Classification of Finite Topological Quandles and Shelves via Posets

Author

Hitakshi Lahrani, University of South Florida

Graduation Year

2023

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Mohamed Elhamdadi, Ph.D.

Committee Member

Masahico Saito, Ph.D.

Committee Member

Boris Shekhtman, Ph.D.

Committee Member

Razvan Teodorescu, Ph.D.

Committee Member

Dmytro Savchuk, Ph.D.

Keywords

Quandles, Shelves, Posets

Abstract

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, ... , X_n. Then any right continuous poset on X is n-partite with vertex sets X₁, ... , X_n.

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 Theorem

Proposition 0.0.3. Let S be a shelf such that R_a1 = R_a2 = R_a3 = ... = Ra_m and L_a1 (x) = a1, L_a2 (x) = a₂, ... , La_m (x) = a_m, for some a1, ... , a_m ∈ S and for all x ∈ S. Then S has at least γ(m) partial orders continuous on S, where γ(m) is number of T_o 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.

Scholar Commons Citation

Lahrani, Hitakshi, "Classification of Finite Topological Quandles and Shelves via Posets" (2023). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/9981