Category theoretic properties of the A. R\'enyi and C. Tsallis entropies

摘要

The problem of embedding the Tsallis and R\'{e}nyi entropies in the frameworkof category theory and their axiomatic foundation is studied. To this end, weconstruct a special category MES related to measured spaces. We prove that bothof the R\'{e}nyi and Tsallis entropies can be imbedded in the formalism ofcategory theory by proving that the same basic functional that appears in theirdefinitions, as well as in the associated Lebesgue space norms, has goodalgebraic compatibility properties. We prove that this functional is bothadditive and multiplicative with respect to the direct product and the disjointsum (the coproduct) in the category MES, so it is a natural candidate for themeasure of information or uncertainty. We prove that the category MES can beextended to monoidal category, both with respect to the direct product as wellas to the coproduct. The basic axioms of the original R\'{e}nyi entropy theoryare generalized and reformulated in the framework of category MES and we provethat these axioms foresee the existence of an universal exponent having thesame values for all the objects of the category MES. In addition, thisuniversal exponent is the parameter, which appears in the definition of theTsallis and R\'{e}nyi entropies.