down_action: The down action for a birack or biquandle.
In quhomology: Calculation of Homology of Quandles, Racks, Biquandles and Biracks

Description Usage Arguments Details Value References See Also Examples

This functions defines the down action for a birack or biquandle. In the case of a quandle or rack, it is the identity. The definition of this functions is $f_b(a)$, that is, b acting on a from below.

1	down_action(a, b, k)

`a`	This is the elements that is acted upon. An integer.
`b`	This is the element that acts. An integer.
`k`	This is the order of the biquandle. It is not always required, but passed on nevertheless. An integer.

This can (and should) be changed by the user if s/he requires a different down action. It could be implemented as a matrix lookup, a function or some other way. Examples for the first two options are below.

An integer, representing an element in the birack or rack.

http://en.wikipedia.org/wiki/Biquandle http://en.wikipedia.org/wiki/Racks_and_quandles

up_action

## Example for version with function (for a dihedral quandle)
down_action <- function (a, b, k){

    result <- (2 * b - a)%%k
    return(as.integer(result))
}


##Example for matrix lookup (for commutative quandle over S_3, in which case k = 6)
down_action <- function (a, b, k){
    #first define the action matrix
    action_matrix <- rbind(c(0,0,0,0,0,0),c(1,1,5,5,2,2),c(2,5,2,1,5,1),
    c(3,4,4,3,4,4),c(4,3,3,3,4,3),c(5,2,1,2,1,5))
    result <-action_matrix[a + 1, b + 1]
    return(as.integer(result))
}

##example for quandles/racks
down_action <- function (a, b, k){

    return(a)
}