I've tried to do this with a sheet of paper since I knew about this:
$$x+x = 2x$$ $$x\cdot x = x^2$$ $$x^x = {^{2}x} $$
So based on logic, I assume that:
$$x.x=x+2$$
The dot in between the two xs is just an arbitrary symbol that I'm using. Please don't mind it. It was recently editted to be a multiplication dot, but I want it to be a period so that it's not confused.
So what is this "operation" if it exists? What it is called?
On
Perhaps you are looking for the Successor function, as defined by the Peano axioms. The Successor function $S(x) = x+1$ is used to define the Natural numbers, ie: $n \in \Bbb{N} = S(S(S(\dots (S(0))\dots))) = S^n(0)$
On
It's called "adding two".
It's not actually a lower operation as the operations below manipulate two instance of x. "adding two" doesn't do anything with the tw instances of x but just adds a 2 (which is independent of x) to one of them. A lower operation needs to take two x's and manipulate them.
So the actual lower operation is:
$x \cdot x = 2$. This operation is called "counting".
This function is either the successor function or zeration. Zeration is my personal favorite... it is a deep topic, but fascinating. Here is a link to the tetration forum about it. It's too deep for me to hope to explain here... just read through the post.
Edit: A basic bit of info is found in one of the initial definitions given (there are many different definitions as time has gone alone trying to plug up the ever increasing holes): $$a ° b = a + 1 \iff a > b$$ $$a ° b = b + 1 \iff a < b$$ $$a ° b = a + 2 = b + 2 \iff a = b$$