Can I read Subrack Lattices of Conjugation Racks on EtoBox?

Subrack Lattices of Conjugation Racks by Kayacan, Selçuk is a scholarly article available to read on EtoBox.

What is Subrack Lattices of Conjugation Racks about?

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 $S$ be a subset of a finite group $G$ which is closed under the conjugation operation $a\triangleright b := aba^{-1}$. The set $S$ with the conjugation operation $\triangleright$ is a quandle. We call those objects \emph{conjugation racks}. The prime examples are \begin{itemize} \item the group rack $(G,\triangleright)$, \item the conjugacy class rack $(C,\triangleright)$, where $C$ is a conjugacy class in $G$, and \item the $p$-power rack $(G_p,\triangleright)$, where $p$ is a prime and $G_p$ is the set of all elements in $G$ whose order is a power of $p$. \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 particular, we show that the subrack lattice can be associated with a subposet of a partition lattice as well as with a subposet of an integer partition lattice in a canonical way if the rack is connected. And, if the rack is not connected, the study of the homotopy properties of the subrack lattice can be reduced

Author

Kayacan, Selçuk

Published

2024

Language

EN