Some remarks on the subrack lattice of finite racks

Abstract

The set of all subracks $\mathcal{R}(X)$ of a finite rack $X$ form a lattice under inclusion. We prove that if a rack $X$ satisfies a certain condition then the homotopy type of the order complex of $\mathcal{R}(X)$ is a $(m-2)$-sphere, where $m$ is the number of maximal subracks of $X$. The rack $X$ satisfying the condition of this general result is necessarily decomposable. Two particular instances occur when \begin{itemize} \item $X=G$ is a group rack, and when \item $X=C$ is a conjugacy class rack of a nilpotent group. \end{itemize} We also studied the subrack lattices of indecomposable racks by focusing on the conjugacy class racks of symmetric or alternating groups and determined the homotopy types of the corresponding order complexes in some cases.