quhomology (version 1.1.1)

up_action: The up action for a birack or biquandle.

Description

This function defines the up action for a birack or biquandle. In the case of a quandle or rack, it is the rack or quandle action. The definition of this functions is $f^b(a)$, that is, b acting on a from above.

Usage

up_action(a, b, k)

Arguments

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.

Value

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

Details

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

References

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

See Also

down_action

Examples

# NOT RUN {
## Example for version with function (for a dihedral quandle)
up_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)
up_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))
}

# }