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 α,β,γ we
formally define A=αx2yx+βzy and B=−βzy+γy4. Addition is commutative so A+B=B+A. However, multiplication is not
commutative so AB≠BA in
general; both are associative. We also have consistency in that α(βP)=(αβ)P for any
expression P. Then:
A+B=(αx2yx+βzy)+(−βzy+γy4)=αx2yx+γy4
AB=(αx2yx+βzy)(−βzy+γy4)=−αβx2yxzy+αγx2yxy4−β2zyzy+βγzy5
BA=(−βzy+γy4)(αx2yx+βzy)=−αβzyx2yx−β2zyzy+αγy4x2yx+βγy4zy
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.
freealg
package in useThe above examples are a little too general for the
freealg
package; the idiom requires that we have specific
numerical values for the coefficients α,β,γ. Here we will use
1,2,3 respectively.
## free algebra element algebraically equal to
## + xxyx + 2zy
## free algebra element algebraically equal to
## + 3yyyy - 2zy
## free algebra element algebraically equal to
## + xxyx + 3yyyy
## free algebra element algebraically equal to
## + 3xxyxyyyy - 2xxyxzy + 6zyyyyy - 4zyzy
## 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.
## 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:
## free algebra element algebraically equal to
## + 3 + 5X - 2Xyx
## free algebra element algebraically equal to
## + 5xxy + 3xxyx - 2xxyyx + 6zy + 10zyX - 4zyXyx
## 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:
## [1] TRUE
## [1] TRUE
## [1] TRUE
Various utilities are included in the package. For example, the commutator bracket is represented by reasonably concise idiom:
## free algebra element algebraically equal to
## + ab - ba
Using rfalg()
to generate random free algebra objects,
we may verify the Jacobi identity:
## free algebra element algebraically equal to
## 0
The package includes functionality for substitution:
## free algebra element algebraically equal to
## + aaccc + 3aaxccc
## free algebra element algebraically equal to
## + 4accc + 3adccc + 6aeccc
It is possible to extract components of freealg
objects
using reasonably standard idiom:
## free algebra element algebraically equal to
## + aaa + 2aaba + 3abbbba - 3abc + 9xyzabc
## free algebra element algebraically equal to
## + 3abbbba + 9xyzabc
## free algebra element algebraically equal to
## + aaa + 2aaba + 3abbbba + 99abc + 9xyzabc
There is even some experimental functionality for calculus:
## 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).
disordR
Package.”
arXiv. https://doi.org/10.48550/ARXIV.2210.03856.