In RobinHankin/freegroup: The Free Group

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Overview

The freegroup package provides functionality for working with the free group in R. A detailed vignette is provided in the package. Informally, the free group is the set $X$ of words that are objects like $W=c^{-4}bb^2aa^{-1}ca$, with a group operation of string juxtaposition. Usually one works only with words that are in ``reduced form'', which has successive powers of the same symbol combined, so $W$ would be equal to $c^{-4}b^3ca$; see how $b$ appears to the third power and the $a$ term in the middle has vanished.

The group operation of juxtaposition is formally indicated by $\circ$, but this is often omitted in algebraic notation; thus, for example $a^2b^{-3}c^2\circ c^{-2}ba =a^2b^{-3}c^2c{^-2}ba =a^2b^{-2}ba$.

Installation

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

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

The `freegroup` package in use

Function rfree() generates a vector of random free group elements, giving quick "get you going" examples:

set.seed(0)

a <- rfree(10,5)
a
b <- as.free('x')

Then we can perform various operations on these vectors:

a+b
a-b
a^b

There are a number of package functions that work in a vectorized way:

sum(a)

The package also supports extraction and replacement:

a[3:9] <- as.free('xy')
a

Various simple elements can be created:

alpha(1:10)
abc(1:5)

References

Wikipedia contributors, "Free group," Wikipedia, the free encyclopedia, 4 June 2019. [https://en.wikipedia.org/w/index.php?title=Free_group&oldid=900295259]