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Classification of Finite Topological Quandles and Shelves via Posets.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Classification of Finite Topological Quandles and Shelves via Posets./
Author:	Lahrani, Hitakshi.
Description:	1 online resource (66 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-01, Section: B.
Contained By:	Dissertations Abstracts International85-01B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798379953614

Classification of Finite Topological Quandles and Shelves via Posets.
Lahrani, Hitakshi.

Classification of Finite Topological Quandles and Shelves via Posets. - 1 online resource (66 pages)

Source: Dissertations Abstracts International, Volume: 85-01, Section: B.

Thesis (Ph.D.)--University of South Florida, 2023.

Includes bibliographical references

The objective of this dissertation is to investigate finite topological quandles and topological shelves. Precisely, we give a classification of both finite topological quandles and topological shelves using the theory of posets. For quandles with more than one orbit, we prove the following Theorem.Proposition 0.0.1. Let X be a finite quandle with n orbits X1, ... , Xn. Then any right continuous poset on X is n-partite with vertex sets X1, ... , Xn.For connected quandles, we prove the following Theorem.Theorem 0.0.2. There is no T0-topology on a finite connected quandle X that makes X into a right topological quandle.We used computer programming to help in exploring and analyzing various finite topological quandles and shelves. Since right multiplications in topological quandles are homeomorphisms, this implies that there is no T0-topology on a finite connected quandle X that makes X into a right topological quandle. By ignoring this condition and investigating topological shelves in which right multiplications are not homeomorphisms, we were able to obtain T0-topology on a finite connected shelves. For example we obtained the following TheoremProposition 0.0.3. Let S be a shelf such that Ra1 = Ra2 = Ra3 = ... = Ram and La1 (x) = a1, La2 (x) = a2, ... , Lam (x) = am, for some a1, ... , am ∈ S and for all x ∈ S. Then S has at least γ(m) partial orders continuous on S, where γ(m) is number of To topologies on m symbols.The results of this study reveal the significance of topological quandles and shelves as important mathematical structures that can be used to study a wide range of problems in topology and knot theory. One of the key findings is that there is no connected topological quandle, while there exist onnected topological shelves. Additionally, an enumeration of topological quandles and shelves was provided, which has important implications for the study of these structures.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798379953614Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

Knot theoryIndex Terms--Genre/Form:

554714
Electronic books.

Classification of Finite Topological Quandles and Shelves via Posets.
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502 $a Thesis (Ph.D.)--University of South Florida, 2023.
504 $a Includes bibliographical references
520 $a The objective of this dissertation is to investigate finite topological quandles and topological shelves. Precisely, we give a classification of both finite topological quandles and topological shelves using the theory of posets. For quandles with more than one orbit, we prove the following Theorem.Proposition 0.0.1. Let X be a finite quandle with n orbits X1, ... , Xn. Then any right continuous poset on X is n-partite with vertex sets X1, ... , Xn.For connected quandles, we prove the following Theorem.Theorem 0.0.2. There is no T0-topology on a finite connected quandle X that makes X into a right topological quandle.We used computer programming to help in exploring and analyzing various finite topological quandles and shelves. Since right multiplications in topological quandles are homeomorphisms, this implies that there is no T0-topology on a finite connected quandle X that makes X into a right topological quandle. By ignoring this condition and investigating topological shelves in which right multiplications are not homeomorphisms, we were able to obtain T0-topology on a finite connected shelves. For example we obtained the following TheoremProposition 0.0.3. Let S be a shelf such that Ra1 = Ra2 = Ra3 = ... = Ram and La1 (x) = a1, La2 (x) = a2, ... , Lam (x) = am, for some a1, ... , am ∈ S and for all x ∈ S. Then S has at least γ(m) partial orders continuous on S, where γ(m) is number of To topologies on m symbols.The results of this study reveal the significance of topological quandles and shelves as important mathematical structures that can be used to study a wide range of problems in topology and knot theory. One of the key findings is that there is no connected topological quandle, while there exist onnected topological shelves. Additionally, an enumeration of topological quandles and shelves was provided, which has important implications for the study of these structures.
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538 $a Mode of access: World Wide Web
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650 4 $a Statistics. $3 556824
650 4 $a Applied mathematics. $3 1069907
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653 $a Topology
653 $a Finite quandle
653 $a Poset
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710 2 $a University of South Florida. $b Mathematics and Statistics. $3 1188070
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