Generalized algebra within a nonextensive statistics

TL;DR

This paper develops a generalized algebra based on nonextensive statistics by extending standard logarithm and exponential functions, analyzing its properties and identifying which classical algebraic laws are preserved or altered.

Contribution

It introduces a new generalized algebra framework within nonextensive statistics, exploring its properties and deviations from classical algebraic laws.

Findings

Standard properties like associativity and commutativity are preserved.

Distributivity law is not generally valid in the generalized algebra.

Neutral elements exist for the generalized operators.

Abstract

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.