Shannon Entropy Reinterpreted

Introducing a new concept of independence of two systems, the Shannon additivity is replaced by a noncommutative and nonassociative law the limit of which is the usual addition. The main properties associated with the generalized entropy are established, particularly those corresponding to statistical ensembles. The Boltzmann-Gibbs statistics is recovered as a limit. The connection with thermodynamics is also studied. We also provide a guideline for systematically defining a deformed algebra the limit of which is the classical linear algebra. As an illustrative example we study a generalized entropy based on Tsallis self-information. We point out possible connections between deformed algebra and fuzzy logics. Finally, noticing that the new concept of independence is based on t-norm, the one-parameter deformation of the logarithm is interpreted as an additive generator of t-norms.