras associated with the topology of pointwise convergence of the associated
construct, and Strictly completely regular bitopological spaces.
The Motivation for part of the first part (Chapters 2, 3 and 4) comes from the
recent study of function spaces for bitopological spaces in [44] and [45]. In these
papers we see a clear generalisation of classical results in function spaces ( [14] and
[55]) to bi-topological spaces. The well known definitions of the pointwise topology and
the compact open topology in function spaces are generalized to bitopological spaces,
and then familiar results such as Arens' theorem are generalised. We will use the same
approach in chapters 2, 3 and 4 to formulate analogous definitions in the setting of
ordered spaces. Well known results, including Arens' theorem, are also generalised
to ordered spaces. In these chapters we will also compare function spaces in the
category of topological spaces and continuous functions, the category of bi topological
spaces and bicontinuous functions, and the category of ordered topological spaces and
continuous order-preserving functions. This work has resulted in the publication of
[30] and [31].
Continuing our study of Function Spaces, we oonsider in Chapters 5 and 6 some
Categorical aspects of the construction, motivated by a series of papers which includes
[39], [40], [41] and [50]. In these papers the Eilenberg-Moore Category of algebras of
the monad induced by the Hom-functor on the categories of sets and categories of
topological spaces are classified. Instead of looking at the whole product topology we
will restrict ourselves to the pointwise topology and give examples of the EilenbergMoore Algebras arising from this restriction. We first start by way of motivation, with
the discussion of the monad when the range space is the real line with the usual topology.
We then restrict our range space to the two point Sierpinski space, with the aim
of discovering a topological analogue of the well known characterization of Frames as
the Eilenberg-Moore Category of algebras associated with the Hom-F\mctor of maps
into the Sierpinski space [11]. In this case the order structure features prominently, resulting in the category Frames with a special property called "balanced" and Frame
homomorphisms as the Eilenberg-Moore category of M-algebras. This has resulted
in [34].
The Motivation for the second part comes from [20] and [15]. In [20], J. D. Lawson
introduced the notion of strict complete regularity in ordered spaces. A detailed study
of this notion was done by H-P. A. Kiinzi in [15]. We shall introduce an analogous
notion for bitopological spaces, and then shall also compare the two notions in the categories
of bi topological spaces and bicontinuous functions, and of ordered topological
spaces and continuous order-preserving functions via the natural functors considered
in the previous chapters. We further study the Stone-Cech bicompactification and
Stone-Cech ordered compactification in the two categories. This has resulted in [32] and [33] / Mathematical Sciences / D. Phil. (Mathematics)
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