Hierarchical structure of operations defined in nonextensive algebra

In the past few years, several generalized algebras were developed from physical background associated with the so-called nonextensive statistical mechanics. One of which, the q-generalized algebra, is a functional mimicking the morphisms between the standard algebraic operations through generalized exponential eax = (1+ax)1/a and logarithm l n a ( x ) = x a − 1 a . These functions and the resulting generalized operations possess very interesting mathematical properties and have been used in statistical physics for finite systems and nonextensive systems in general. We establish that the link between the two different operations can be either of functional or iterative nature. Both methods can be combined to introduce new nonextensive operations. The complete set of operations can be represented on a plane structured diagram. The generalized operations can be distributed into two classes, namely the “up” and “down” operations, depending on their localization in the diagram. The properties of generalized operations naturally arise from functional relations and equivalent properties of standard operations.