Thinking about whether axes are fundamental to unary operators. I do not think so. I think axes are fundamental to geometry and even though euclid did not particularly mention them they are implicit in what he was describing. Their implicit nature in geometry was made explicit by algebra. This is because algebra is about the syntax and syntax forms of any ARITHMETIC and the syntax of the outcome of any manipulation / transform of that arithmetic. Because of algebra we can communicate alternative arithmetics and introduce and define new transforms as well as describe in revealing ways old arithmetic schemes.

So mod(0) define as {0,1,2,3,....

} : addition and subtraction (multiple addition and multiple subtraction methods) as in everyday early centuries arithmetic.

mod(1) define as {0} : addition and subtraction etc as in modern definition of nullity

mod(2) define as {0,1} : addition and subtraction as in binary arithmetics and boolean algebras and modifications.

mod(3) define as {0,1,2} :

http://en.wikipedia.org/wiki/Modular_arithmeticmod(4) define as {0,1,2,3}

mod(5) define as {0,1,2,3,4} Just in passing these sets may be useful for describing truncated iterations.

mod(6) define as {0,1,2,3,4,5}

mod(n) define as {0,1,2,3,.......,n-1}

All of these are algebraic definitions from which the arithmetics are prescribed for integers.

Of course we can extend this to the real numbers and then we would have formal partitioning of the reals into equivalence sets. This may be useful in describing fractal relationships at different levels, and may be what Musean numbers are attempting to describe.

Now the term number can be seen as a hangover from the historic arithmetic connection, but when these structures are aplied outside of a quantitative scheme the term number becomes misleading. So i generalise it to numeral.

I also generalise the notion of sign to signal. IT IS EXACTLY THE SAME AS using superscription subscription index tilde etc to distinguish a different but related denotation/ definition. In kujonai method he uses the mod set numerals as signals placing them in [], amongst other notations.

So in the case of mod(2)

define addition 0 + 0 = 0 define multiplication 0 * 0 = 0

0 + 1 = 1 0 * 1 = 0

1 + 0 = 1 1 * 0 = 0 These used for logic applications.

1 + 1 = 0 1 * 1 = 1

I will use the add definition for the signal manipulation.

write u

_{2}, v

_{2} {0,1} for the general unary operator from mod(2).

u

_{2} therefore has the numeral values 0 or 1.

Now for any a,b

the natural numbers or a,b

lines in Euclidean space, a || b, a units and b units in length.

define u

_{2}a = 0]a or 1]a geometrically u

_{2}a = a or a rotated

about a point O

and u

_{2}0]a =1]a or 0]a u

_{2}a = a rotated

or a

and u

_{2}1]a =0]a or 1]a... u

_{2}a rotated

= a or a rotated

specifically

if u

_{2}a = 0]a then u

_{2}0]a =(u

_{2} + u

_{2})a =(0 + 0)]a

if u

_{2}a = 1]a then u

_{2}0]a =(u

_{2} + v

_{2})a =(1 + 0)]a

if u

_{2}a = 0]a then u

_{2}1]a =(u

_{2} + v

_{2})a =(0 + 1)]a

if u

_{2}a = 1]a then u

_{2}1]a =(u

_{2} + u

_{2})a =(1 + 1)]a

Define a binary operation for addition as follows

u

_{2}a + v

_{2}b u

_{2}a + v

_{2}b = length a followed by length b

if u

_{2} = v

_{2} then = u

_{2}(a + b) lengths added in direction a or in a rotated

if u

_{2} v

_{2} then = u

_{2}a + v

_{2}b lengths added in opposite directions by moving in direction a followed by a rotated

or vice versa

There is a procedure that collects like things together which applies in algebraic manipulation which allows the first type of addition to be applied to collections of the same types reducing any form of addition to the second type .

Kujonai now adds another rule

pick the smallest of {a,b} call it s

Perform a binary subtraction on the numerals a,b to give a value to u

_{2}a + v

_{2}b

u

_{2}a + v

_{2}b = u

_{2}(a-s) + v

_{2}(b-s)

= either u

_{2}(a-s) or v

_{2}(b-s)

Define multiplication as follows

u

_{2}a * v

_{2}b = u

_{2}+ v

_{2}]ab

if u

_{2} = v

_{2} then = 0]ab

if u

_{2} v

_{2} then = 1]ab,

There is a logarithmic relationship to unary operators acting on each other in that apparent products have indices that are added mod(n).

So the algebraic description does not rely on axes, and so has a wider application. The signals could be tones or radio signals for example.