To cite the freealg package in publications, please use Hankin (2022b). In this short document I show how free algebras may be studied using the freealg package. The free algebra is best introduced by an example: with an alphabet of \(\{x,y,z\}\), and real numbers \(\alpha,\beta,\gamma\) we formally define \(A=\alpha x^2yx + \beta zy\) and \(B=-\beta zy + \gamma y^4\). Addition is commutative so \(A+B=B+A\). However, multiplication is not commutative so \(AB\neq BA\) in general; both are associative. We also have consistency in that \(\alpha(\beta P)=(\alpha\beta)P\) for any expression \(P\). Then:

\[ A+B=(\alpha x^2yx + \beta zy) + (-\beta zy + \gamma y^4) = \alpha x^2yx + \gamma y^4 \]

\[ AB= (\alpha x^2yx + \beta zy) (-\beta zy + \gamma y^4) = -\alpha\beta x^2yxzy +\alpha\gamma x^2yxy^4 -\beta^2zyzy +\beta\gamma zy^5 \]

\[ BA=(-\beta zy + \gamma y^4)(\alpha x^2yx + \beta zy) = -\alpha\beta zyx^2yx -\beta^2 zyzy + \alpha\gamma y^4x^2yx + \beta\gamma y^4zy \]

This is a natural set of objects to consider. Formally, we consider the free R-module with a basis consisting of all words over an alphabet of symbols [conventionally lower-case letters] with multiplication of words defined as concatenation. The system inherits associativity from associativity of concatenation; distributivity follows from the definition of R-module. However, the free algebra is not commutative in general.

The `freealg` package in use

The above examples are a little too general for the freealg package; the idiom requires that we have specific numerical values for the coefficients \(\alpha,\beta,\gamma\). Here we will use \(1,2,3\) respectively.

(A <- as.freealg("xxyx + 2zy"))

## free algebra element algebraically equal to
## + xxyx + 2zy

(B <- as.freealg("-2zy + 3yyyy"))

## free algebra element algebraically equal to
## + 3yyyy - 2zy

A+B

## free algebra element algebraically equal to
## + xxyx + 3yyyy

A*B

## free algebra element algebraically equal to
## + 3xxyxyyyy - 2xxyxzy + 6zyyyyy - 4zyzy

B*A

## free algebra element algebraically equal to
## + 3yyyyxxyx + 6yyyyzy - 2zyxxyx - 4zyzy

Note that the terms are stored in an implementation-specific order. For example, A might appear as xxyz + 2*zy or the algebraically equivalent form 2*zy + xxyz. The package follows disordR discipline (Hankin 2022a).

Inverses are coded using upper-case letters.

A*as.freealg("X") # X = x^{-1}

## free algebra element algebraically equal to
## + xxy + 2zyX

See how multiplying by \(X=x^{-1}\) on the right cancels one of the x terms in A. We can use this device in more complicated examples:

(C <- as.freealg("3 + 5X - 2Xyx"))

## free algebra element algebraically equal to
## + 3 + 5X - 2Xyx

A*C

## free algebra element algebraically equal to
## + 5xxy + 3xxyx - 2xxyyx + 6zy + 10zyX - 4zyXyx

C*A

## free algebra element algebraically equal to
## - 2Xyxxxyx - 4Xyxzy + 10Xzy + 3xxyx + 5xyx + 6zy

With these objects we may verify that the distributive and associative laws are true:

A*(B+C) == A*B + A*C

## [1] TRUE

(A+B)*C == A*C + B*C

## [1] TRUE

A*(B*C) == (A*B)*C

## [1] TRUE

Various utilities are included in the package. For example, the commutator bracket is represented by reasonably concise idiom:

a <- as.freealg("a")
b <- as.freealg("b")
.[a,b] # returns ab-ba

## free algebra element algebraically equal to
## + ab - ba

Using rfalg() to generate random free algebra objects, we may verify the Jacobi identity:

x <- rfalg()
y <- rfalg()
z <- rfalg()

.[x,.[y,z]] + .[y,.[z,x]] + .[z,.[x,y]]

## free algebra element algebraically equal to
## 0

The package includes functionality for substitution:

subs("aabccc",b="1+3x")  # aa(1+3x)ccc

## free algebra element algebraically equal to
## + aaccc + 3aaxccc

subs("abccc",b="1+3x",x="1+d+2e")

## free algebra element algebraically equal to
## + 4accc + 3adccc + 6aeccc

Experimental functionality

It is possible to extract components of freealg objects using reasonably standard idiom:

(a <- as.freealg("aaa + 2*aaba + 3*abbbba + 9*xyzabc - 3*abc"))

## free algebra element algebraically equal to
## + aaa + 2aaba + 3abbbba - 3abc + 9xyzabc

a[coeffs(a) > 2]

## free algebra element algebraically equal to
## + 3abbbba + 9xyzabc

a[coeffs(a) < 0] <- 99
a

## free algebra element algebraically equal to
## + aaa + 2aaba + 3abbbba + 99abc + 9xyzabc

There is even some experimental functionality for calculus:

deriv(as.freealg("aaaxaa"),"a")

## free algebra element algebraically equal to
## + aaaxa(da) + aaax(da)a + aa(da)xaa + a(da)axaa + (da)aaxaa

Above, “da” means the differential of a. Note how it may appear at any position in the product, not just the end (cf matrix differentiation).

The free algebra with R

Robin K. S. Hankin

The `freealg` package in use

Experimental functionality

References

The free algebra with R

Robin K. S. Hankin

The freealg package in use

Experimental functionality

References

The `freealg` package in use