On the entropy of dynamical systems in product MV algebras. (English) Zbl 0983.37007

From the introduction: The notion of the entropy of a dynamical system has been defined and studied for distinguishing non-isomorphic dynamical systems. If two dynamical systems are isomorphic, they have the same entropy. Therefore, systems with different entropies cannot be isomorphic. If one substitutes in the definition of entropy, the notion of a set partition by the notion of fuzzy partition, a larger class of invariants can be obtained.
We generalize these results considering the general notion of the product MV algebra and obtain some standard assertions, the Kolmogorov theorem on generators being omitted. Namely, in the general situation we have no satisfactory version of the martingale convergence theorem.


37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
06D35 MV-algebras
Full Text: DOI


[1] Ban, A.I., Entropy of a fuzzy \(T\)-dynamical systems, J. fuzzy math., 6, 351-362, (1998) · Zbl 0921.28011
[2] Dumitrescu, D., Entropy of a fuzzy process, Fuzzy sets and systems, 55, 169-177, (1993) · Zbl 0818.28008
[3] Dumitrescu, D., Entropy of fuzzy dynamical systems, Fuzzy sets and systems, 70, 45-57, (1995) · Zbl 0876.28029
[4] Grošek, O., Entropia na algebraičeskich strukturach, Math. slovaca, 29, 411-424, (1979)
[5] Hudetz, T., Space – time dynamical entropy of quantum systems, Lett. math. phys., 16, 151-161, (1988) · Zbl 0674.46040
[6] J. Jakubı́k, On the product of \(MV\) algebras, Czech Math. J., to appear.
[7] P. Maličký, B. Riečan, On the entropy of dynamical systems in: Proceedings Conference on Ergodic Theory and Related Topics ll, Georgenthal, 1986, Teubner, Leipzig, 1987, 135-138.
[8] Markechová, D., A note to the kolmogorov – sinaj entropy of fuzzy dynamical systems, Fuzzy sets and systems, 64, 87-90, (1994) · Zbl 0845.93054
[9] Mundici, D., Interpretation of \(AFC\^{}\{*\}\)-algebras in lukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059
[10] Mundici, D., Nonboolean partitions and their logic, Soft comput. (special issue containing the Proceedings of the first Springer-verlag forum on soft computing), 2, 18-22, (1998)
[11] Mundici, D., Tensor products and the loomir – sikorski theorem for \(MV\)-algebras, Adv. appl. math., 22, 227-248, (1999) · Zbl 0926.06004
[12] D. Mundici, Many-valued logic: from coding theory to operator algebras, in: Information Technology, INFOREC Printing House, Bucharest, 1999, 1011-1022.
[13] J. Petrovičová, On the entropy of partitions in product \(MV\) algebras, Soft Comput., to appear.
[14] Riečan, B., On the product \(MV\) algebras, Tatra mt. math. publ., 16, 143-149, (1999) · Zbl 0951.06013
[15] Riečan, B., On the probability theory on product \(MV\) algebras, (), 445-450 · Zbl 0930.06009
[16] Riečan, B., On a type of \(MV\) algebras, (), 80-84
[17] B. Riečan, T. Neubrunn, Integral, Measure, and Ordering, Kluwer, Dordrecht, and Ister Science, Bratislava, 1997.
[18] Riečan, B.; Markechová, D., The entropy of fuzzy dynamical systems, general scheme and generators, Fuzzy sets and systems, 96, 191-199, (1998) · Zbl 0926.94012
[19] Rybárik, J., The entropy based on pseudo-arithmetical operations, Tatra mt. math. publ., 6, 157-164, (1995) · Zbl 0859.28011
[20] M. Vrábelová, A note on the conditional probability on product \(MV\) algebras, Soft Comput., to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.