AXIOM

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

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

Define "number system". Generally when we refer to a number system, we mean a set along with certain axioms relating its elements; at the very least, we mean a group, and the equivalence relation (which requires symmetry) is a fundamental requirement of a group.

tl;dr = no