Distributivity and deformation of the reals from Tsallis entropy

We propose a one-parameter family R_q of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the generalized addition of the Tsallis entropy. These operations establish a one-parameter family of field isomorphisms τ_q between R and R_q through which an absolute value on R_q is introduced. This turns out to be a quasisymmetric map, whose metric and measure-theoretical implications are pointed out.