Thesis (M. Phil.)--University of Hong Kong, 1981.

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

School of Mathematics University of the Witwatersrand (Wits), Johannesburg 31 August 2015 Submitted in partial fulflment of a Masters degree at Wits / The operation defined on a discrete semigroup S can be extended to the Stone- Cech compactification S of S so that for all a 2 S, the left translation S 3 x 7! ax 2 S is continuous, and for all q 2 S, the right translation S 3 x 7! xq 2 S is continuous. Because any compact right topological semigroup, S contains a smallest two-sided ideal K( S) which is a completely simple semigroup. We give an exposition of some basic results related to the semigroup S and to the semigroup algebra `1( S). In particular, we review the result that `1( N) is semisimple if and only if `1(K( N)) is semisimple. We also review the reduction of the question whether `1(K( N)) is semisimple to a question about K( N).

Diss. (sammanfattning) Uppsala : Univ., 2003. / Härtill 3 uppsatser.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1971. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 54-55).

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemerédi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length.

Assumindo a existência de $\\mathfrak c$ ultrafiltros seletivos dois a dois incomparáveis (segundo a ordem de Rudin-Keisler) provamos que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta com uma seqüência não trivial convergente. Sob as mesmas hipóteses, mostramos que um grupo topológico abeliano quase livre de torção $(G, +, \\tau)$ com $|G| = |\\tau| = \\mathfrak c$ admite uma topologia independente de $\\tau$ que o torna um grupo topológico e caracterizamos algebricamente os grupos abelianos de não torção que têm cardinalidade $\\mathfrak c$ e que admitem uma topologia de grupo enumeravelmente compacta (sem seqüências não triviais convergentes). Provamos, ainda, que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo que torna seu quadrado enumeravelmente compacto e construímos um semigrupo de Wallace cujo quadrado é, também, enumeravelmente compacto. Por fim, assumindo a existência de $2^{\\mathfrak c}$ ultrafiltros seletivos, garantimos que se um grupo abeliano de não torção e cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta, então o mesmo admite $2^{\\mathfrak c}$ topologias de grupo enumeravelmente compactas (duas a duas não homeomorfas). / Assuming the existence of $\\mathfrak c$ pairwise incomparable selective ultrafilters (according to the Rudin-Keisler ordering) we prove that the free abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology that contains a non-trivial convergent sequence. Under the same hypothesis, we show that an abelian almost torsion-free topological group $(G, +, \\tau)$ with $|G| = |\\tau| = \\mathfrak c$ admits a group topology independent of $\\tau$ and we algebraically characterize the non-torsion abelian groups of cardinality $\\mathfrak c$ which admit a countably compact group topology (without non-trivial convergent sequences). We also prove that the free abelian group of cardinality $\\mathfrak c$ admits a group topology that makes its square countably compact and we construct a Wallace\'s semigroup whose square is countably compact. Finally, assuming the existence of $2^$ selective ultrafilters, we ensure that if a non-torsion abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology, then it admits $2^$ (pairwise non-homeomorphic) countably compact group topologies.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.