Arithmetic Ops methods for the free group

Allows arithmetic operators to be used for manipulation of free group elements such as addition, multiplication, powers, etc

Usage

# S3 method for class 'free'
Ops(e1, e2)
free_equal(e1,e2)
free_power(e1,e2)
free_repeat(e1,n)
juxtapose(e1,e2)
# S3 method for class 'free'
inverse(e1)
# S3 method for class 'matrix'
inverse(e1)

Arguments

e1,e2

Objects of class free

n

An integer, possibly non-positive

Details

The function Ops.free() passes binary arithmetic operators (“+”, “-”, “*”, “^”, and “==”) to the appropriate specialist function.

There are two non-trivial basic operations: juxtaposition, denoted “a+b”, and inversion, denoted “-a”. Note that juxtaposition is noncommutative and a+b will not, in general, be equal to b+a.

All operations return a reduced word.

The caret, as in a^b, denotes group-theoretic exponentiation (-b+a+b); the notation is motivated by the identities x^(yz)=(x^y)^z and (xy)^z=x^z*y^z, as in the permutations package.

Multiplication between a free object a and an integer n (as in a*n or n*a) is defined as juxtaposing n copies of a and reducing. Zero and negative values of n work as expected.

Comparing a free object with a numeric does not make sense and idiom such as rfree() > 4 will return an error. Comparing a free object with another free object might be desirable [specifically, lexicographic ordering], but is not currently implemented.

Author

Robin K. S. Hankin

Note

The package uses additive notation but multiplicative notation might have been better.

Examples

x <- as.free(c("a","ab","aaab","abacc"))
y <- as.free(c("aa","BA","Bab","aaaaa"))
x
#> [1] a         a.b       a^3.b     a.b.a.c^2
y
#> [1] a^2       b^-1.a^-1 b^-1.a.b  a^5      


x + x
#> [1] a^2                 a.b.a.b             a^3.b.a^3.b        
#> [4] a.b.a.c^2.a.b.a.c^2
x + y
#> [1] a^3           0             a^4.b         a.b.a.c^2.a^5
x + as.free("xyz")
#> [1] a.x.y.z         a.b.x.y.z       a^3.b.x.y.z     a.b.a.c^2.x.y.z

x+y == y+x    # not equal in  general
#> [1]  TRUE  TRUE FALSE FALSE

x*5 == x+x+x+x+x      # always true
#> [1] TRUE TRUE TRUE TRUE

x + alpha(26)
#> [1] a.z         a.b.z       a^3.b.z     a.b.a.c^2.z

x^y
#> [1] a                 a.b               b^-1.a^-1.b.a^4.b a^-4.b.a.c^2.a^5