  We set out to demonstrate that the R\'enyi entropies with parameter $\alpha$
are better thought of as operating in a type of non-linear semiring called a
positive semifield. We show how the R\'enyi's postulates lead to Pap's
g-calculus where the functions carrying out the domain transformation are
Renyi's information function and its inverse. In its turn, Pap's g-calculus
under R\'enyi's information function transforms the set of positive reals into
a family of semirings where "standard" product has been transformed into sum
and "standard" sum into a power-deformed sum. Consequently, the transformed
product has an inverse whence the structure is actually that of a positive
semifield. Instances of this construction lead into idempotent analysis and
tropical algebra as well as to less exotic structures. Furthermore, shifting
the definition of the $\alpha$ parameter shows in full the intimate relation of
the R\'enyi entropies to the weighted generalized power means. We conjecture
that this is one of the reasons why tropical algebra procedures, like the
Viterbi algorithm of dynamic programming, morphological processing, or neural
networks are so successful in computational intelligence applications. But
also, why there seem to exist so many procedures to deal with "information" at
large.
