Ops.free {freegroup}

Arithmetic Ops methods for the free group

Description

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 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 is defined as juxtaposing n copies of a and reducing. Zero and negative values of n work as expected.

Note

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

Author(s)

Robin K. S. Hankin

Examples

x <- as.free(c("a","ab","aaab","abacc"))
y <- as.free(c("aa","BA","Bab","aaaaa"))
x
y


x + x
x + y
x + as.free("xyz")

x+y == y+x    # not equal in  general

x*5 == x+x+x+x+x      # always true

x + alpha(26)

x^y