Entropy of MV-algebraic dynamical systems: An example
-
Giuseppina Barbieri
und
Giacomo Lenzi
Abstract
We give examples showing that the Kolmogorov-Sinai entropy generator theorem is false for both upper and lower Riesz entropy of MV-algebraic dynamical systems, both two sided (i.e., analogous to two sided Bernoulli shifts) and one sided (i.e., analogous to one sided Bernoulli shifts).
(Communicated by Anatolij Dvurečenskij)
References
[1] Barbieri, G.—Weber, H.: Measures on clans and on MV-algebras. In: Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam, 2002, pp. 911–945.10.1016/B978-044450263-6/50023-3Suche in Google Scholar
[2] Cignoli, R.— D′Ottaviano, I.—Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Trends Log. Stud. Log. Libr., Kluwer, Dordrecht, 2000.10.1007/978-94-015-9480-6Suche in Google Scholar
[3] Di Nola, A.—Dvurečenskij, A.— Hyčko, M.— Manara, C.: Entropy on effect algebras with the Riesz decomposition property I: Basic properties, Kybernetika 41 (2005), 143–160.Suche in Google Scholar
[4] Di Nola, A.—Dvurečenskij, A.—Hyčko, M.—Manara, C.: Entropy on effect algebras with the Riesz decomposition property II: MV-algebras, Kybernetika 41 (2005), 161–176.Suche in Google Scholar
[5] Kolmogorov, A.: A new invariant for transitive dynamical systems, Dokl. Akad. Nauk SSSR 119 (1958), 861–864.Suche in Google Scholar
[6] Marra, V.—Mundici, D.: MV-algebras and abelian l-groups: a fruitful interaction. In: Ordered Algebraic Structures, Dev. Math. 7, Kluwer Acad. Publ., Dordrecht, 2002, pp. 57–88.10.1007/978-1-4757-3627-4_4Suche in Google Scholar
[7] McNAUGHTON, R.: A theorem about infinite-valued sentential logic, J. Symbolic Logic 16 (1951), 1–13.10.2307/2268660Suche in Google Scholar
[8] Ornstein, D.: Ergodic Theory, Randomness and Dynamical Systems, Yale Univ. Press, New Haven, 1973.Suche in Google Scholar
[9] Shields, P.: The theory of Bernoulli Shifts. Chicago Lectures in Math., The University of Chicago Press, Chicago, Ill.-London, 1973.Suche in Google Scholar
[10] Sinai, Y.: On the Notion of Entropy of a Dynamical System, Dokl. Akad. Nauk SSSR 124 (1959), 768–771.10.1007/978-0-387-87870-6_1Suche in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces
Artikel in diesem Heft
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces
