# Shannon Entropy Reinterpreted

## TL;DR

This paper reinterprets Shannon entropy using the Lambert W function, introduces a generalized entropy through a one-parameter deformation of the logarithm, and explores its implications for statistical mechanics, algebra, and fuzzy logic.

## Contribution

It presents a novel reinterpretation of Shannon entropy, defines a generalized entropy with a deformation parameter, and introduces a new concept of independence with non-commutative, non-associative addition.

## Key findings

- Generalized entropy converges to Shannon entropy in a limit.
- New independence concept leads to non-commutative, non-associative addition laws.
- Connections established between deformed algebra, thermodynamics, and fuzzy logics.

## Abstract

In this paper we remark that Shannon entropy can be expressed as a function of the self-information (i.e. the logarithm) and the inverse of the Lambert $W$ function. It means that we consider that Shannon entropy has the trace form: $-k \sum_{i} W^{-1} \circ \mathsf{ln}(p_{i})$. Based on this remark we define a generalized entropy which has as a limit the Shannon entropy. In order to facilitate the reasoning this generalized entropy is obtained by a one-parameter deformation of the logarithmic function. Introducing a new concept of independence of two systems the Shannon additivity is replaced by a non-commutative and non-associative law which limit 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 which limit 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.