The free algebra with R

In freealg: The Free Algebra

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library("freealg")
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![](`r system.file("help/figures/freealg.png", package = "freealg")`){width=10%}

To cite the freealg package in publications, please use @hankin2022_freealg.

The free algebra is best defined 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 $x+y=y+x$ (and so $A=\beta zy + \alpha x^2yx$) but multiplication is not commutative, so $xy\neq yx$; 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"))
(B <- as.freealg("-2zy + 3yyyy"))
A+B
A*B
B*A

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 [@hankin2022_disordR].

Inverses are coded using upper-case letters.

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

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"))
A*C
C*A

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

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

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

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]]

The package includes functionality for substitution:

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

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

There is even some experimental functionality for calculus:

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

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).

References

freealg documentation built on March 31, 2023, 7:13 p.m.