The Free Algebra in R

Overview

The free algebra is an interesting and useful object. Here I present the freealg package which provides some functionality for free algebra in the R programming environment. The package uses the C++ map class for efficiency and conforms to disordR discipline. Several use-cases are provided.

Installation

You can install the released version of freealg from CRAN with:

# install.packages("freealg")  # uncomment this to install the package
library("freealg")

The free algebra

The free algebra is the free R-module with a basis consisting of all words over an alphabet of symbols with multiplication of words defined as concatenation. Thus, with an alphabet of and

A=x^2yx + zy

and

B=z + y^4

we would have

AB=(x^2yx+zy)(z+y4)=x^2yxz+x2yxy^4+zyz+zy5

and

BA=(z+y^4)(x2yx+zy)=zx^2yx + y^4 x^2yx + z^2y + y^4zy.

A natural and easily implemented extension is to use upper-case symbols to represent multiplicative inverses of the lower-case equivalents (formally we would use the presentation ). Thus if

C=(x^{-1})2=X^2

we would have

AC=(x^2yx+zy)X2= x^2yX + zyX^2

and

CA=X^2(x2yx+zy)= yx + X^2zy.

The system inherits associativity from associativity of concatenation, and distributivity is assumed, but it is not commutative.

The freealg package in use

Creating a free algebra object is straightforward. We can coerce from a character string with natural idiom:

X <- as.freealg("1 + 3a + 5b + 5abba")
X
#> free algebra element algebraically equal to
#> + 1 + 3*a + 5*abba + 5*b

or use a more formal method:

freealg(sapply(1:5,seq_len),1:5)
#> free algebra element algebraically equal to
#> + a + 2*ab + 3*abc + 4*abcd + 5*abcde

Y <- as.freealg("6 - 4a +2aaab")
X+Y
#> free algebra element algebraically equal to
#> + 7 - a + 2*aaab + 5*abba + 5*b
X*Y
#> free algebra element algebraically equal to
#> + 6 + 14*a - 12*aa + 6*aaaab + 2*aaab + 30*abba - 20*abbaa + 10*abbaaaab + 30*b
#> - 20*ba + 10*baaab
X^2
#> free algebra element algebraically equal to
#> + 1 + 6*a + 9*aa + 15*aabba + 15*ab + 10*abba + 15*abbaa + 25*abbaabba +
#> 25*abbab + 10*b + 15*ba + 25*babba + 25*bb

We can demonstrate associativity (which is non-trivial):

set.seed(0)
(x1 <- rfalg(inc=TRUE))
#> free algebra element algebraically equal to
#> + 7*C + 6*Ca + 4*B + 3*BC + a + 5*aCBB + 2*bc
(x2 <- rfalg(inc=TRUE))
#> free algebra element algebraically equal to
#> + 6 + CAAA + 2*Ca + 3*Cbcb + 7*aaCA + 4*b + 5*c
(x3 <- rfalg(inc=TRUE))
#> free algebra element algebraically equal to
#> + 3*C + 5*CbAc + BACB + 2*a + 10*b + 7*cb

(function rfalg() generates random freealg objects). Then

x1*(x2*x3) == (x1*x2)*x3
#> [1] TRUE

Further information

For more detail, see the package vignette

vignette("freealg")