Algebraic structures arising in statistical mechanics

Complex systems have statistical proprieties that differ greatly from those of classical systems governed by the Boltzmann-Gibbs entropy. Often, the probability distribution observed in these systems deviate from the Gibbs one, showing asymptotic behavior with stretched exponential, power law or log-oscillating tails. Recently, several entropic forms have been introduced in literature to take into account the new phenomenologies observed in such systems. In this paper, we show that for any trace-form entropy one can introduce a pair of algebraic structures with a generalized sum and a generalized product, each forming a commutative group. These generalized operations may be useful in developing the corresponding statistical theory. We specify our results to some entropic forms already known in literature presenting the related algebraic structures.