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.

The package uses C++’s STL map class for efficiency, which has the downside that the order of the terms is undefined. This does not matter as the mathematical value is unaffected by reordering; and the print method does a good job in producing human-readable output.

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

and

we would have

Note that multiplication is not commutative, but it is associative. A natural and easily implemented extension is to use upper-case symbols to represent multiplicative inverses of the lower-case equivalents. Thus if

we would have

and

The system inherits power associativity from distributivity and associativity of concatenation, but 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
#>  + 1*a + 2*ab + 3*abc + 4*abcd + 5*abcde

Y <- as.freealg("6 - 4a +2aaab")
X+Y
#> free algebra element algebraically equal to
#>  + 7 - 1*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 + 1*a + 5*aCBB + 2*bc
(x2 <- rfalg(inc=TRUE))
#> free algebra element algebraically equal to
#>  + 6 + 1*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 + 1*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")