Class “dot”

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

Robin K. S. Hankin

Examples

.[as.free("x"), as.free("y")]
#> [1] x^-1.y^-1.x.y

.[abc(1:6), "z"]
#> [1] a^-1.z^-1.a.z                                   
#> [2] b^-1.a^-1.z^-1.a.b.z                            
#> [3] c^-1.b^-1.a^-1.z^-1.a.b.c.z                     
#> [4] d^-1.c^-1.b^-1.a^-1.z^-1.a.b.c.d.z              
#> [5] e^-1.d^-1.c^-1.b^-1.a^-1.z^-1.a.b.c.d.e.z       
#> [6] f^-1.e^-1.d^-1.c^-1.b^-1.a^-1.z^-1.a.b.c.d.e.f.z

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

.[x, y] == -x-y+x+y   # should be TRUE
#> [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE

abelianize(.[x, y])
#> [1] 0 0 0 0 0 0 0

## Jacobi identity _not_ satisfied with this definition:
is.id(.[x, .[y, z]] + .[y, .[z, x]] + .[z, .[x, y]])
#> [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE

## But the Hall-Witt identity is:
all(is.id(.[.[x, -y], z]^y + .[.[y, -z], x]^z + .[.[z, -x], y]^x))
#> [1] TRUE