This package provides some functionality for manipulating the free group on a finite list of symbols.
There is an excellent wikipedia page.
Basically, the free group~\(\left(X,\circ\right)\) on a set \(S=\{a,b,c,...,z\}\) is the set \(X\) of words that are things like \(c^{-4}bb^2aa^{-1}ca\). Usually one works only with words that are in reduced form, which has successive powers of the same symbol combined, in this case it would be \(c^{-4}b^3ca\); see how the \(a\) term in the middle has vanished.
The group operation is juxtaposition; thus, for example \(a^2b^{-3}c^2\circ c^{-2}ba =a^2b^{-3}c^2c^-2ba =a^2b^{-2}ba\).
A word is represented by a two-row integer matrix: The top row is the integer representation of the symbol and the second row is the corresponding power. For example, say we want to represent~\(a^2b^{-3}ac^2a^{-2}\) we would identify \(a\) as 1, \(b\) as 2, etc and write
(M <- rbind(c(1,2,3,3,1),c(2,-3,2,3,-2)))
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1 2 3 3 1
#> [2,] 2 -3 2 3 -2(see how negative entries in the second row correspond to negative powers). Then to convert to a more useful form we would have
library(freegroup)
(x <- free(M))
#> [1] a^2.b^-3.c^5.a^-2The representation for R object x is still a two-row matrix, but the print method is active and uses a more visually appealing scheme.
We can coerce strings to free objects:
(y <- as.free("aabbbcccc"))
#> aabbbcccc
#> a^2.b^3.c^4The free group operation is simply juxtaposition, represented here by the plus symbol, “+”:
x+y
#> [1] a^2.b^-3.c^5.b^3.c^4See how the \(a\) “cancels out” in the juxtaposition. Note that concatenation is associative. Multiplication is also defined. Suppose we want to concatenate 5 copies of x:
x*5
#> [1] a^2.b^-3.c^5.b^-3.c^5.b^-3.c^5.b^-3.c^5.b^-3.c^5.a^-2The package is vectorized:
x*(0:3)
#> [1] 0 a^2.b^-3.c^5.a^-2
#> [3] a^2.b^-3.c^5.b^-3.c^5.a^-2 a^2.b^-3.c^5.b^-3.c^5.b^-3.c^5.a^-2There are a few methods for creating free objects:
abc(1:9)
#> [1] a a.b a.b.c a.b.c.d
#> [5] a.b.c.d.e a.b.c.d.e.f a.b.c.d.e.f.g a.b.c.d.e.f.g.h
#> [9] a.b.c.d.e.f.g.h.i
rfree(10,4)
#> [1] d^-2.c^3.b^-1 a^4.d.c^-4 d^2.b^4.c^-1.b^2 d^2.a.c^3.d^-4
#> [5] d^-3 b^-2.c^-2.b^-4.a b^5 c^-1.a^-1
#> [9] c^-2.a^-2.c c^-5.d^6Inverses are calculated using unary or binary minus:
(u <- rfree(10,4))
#> [1] a^3.b^-3.c^4.b^-4 c^-1 c^2.d^3.b^3.a^-4 a^2
#> [5] a^3.b^4.c.d^-4 d^3.c^2 c^2.b^-3 a^-2.b^-4.a^-4
#> [9] a^-3.c^3.d^3 b.d^-6
-u
#> [1] b^4.c^-4.b^3.a^-3 c a^4.b^-3.d^-3.c^-2 a^-2
#> [5] d^4.c^-1.b^-4.a^-3 c^-2.d^-3 b^3.c^-2 a^4.b^4.a^2
#> [9] d^-3.c^-3.a^3 d^6.b^-1
u-u
#> [1] 0 0 0 0 0 0 0 0 0 0We can take the “sum” of a vector of free objects simply by juxtaposing the elements:
sum(u)
#> [1] a^3.b^-3.c^4.b^-4.c.d^3.b^3.a.b^4.c.d^-1.c^4.b^-3.a^-2.b^-4.a^-7.c^3.d^3.b.d^-6Powers are defined as per group conjugation: x^y == y^{-1}xy (or, written in additive notation, -y+x+y):
u
#> [1] a^3.b^-3.c^4.b^-4 c^-1 c^2.d^3.b^3.a^-4 a^2
#> [5] a^3.b^4.c.d^-4 d^3.c^2 c^2.b^-3 a^-2.b^-4.a^-4
#> [9] a^-3.c^3.d^3 b.d^-6
z <- alpha(26)
u^z
#> [1] z^-1.a^3.b^-3.c^4.b^-4.z z^-1.c^-1.z z^-1.c^2.d^3.b^3.a^-4.z
#> [4] z^-1.a^2.z z^-1.a^3.b^4.c.d^-4.z z^-1.d^3.c^2.z
#> [7] z^-1.c^2.b^-3.z z^-1.a^-2.b^-4.a^-4.z z^-1.a^-3.c^3.d^3.z
#> [10] z^-1.b.d^-6.zThus:
sum(u^z) == sum(u^z)
#> [1] TRUEIf we have more than 26 symbols the print method runs out of letters:
free(rbind(1:30,1))
#> [1] a.b.c.d.e.f.g.h.i.j.k.l.m.n.o.p.q.r.s.t.u.v.w.x.y.z.NA.NA.NA.NAIf this is a problem (it might not be: the print method might not be important) it is possible to override the default symbol set:
options(symbols = state.abb)
free(rbind(1:50,1))
#> [1] AL.AK.AZ.AR.CA.CO.CT.DE.FL.GA.HI.ID.IL.IN.IA.KS.KY.LA.ME.MD.MA.MI.MN.MS.MO.MT.NE.NV.NH.NJ.NM.NY.NC.ND.OH.OK.OR.PA.RI.SC.SD.TN.TX.UT.VT.VA.WA.WV.WI.WY