I'm considering an operator that looks like this:
This grid means for instance 9$3 = 33 and 3$9 = 33 and 3$(-9) = -33 and -3$(-9) = 33 and -3$(9) = -33 etc. In general:
a$b = b$a =(-b$-a) = (-a$-b) = -(-b$a) = -(b$-a) = -(-a$b) = -(a$-b)
An interesting feature is that c = a$b for a unique pair of a and b. For instance 30 is 6$7 or -6$-7 and there is no other way to produce 30 from a pair of a and b, aka b and a.
The operation is transitive over addition:
a$(b+c) = a$b + a$c
but it is not commutative:
a$(b$c) != a$(b$c)
a$b is not f(a,b,+,-,*,/)
simply because x$x = x^2 and x$0 = x
the only polynomial with rational coefficients that equals x^2 is x^2 itself and x*0 is 0 not x.
So in a way f(a,b,+,-,*,/,$) extends the usual algebra.
Every positive integer can form a binary tree like this:
30 = 6$7
but 6 is 2$3 and 7 is 1$5
2 and 3 are both 1$2 and 1$3
the 1's can be removed.
Two obvious extensions of the idea are to define $ over the rationals which looks to be possible as $ has nice continuity properties, for instance it is always increasing in the a and b directions. Also to make another operator say, #, that acts as the inverse of $. for instance (a$b)#a = b and (a$b)#b = a.