Extended xor to Tetrion space

[Xorter]

Therefore, the only trion, 5-ion, etc. is (0;0;0) for non-negative integers. (Above, maybe you were skimming or something.)
However, this does not address negative integers nor non-integers. I assume negative integers are not defined bitwise, however. Taking the 2's complement, however, does define negative integers. In binary, 1 xor -1 = 2. Thus, for example, a negative triad would be (-1;1;2) (1).

The 2's complement, by the way, is a way to define bitwise negatives. (If you already know this, then skip this paragraph.) Take the number -5, for example:
dec = bin
5 = 101
Not 5 = ...11010
(Not 5) + 1 = ...11011 = -5
Here is the intuition behind this:
0 = 0
Not 0 = ...111
(Not 0) + 1 = ...111 + 1 = 0 (wrap-around) = -0
Basically, 2's complement assumes ...111 + 1 = 0, which works if you assume 2^infinity = 0, which works in some contexts like this.

For non-integers, it is important to define what our current version of an integer is. Right now we are dealing with bitwise integers, so really, it might be more useful to think of individual digits as our fundamental number system rather than the numbers themselves. For example, (please don't assume I know group theory, I am just using it as a reference here), a cyclic group of order 2 (bits under xoring) has the property that x=-x is true for all (both) of its members. A cyclic group of order 2k+1 has it that x=-x iff x=0 and a cyclic group of order 2k has it that x=-x iff x=0 or x=k. All of these properties explain why, as stated previously, only even-ads exist (besides (0;0;0)) and they are all of only two digits in their respective bases. Only if a digit is its own inverse can it be in a 2k-ad. If you want to explore other number systems, I suggest considering the above.

What are you using this for, anyway? What spooky number systems can you create with sexads?

*EDIT: Rings are probabaly better for describing digits, not groups.

Well, according to pari gp binary operator xor, 1 xor -1 = -2, however according to my primitive xor extension by simple addition and mod it returns 1 xor -1 = 0. But I guess this last result is wrong, because the original xored function does not look like extended xored function to other number systems at negatives. I suppose x xor y should be extended with more precisely than a simple (x+y) mod 2. We should complete this extension because of the negatives.
You mentioned a logical thesis at cyclic groups in where you had - I think - a preconception: not not x = x. I sign the operator negation with not. I guess you think of it or something like this at that passage. So, what if we construct numbers, or operated combination of numbers whose triple negation are itselves. So not not not x = x while not not x isn't x as usual, and this x will make a trionic triad whose 1st is: (x/2, x^2/2,(x+x^2)/2) or something like this. But I am sure that we should investigate higher dimensional numbers.

We know Bions, the set of Bions is {reals, complexes, quaternions, octonions, sedenions, and so on}. These are bionic because the number system base is binary.
What if the base is not binary? If it is tetranary, we get the Tetrions.
If it is hexanary, we get the Hexions or Sexions.
If it is octanary, we get the Oktions/Octions.
... etc.
These are infinite sets of higher dimensional number sets. The composition of imaginary units and the type of the multiplication make the difference. I must investigate them, because this is the key for the understanding of the Multiverse and its patterns. I have been writing a paper about it for weeks. And I need others' help like yours. So thank you for replying.