03/16/2011, 05:47 PM

x {1.5} 2 = x {0.5} x

This is the general requirement that rational operators be recursive.

Consider,

x {0} y = x + y

x {1} y = x * y

x {2} y = x ^ y

x {3} y = x ^^ y

x * 2 = x + x

x {1} 2 = x {0} x

x ^ 2 = x * x

x {2} 2 = x {1} x

x ^^ 2 = x ^ x

x {3} 2 = x {2} x

etc etc...

It only be natural that this law holds for rational operators.

Generally, if {r} is any operator, than {r+1} is the superfunction of {r}.

the law stated mathematically is:

(x {r+1} (n-1)) {r} x = x {r+1} n

This is the general requirement that rational operators be recursive.

Consider,

x {0} y = x + y

x {1} y = x * y

x {2} y = x ^ y

x {3} y = x ^^ y

x * 2 = x + x

x {1} 2 = x {0} x

x ^ 2 = x * x

x {2} 2 = x {1} x

x ^^ 2 = x ^ x

x {3} 2 = x {2} x

etc etc...

It only be natural that this law holds for rational operators.

Generally, if {r} is any operator, than {r+1} is the superfunction of {r}.

the law stated mathematically is:

(x {r+1} (n-1)) {r} x = x {r+1} n