Subrack Lattices of Conjugation Racks
Sel\c{c}uk Kayacan

TL;DR
This paper investigates the structure of subrack lattices in conjugation racks derived from finite groups, revealing connections to partition lattices and properties related to Sylow p-subgroups.
Contribution
It establishes a canonical association between subrack lattices of conjugation racks and partition lattices, and explores homotopy and Euler characteristic properties in these structures.
Findings
Subrack lattice can be associated with a subposet of a partition lattice.
Homotopy properties of non-connected racks relate to parabolic subracks.
Order of Sylow p-subgroup divides the reduced Euler characteristic of the subrack lattice.
Abstract
A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. Let be a subset of a finite group which is closed under the conjugation operation . The set with the conjugation operation is a quandle. We call those objects \emph{conjugation racks}. The prime examples are \begin{itemize} \item the group rack , \item the conjugacy class rack , where is a conjugacy class in , and \item the -power rack , where is a prime and is the set of all elements in whose order is a power of . \end{itemize} The set of all subracks of a finite rack form a lattice under inclusion. In this paper we study the subrack lattices of the conjugation racks. In…
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Taxonomy
TopicsStructural Analysis of Composite Materials · Mechanical Behavior of Composites
