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.

You can install the released version of `freealg`

from CRAN with:

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

[ C=\epsilon X^2](https://latex.codecogs.com/png.latex?%0AC%3D%5Cepsilon%20X%5E2 " C=\epsilon X^2")

we would have

and

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

`freealg`

package in useCreating 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

For more detail, see the package vignette

`vignette("freealg")`