AXIOM

Is it possible to have a number system without the symmetry of equality axiom?

e.g
a=b;
b!=a;

If you're talking about something akin to first-order Peano arithmetic, no; axiom #2 is "every natural number is equal to itself", which is roughly the corollary of saying that every natural number has only one unique successor. (If you're talking about _non_-Peano arithmetic, then you're in crazyland, populated by utter loons.)

Non-commutative _operations_, on the other hand, are quite spiffy.