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Operator generation as a motor for an operator function

First the natural numbers are created:
a = 111... [1#a]
Addition of numbers is accomplished simply by writing numbers next to each other:
a+b = ab
eg. 3+2 = 111+11 = 11111 = 5

Consider an operator * for repetitions:
a*b = aa...a [a#b] (multiplication)
a**b = a*a*...*a [a#b] (powers with one exponent)
a***b = a**a**...**a [a#b] (powers with repeated exponents)
And so on, eg. 3****2 = 3***3 = 3**3**3 = 3**27 = 7625597484987 (my Mann number M3)

Let this define an operator function O(a,b,c),
where c is the number of **... in an expression a**...b,
and O(a,b,0) = O(a,b) = a+b = ab

A higher operator *1 can be specified as:
a*1b = a**...a [*#b] = O(a,a,b)
a*1*1b = a*1a*1..*1a [a#b]
a*1*1*1b = a*1*1a*1*1...*1*1a [a#b]
etc.
A second higher operator *2 as:
a*2b = a*1*1...a [*1#b]
a*2*2b = a*2a*2..*2a [a#b]
etc.
And further higher operators *d as:
a*d1b = a*d*d...a [*d#b]
a*d*db = a*da*d..*da [a#b]
etc.
Defining an operator function O(a,b,c,d),
where d is the type of a higher operator *d in an expression a*d*d...b,
and O(a,b,c,0) = O(a,b,c), and O(a,b,0,d) = a

Further types of operators *d,e can be specified similarly:
a*d,1b = a*d*d...a [*d#b] = O(a,a,b,d)
a*d,e1b = a*d,e*d,e...a [*d,e#b]
a*d,e*d,eb = a*d,ea*d,e..*d,ea [a#b]
etc.
Defining an operator function O(a,b,c,d,e),
where d,e is a type of higher operator *d,e in an expression a*d,e*d,e...b

We can specify an arbitrary number of coefficients for an operator *d,e,f,... similarly
and use it to extend the definition to the first row of parameters of an operator function bigO:
O(a1,...,an)

We can append a double comma ,, to this row and have it followed by a second row of parameters:
O(a,,b) = O(a,a,...,a) [a#b]

We can append a triple comma ,,, to specify a second dimensional parameter list:
O(a,,,b) = O(a,,a,,...,,a) [a#b]
And so on until bigO can have a parameter list with multiple dimensions ,,,...
Eg. my Lady cube L3 = O(3,3,3,,3,3,3,,3,3,3,,,3,3,3,,3,3,3,,3,3,3,,,3,3,3,,3,3,3,,3,3,3)

Let this define a higher operator function O2(a,b,c),
where c is the number of commas in an operator function O(a,,,...b) [,#c]
and O2(a,b,0) = O(ab) = ab

A higher separator ,2 can be specified as:
a,2b = a,1,1...a [,1#b] = O2(a,a,b)
a,2,2b = a,2a,2..,2a [a#b]
a,2,2,2b = a,2,2a,2,2...,2,2a [a#b]
etc.
And further higher separators ,d as:
a,d1b = a,d,d...a [,d#b]
a,d,db = a,da,d..,da [a#b]
etc.
Defining O2(a,b,c,d)

We can specify an arbitrary number of coefficients for a separator ,d,e,f,... similarly,
and use it to extend the definition of the operator-separator function bigO2 as O2(a1,...,an)

Append double, triple, multiple commas in bigO2 as above to create a multi-dimensional parameter list.
Then define a higher operator function O3(a,b,c) = O2(a,,,...b) [,#c]
and develop O3() and its successors On() similarly as O2() and O() before.

Create a next type of operator function bigO1,1 where:
O1,1(a,b,0) = O0(a,,...a) [,#b] = a**...a [*#b] = O(a,a,b)
O1,1(a,b,1) = O(a,,...a) [,#b] = O2(a,a,b)
O1,1(a,b,2) = O2(a,,...a) [,#b] = O3(a,a,b)
O1,1(a,b,c) = Oc(a,,...a) [,#b] = Oc1(a,a,b)

I know this sounds crazy, but I'm under the impression that an a,b = a*b type construction of bigO is less fundamental than an a,b = a*2^b type construction. Considering the quantum-world this may come as no surprise...

posted on Monday, August 25, 2008 10:10 AM