Published January 1, 2024 | Version v1
Journal article Open

Some remarks on the subrack lattice of finite racks

  • 1. Bahcesehir Univ, Fac Engn & Nat Sci, Istanbul, Turkiye

Description

The set of all subracks R(X) of a finite rack X forms a lattice under inclusion. We prove that if a rack X satisfies a certain condition then the homotopy type of the order complex of 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

center dot X=G is a group rack, and when

center dot X=C 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.

Files

bib-c6fd5f21-21ae-4893-9f14-fa5f3b0bb3ab.txt

Files (121 Bytes)

Name Size Download all
md5:5ea2f3fb9718371b6466c3f37136ba65
121 Bytes Preview Download