On the Negation of a Hyperbolic-Valued Probability Distribution

Abstract

In the context of hyperbolic numbers we define the concept of negation of finite hyperbolicvalued probability distributions that is based on the partial order induced by the idempotent structure of hyperbolic numbers. Then, a hyperbolic majorization and general hyperbolic negators are introduced. For a broad class of generated negators, we prove that the original distribution majorizes its negation. This comparison yields that entropy increase for the strong hyperbolic Shannon entropy and the hyperbolic Gini-Simpson entropy, and it implies component-wise uniformization of the iterated negation. Finally, we analyze involutive property of hyperbolic negators and prove that are structurally distinct from the generated negators responsible for the entropy increase. These results show that hyperbolic probabilistic negation is not merely a component-wise copy of the real case, but a theory governed by the interaction between idempotent decomposition, partial order, and entropy measure.