Some remarks on the subrack lattice of finite racks

Abstract

The set of all subracks [Formula: see text] of a finite rack [Formula: see text] forms a lattice under inclusion. We prove that if a rack [Formula: see text] satisfies a certain condition then the homotopy type of the order complex of [Formula: see text] is a [Formula: see text]-sphere, where [Formula: see text] is the number of maximal subracks of [Formula: see text]. The rack [Formula: see text] satisfying the condition of this general result is necessarily decomposable. Two particular instances occur when [Formula: see text] is a group rack, and when [Formula: see text] is a conjugacy class rack of a nilpotent group. 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.