  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.
