Valverde-Albacete, Francisco JoséPeláez-Moreno, Carmen2025-01-312025-01-312019-08-08Valverde-Albacete, F.J.; Peláez-Moreno, C. The Rényi Entropies Operate in Positive Semifields. Entropy 2019, 21, 780. https://doi.org/10.3390/e210807801099-4300https://hdl.handle.net/10115/72257Funding This research was funded by the Spanish Government-MinECo project TEC2017-84395-P. Acknowledgments This paper evolved from a plenary talk “Why are there so many Semifields in Computational Intelligence?” given at the 9th European Symposium on Computational Intelligence and Mathematics (ESCIM 2017) in Faro (Portugal), October 5th, 2017, whose organisers we would like to thank for their kind invitation. We would also like to acknowledge the reviewers of previous versions of their paper for their timely criticism and suggestions, and in particular for the references, previously unknown to us, to Burgin’s and Grossman’s and Katz’s work.We set out to demonstrate that the Rényi entropies are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the Rényi’s postulates lead to Pap’s g-calculus where the functions carrying out the domain transformation are Rényi’s information function and its inverse. In its turn, Pap’s g-calculus under Rényi’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-emphasized sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead to idempotent analysis and tropical algebra as well as to less exotic structures. 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 computational intelligence procedures to deal with “information” at large.enAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/shifted Rényi entropyPap's g-calculusnon-Newtonian calculuspositive commutative semifieldsidempotent semifieldsartificial intelligencemachine learningcomputational intelligenceArticle10.3390/e21080780info:eu-repo/semantics/openAccess