Entropies and Dynamical Systems in Riesz MV-algebras

Abstract

Kolmogorov and Sinai, using Shannon entropy, defined the entropy of dynamical systems and they proved that the entropy is invariant under isomorphisms of dynamical systems. Amongst entropies, the logical entropy was suggested by Ellerman as a new information measure. In this paper we define partitions of unit that serve as a mathematical model of the random experiment whose results are vaguely defined events. Then we study Entropies and Dynamical Systems, in particular we give different definitions of entropy and we focus our attention on logical entropy. Finally, we prove that the logical entropy of a dynamical system is invariant under isomorphisms of dynamical systems and we give an example which shows that logical entropy allows to distinguish non-isomorphic dynamical systems.