In RobinHankin/freealg: The Free Algebra

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

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"))
a[coeffs(a) > 2]
a[coeffs(a) < 0] <- 99
a

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

RobinHankin/freealg documentation built on Dec. 24, 2024, 3:16 a.m.