dot: Class "dot"

In freegroup: The Free Group

Class “dot”

Description

The dot object is defined in the freealg package, and imported here, so that idiom like .[x,y] returns the commutator, that is, x^-1 y^-1 xy.

The dot object is generated by running script inst/dot.Rmd, which includes some further discussion and technical documentation, and creates file dot.rda which resides in the data/ directory.

Usage

## S4 method for signature 'dot,ANY,ANY'
x[i, j, drop]
## S4 method for signature 'dot,free,ANY,ANY'
x[i, j, drop]

Arguments

x	Object of any class
i, j	elements to commute
drop	Boolean, indicates whether to drop attributes

Value

Always returns an object of the same class as xy.

Author(s)

Robin K. S. Hankin

Examples

.[as.free("x"), as.free("y")]

.[abc(1:6), "z"]

x <- rfree()
y <- rfree()
z <- rfree()

.[x, y] == -x-y+x+y   # should be TRUE

abelianize(.[x, y])

## Jacobi identity _not_ satisfied with this definition:
is.id(.[x, .[y, z]] + .[y, .[z, x]] + .[z, .[x, y]])

## But the Hall-Witt identity is:
all(is.id(.[.[x, -y], z]^y + .[.[y, -z], x]^z + .[.[z, -x], y]^x))

freegroup documentation built on July 15, 2026, 5:07 p.m.