On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift
On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift
View/Type of Scholarly Publication
DissertationDate of Exam
01.02.2016Date of Publication
30.05.2016Advisor
Geschke, StefanCo-Referee
Koepke, PeterMetadata
Show full item record
Citable Links 
Abstract
The first part of the dissertation concerns the space ω*:=βω\ω and its quotients. A 2002 result from Bella, Dow, Hart, Hrusak, van Mill and Ursino implies that homeomorphisms between 0-dimensional second-countable Hausdorff quotients of ω* can always be lifted to self-homeomorphisms of ω*. We show that 0-dimensionality can be dropped and also prove that if the map between the quotients is continuous, but not necessarily a homeomorphism, then it can be lifted to a continuous map from ω* into itself. We prove that all quotient maps from ω* onto products of compact metrizable spaces are restrictions of quotient maps from βω (with the same range). We then defend our choice of hypotheses for this last result by showing that there is a quotient map from ω* onto the double arrow space (which is separable, 0-dimensional, first-countable, compact and Hausdorff) with no continuous extension from βω.
In the second part, the focus is on the Boolean algebra P(ω)/fin and the automorphisms of P(ω)/fin called the shift, which is induced by the map n → n+1 in ω. We show that the automorphisms sm for m ∈ℤ are the only trivial automorphisms of P(ω)/fin which commute with the shift. Together with a 1993 result of Velickovic, this characterizes all automorphisms of (P(ω)/fin, s) under OCA+MAℵ1. We study the algebra Per of all elements of P(ω)/fin with finite orbit under the action of the shift, and characterize all the automorphisms of Per which commute with the shift (many of which are not powers of the shift) or conjugate the shift to its inverse. Then, we show that every automorphism of the group of trivial automorphisms of P(ω)/fin either preserves or inverts (additively) the index function introduced by van Douwen in 1990. We also construct a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise isomorphic, and a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise non-isomorphic. We finish with a method for constructing automorphisms of P(ω)/fin which conjugate the shift to itself or to its inverse on any given countable subalgebra.
In the second part, the focus is on the Boolean algebra P(ω)/fin and the automorphisms of P(ω)/fin called the shift, which is induced by the map n → n+1 in ω. We show that the automorphisms sm for m ∈ℤ are the only trivial automorphisms of P(ω)/fin which commute with the shift. Together with a 1993 result of Velickovic, this characterizes all automorphisms of (P(ω)/fin, s) under OCA+MAℵ1. We study the algebra Per of all elements of P(ω)/fin with finite orbit under the action of the shift, and characterize all the automorphisms of Per which commute with the shift (many of which are not powers of the shift) or conjugate the shift to its inverse. Then, we show that every automorphism of the group of trivial automorphisms of P(ω)/fin either preserves or inverts (additively) the index function introduced by van Douwen in 1990. We also construct a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise isomorphic, and a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise non-isomorphic. We finish with a method for constructing automorphisms of P(ω)/fin which conjugate the shift to itself or to its inverse on any given countable subalgebra.
Subjects
Logik, Mengenlehre, Boolesche Algebra, Kompaktifizierung, Liften, logic, set theory, Boolean algebra, compactification, lifting, Stone space
Classification (DDC)
510 MathematikSilveira Salles, Tomás: On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift. - Bonn, 2016. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-43312
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-43312
@phdthesis{handle:20.500.11811/6749,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-43312,
author = {{Tomás Silveira Salles}},
title = {On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2016,
month = may,
note = {The first part of the dissertation concerns the space ω*:=βω\ω and its quotients. A 2002 result from Bella, Dow, Hart, Hrusak, van Mill and Ursino implies that homeomorphisms between 0-dimensional second-countable Hausdorff quotients of ω* can always be lifted to self-homeomorphisms of ω*. We show that 0-dimensionality can be dropped and also prove that if the map between the quotients is continuous, but not necessarily a homeomorphism, then it can be lifted to a continuous map from ω* into itself. We prove that all quotient maps from ω* onto products of compact metrizable spaces are restrictions of quotient maps from βω (with the same range). We then defend our choice of hypotheses for this last result by showing that there is a quotient map from ω* onto the double arrow space (which is separable, 0-dimensional, first-countable, compact and Hausdorff) with no continuous extension from βω.
In the second part, the focus is on the Boolean algebra P(ω)/fin and the automorphisms of P(ω)/fin called the shift, which is induced by the map n → n+1 in ω. We show that the automorphisms sm for m ∈ℤ are the only trivial automorphisms of P(ω)/fin which commute with the shift. Together with a 1993 result of Velickovic, this characterizes all automorphisms of (P(ω)/fin, s) under OCA+MAℵ1. We study the algebra Per of all elements of P(ω)/fin with finite orbit under the action of the shift, and characterize all the automorphisms of Per which commute with the shift (many of which are not powers of the shift) or conjugate the shift to its inverse. Then, we show that every automorphism of the group of trivial automorphisms of P(ω)/fin either preserves or inverts (additively) the index function introduced by van Douwen in 1990. We also construct a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise isomorphic, and a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise non-isomorphic. We finish with a method for constructing automorphisms of P(ω)/fin which conjugate the shift to itself or to its inverse on any given countable subalgebra.},
url = {https://hdl.handle.net/20.500.11811/6749}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-43312,
author = {{Tomás Silveira Salles}},
title = {On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2016,
month = may,
note = {The first part of the dissertation concerns the space ω*:=βω\ω and its quotients. A 2002 result from Bella, Dow, Hart, Hrusak, van Mill and Ursino implies that homeomorphisms between 0-dimensional second-countable Hausdorff quotients of ω* can always be lifted to self-homeomorphisms of ω*. We show that 0-dimensionality can be dropped and also prove that if the map between the quotients is continuous, but not necessarily a homeomorphism, then it can be lifted to a continuous map from ω* into itself. We prove that all quotient maps from ω* onto products of compact metrizable spaces are restrictions of quotient maps from βω (with the same range). We then defend our choice of hypotheses for this last result by showing that there is a quotient map from ω* onto the double arrow space (which is separable, 0-dimensional, first-countable, compact and Hausdorff) with no continuous extension from βω.
In the second part, the focus is on the Boolean algebra P(ω)/fin and the automorphisms of P(ω)/fin called the shift, which is induced by the map n → n+1 in ω. We show that the automorphisms sm for m ∈ℤ are the only trivial automorphisms of P(ω)/fin which commute with the shift. Together with a 1993 result of Velickovic, this characterizes all automorphisms of (P(ω)/fin, s) under OCA+MAℵ1. We study the algebra Per of all elements of P(ω)/fin with finite orbit under the action of the shift, and characterize all the automorphisms of Per which commute with the shift (many of which are not powers of the shift) or conjugate the shift to its inverse. Then, we show that every automorphism of the group of trivial automorphisms of P(ω)/fin either preserves or inverts (additively) the index function introduced by van Douwen in 1990. We also construct a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise isomorphic, and a set of 2א0 many elements of P(ω)/fin such that the substructures of (P(ω)/fin, s) generated by each of these elements individually are pairwise non-isomorphic. We finish with a method for constructing automorphisms of P(ω)/fin which conjugate the shift to itself or to its inverse on any given countable subalgebra.},
url = {https://hdl.handle.net/20.500.11811/6749}
}
The following license files are associated with this item:
