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.
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 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
we would have
and
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
we would have
and
The system inherits associativity from associativity of concatenation, and distributivity is assumed, but it is not commutative.
freealg
package in
useCreating a free algebra object is straightforward. We can coerce from a character string with natural idiom:
<- as.freealg("1 + 3a + 5b + 5abba")
X
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
<- as.freealg("6 - 4a +2aaab")
Y +Y
X#> free algebra element algebraically equal to
#> + 7 - a + 2*aaab + 5*abba + 5*b
*Y
X#> 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
^2
X#> 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)
<- rfalg(inc=TRUE))
(x1 #> free algebra element algebraically equal to
#> + 7*C + 6*Ca + 4*B + 3*BC + a + 5*aCBB + 2*bc
<- rfalg(inc=TRUE))
(x2 #> free algebra element algebraically equal to
#> + 6 + CAAA + 2*Ca + 3*Cbcb + 7*aaCA + 4*b + 5*c
<- rfalg(inc=TRUE))
(x3 #> free algebra element algebraically equal to
#> + 3*C + 5*CbAc + BACB + 2*a + 10*b + 7*cb
(function rfalg()
generates random freealg
objects). Then
*(x2*x3) == (x1*x2)*x3
x1#> [1] TRUE
For more detail, see the package vignette
vignette("freealg")