RAQ/README.md

RAQ, a GAP System package for Racks And Quandles.

• Website: http://code.studioinfinity.org/RAQ/wiki
• Repository: http://code.studioinfinity.org/RAQ
• Authors/maintainers of RAQ: Glen Whitney glen@studioinfinity.org

The RAQ package provides a variety of facilities for constructing and computing with one-sided quasigroups, racks, and quandles in GAP.

RAQ uses no external binaries, so installation consists only of placing its unpacked file tree in a directory in your package search path, e.g. the pkg directory of your GAP installation, or perhaps the .gap/pkg subdirectory of your home directory.

The authors of RAQ would like to acknowledge their debt to the creators of RIG, an earlier package for Racks in GAP; chief among them is Leandro Vendramin. RIG was an inspiration for the creation of RAQ, and using and reading that package suggested many features needed in the development of RAQ.

Perhaps the following GAP interactive session, which constructs the conjugation quandle of the symmetric group on three elements and then performs a few simple computations on that quandle, will give the flavor of RAQ. (It is presumed that the RAQ package has already been loaded with LoadPackage("RAQ"); prior to these example commands being executed.)

gap> S3 := SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> Elements(S3);
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
gap> Q3 := ConjugationQuandle(S3);
<left quandle with 6 generators>
gap> elt := Elements(Q3); # the element ^p: below means conjugation by p in S3 
[ ^():, ^(2,3):, ^(1,2):, ^(1,2,3):, ^(1,3,2):, ^(1,3): ]
gap> elt[4]*elt[3]; # So this will produce (1,2,3)^{-1}(1,2)(1,2,3)
^(2,3):

Note in particular that RAQ generally, unless otherwise specifically requested, produces left quandles and racks. (That is to say, quandles in which for any fixed element l, the "left-multiplication by $l$" operation x\mapsto l*x is a permutation of the quandle.)