Ops.free

0th

Percentile

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

Aliases
  • Ops.free
  • ops
  • free_equal
  • free_power
  • free_repeat
  • juxtapose
  • inverse
  • inverse.free
  • inverse.matrix
Examples
# NOT RUN {
x <- rfree(10,2)
y <- rfree(10,2)
z <- rfree(10,9)    # more complicated than x or y




x+y
x-y

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

x+as.free(0) == x     # always true
as.free(0)+x == x     # always true
x+(y+z)  == (x+y)+z   # always true
x*5 == x+x+x+x+x      # always true

x + alpha(26)

x^alpha(26)

x*12
x*(0:9)


# }
Documentation reproduced from package freegroup, version 1.1-0, License: GPL-2

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