What if we change axiom of arithmetic? I want to experiment with some ideas about changing the axioms of arithemetic. (You know, like changing axioms of Euclidian geometry, we get non-Euclidian geometry.) 
What happens when we change the rules of addition ?
You know that : 


*

*$\forall x\,\forall y\,(x + y = y + x)$

*$∀x\,∀y\,∃z\,(x+y=z)$


If we change that into :


*

*$x + y$ is not always equal to $y + x$ the same way as $x^y$ is not always equal to $y^x$

*$x + y = z$ only if $x \geq y$


Yes, it's pretty different. For example, you're sure that if $x + y = z$ then $z \leq 2x$.
Does this change a lot of things? Does it create new axioms?
 A: The first idea (arithmetics in which addition isn't commutative) has definitely been studied. In abstract algebra, a ring is an object that has two operations we usually call "addition and multiplication." In those, the addition operation is always required to be commutative. 
But there's a name for the same thing without the requirement that $+$ be commutative: they're called near-rings.

The second idea you had is a bit more unusual, since it restricts what things can be added. Usually if you're defining an arithmetic, you want $+$ to be defined everywhere. Nevertheless, people have still studied algebraic objects with partially defined operations.
Division in the real numbers is (barely) partially defined since division by zero isn't permissible. I know, though, there are far stranger (more partially?) defined operations out there.

Actually upon rereading the way you stated the second case, there's a pretty obvious example of an operation with those properties: division on the (positive) natural numbers. There exists $c$ such that $a/b=c$ only if $a\geq b$. This operation also is not commutative!
I would have used division in the integers, but then I'd had to have changed to $|a|\geq|b|$. 
Unfortunately, I don't think it makes a good candidate to be an addition operation for an arithmetic: I'm not sure what kinds of multiplication would distribute over it.
