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
## + 1*xxyx + 2*zy

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

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

A+B

## free algebra element algebraically equal to
## + 1*xxyx + 3*yyyy

A*B

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

B*A

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

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
## + 1*xxy + 2*zyX

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 + 5*X - 2*Xyx

A*C

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

C*A

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

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
## + 1*ab - 1*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
## + 1*aaccc + 3*aaxccc

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

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

There is even some experimental functionality for calculus:

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

## free algebra element algebraically equal to
## + 1*aaaxa(da) + 1*aaax(da)a + 1*aa(da)xaa + 1*a(da)axaa + 1*(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).