README.md

The Free Algebra in R

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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 uses the fact that the order of the terms is algebraically immaterial. The package conforms to disordR discipline.

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 \{x,y,z\} and

A=\alpha x^2yx + \beta zy

and

B=\gamma z + \delta y^4

we would have

AB=\left(\alpha x^2yx+\beta zy\right)\left(\gamma z+\delta y^4\right)=\alpha\gamma x^2yxz+\alpha\delta x^2yxy^4+\beta\gamma zyz+\beta\delta zy^5

and

BA=\left(\gamma z+\delta y^4\right)\left(\alpha x^2yx+\beta zy\right)=\alpha\gamma zx^2yx + \alpha\delta y^4 x^2yx + \beta\gamma z^2y + \beta\delta 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 xX=1). Thus if

C=\epsilon\left(x^{-1}\right)^2=\epsilon X^2

we would have

AC=\left(\alpha x^2yx+\beta zy\right)\epsilon X^2=
\alpha\epsilon x^2yX + \beta\epsilon zyX^2

and

CA=\epsilon X^2\left(\alpha x^2yx+\beta zy\right)=
\alpha\epsilon yx + \beta\epsilon 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
#> + 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")



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freealg documentation built on Dec. 12, 2022, 1:06 a.m.