Virtual-Rack-Classification

GAP program for classifying virtual racks and virtual quandles of a given order. Data up to order 8.

Compiled by Lực Ta

Based on the rack library of Petr Vojtěchovský and Seung Yeop Yang (https://www.cs.du.edu/~petr/libraries_of_algebraic_structures.html)

See OEIS sequences A385040 and A385041 for enumerations.

See the paper "Enumeration of virtual quandles up to isomorphism" for further information.

Usage

virtual-rack-finder.txt exhaustively searches for virtual structures on all racks of a given order n in Vojtěchovský and Yang's library, searches for isomorphisms between them, and outputs a list of all virtual racks of order n up to isomorphism.

For example, to run the search for all n between 1 and 3, load Vojtěchovský and Yang's library and then run the following. The following assumes that you've saved virtual-rack-finder.txt in the same folder as the rack library; if not, then replace "LRQ.path" below and in virtual-rack-finder.txt with the path to wherever you've saved virtual-rack-finder.txt.

racks:=[1, 2, 6, 19, 74, 353, 2080, 16023, 159526, 2093244, 36265070];
for n in [1..3] do
	ReadAsFunction(Concatenation(LRQ.path, "virtual-rack-finder.txt"))()(n,racks[n]);
od;

Here, racks is a list whose n-th entry is the number of isomorphism classes of racks of order n. The entries of racks are taken from OEIS sequence A181770.